MultiblockSBP-SATMethodologyofSymmetricMatrixFormof ...timedomain[48].Alocalhigh-orderABC,aswellasa nonreﬂecting boundary condition (BC), has been widely used for their feasibility,

Research ArticleMultiblock SBP-SAT Methodology of Symmetric Matrix Form ofElastic Wave Equations on Curvilinear Grids

Cheng Sun 1 Zai-Lin Yang 12 Guan-Xi-Xi Jiang1 and Yong Yang12

1College of Aerospace and Civil Engineering Harbin Engineering University Harbin China2Key Laboratory of Advanced Material of Ship and Mechanics Ministry of Industry and Information TechnologyHarbin Engineering University Harbin China

Correspondence should be addressed to Zai-Lin Yang yangzailin00163com

Received 24 August 2019 Revised 23 November 2019 Accepted 12 December 2019 Published 9 January 2020

Academic Editor Cristina Castejon

Copyright copy 2020 Cheng Sun et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A stable and accurate finite-difference discretization of first-order elastic wave equations is derived in this work To simplify theorigin and proof of the formulas a symmetric matrix form (SMF) for elastic wave equations is presented e curve domain isdiscretized using summation-by-parts (SBP) operators and the boundary conditions are weakly enforced using the simultaneous-approximation-term (SAT) technique which gave rise to a provably stable high-order SBP-SATmethod via the energy method Inaddition SMF can be extended to wave equations of different types (SH wave and P-SV wave) and dimensions which can simplifythe boundary derivation process and improve its applicability Application of this approximation can divide the domain into amultiblock context for calculation and the interface boundary conditions of blocks can also be used to simulate cracks and otherstructures Several numerical simulation examples including actual elevation within the area of Lushan China are presentedwhich verifies the viability of the framework present in this papere applicability of simulating elastic wave propagation and theapplication potential in the seismic numerical simulation of this method are also revealed

1 Introduction

Partial differential equations (PDEs) are widely used inmathematics physics and other fields However in practicalengineering it is difficult to obtain the corresponding an-alytical results Hence a number of numerical methods havebeen developed for computational modeling of elastic wavepropagation For example finite-difference methods(FDMs) [1ndash3] finite element methods (FEM) [4 5] spectralelement methods (SEM) [6] boundary element methods(BEM) [7ndash9] finite volume methods (FVM) [10] andpseudospectral methods [11 12] However each method hasits own merits and drawbacks

FDMs are commonly used and extremely efficient forwave propagation in seismic engineering applications due totheir efficiency and ability to simulate complex layeredmedia among others However it is difficult to managecomplex geometries particularly when using a staggeredgrid method [13] An additional problem of this method is

that it is unsatisfactory when simulating cracked mediaGenerally when the traditional FDM is used the method tosimulate the crack is to fill the crack with a low velocitymedium such as water Meanwhile grids close to a lowvelocity mediummust be refined for stability which requiresa high volume of calculations and difficult automaticmodeling Additionally selected FDM methods presentdifficulties in terms of managing long-time stability both interms of spatial- and time-related aspects ese drawbacksgive rise to significant limitations for the simulation of large-scale long-time and undulating media erefore a nu-merical method with time and spatial stability is proposed inthis work

In the context of numerical simulation accuracy willhave a significant effect on results Compared to first- andsecond-order methods higher-order FDM (HOFDM) caneffectively reduce the degree of freedom required for sim-ulating wave propagation [14] However HOFDMs also giverise to additional limitations and difficulties such as discrete

HindawiShock and VibrationVolume 2020 Article ID 8401537 16 pageshttpsdoiorg10115520208401537

approximation of boundary conditions and the establish-ment and simulation of complex structures such as cracksFurthermore a larger amount of computation and longercomputation time are generally required leading to com-putation limitations

In large-scale simulation cases particularly the com-putational load and memory utilization of traditionalmethods will face significant challenges Multiblock simu-lation methods proposed for composite domains can saveprocessing memory by partitioning the region representedin a large-scale model thereby greatly improving computingefficiency by enabling parallel calculations between blocks

A relatively appropriate HOFDM that combines initialboundary value conditions involves summation-by-parts(SBP) operators [15ndash19] is approach is employed forapproximating the derivatives in a spatial domain togetherwith the weak enforcement of boundaries and interfaceconditions using either the simultaneous-approximation-term (SAT) method [20 21] or the projection method[22ndash24] e SBP-SAT method can develop strictly stableproof using the energy method leading to robustness inspatial and time domains Some review papers of SBP op-erators can be found in [25 26] and examples of SBP-SATcan be found in [27ndash30]

e essential characteristics of SBP operators are thatthey mimic the integration-by-parts (IBP) property so that itis possible to mimic the energy dissipation of a continuousproblem e basic theory of first-order SBP operators wasproposed by Kreiss and Scherer [19] in 1974 whereby theenergy method is applied for stability proof leading to theHOFDM known as strict stability [31 32]

e SBP operators introduced above were derived in auniform grid with central-difference stencils in the interiorof the area and using one-sided difference stencils near theboundaries thereby maintaining the properties of SBPoperators ese operators can be referred to as classical SBPoperators Other types of SBP operators are generalized SBP[25 33ndash35] multidimensional SBP [36 37] upwind SBP[38 39] and staggered and upwind SBP operators [40]Generalized SBP operators have one or more characteristicsof nonrepeating interior operators nonuniform nodal dis-tribution and exclude one or both boundary nodes Mul-tidimensional SBP operators remove the limitations ofclassical SBP operators for applying the tensor-productformulation to multidimensional conditions because mostclassical SBP operators are one-dimensional Upwind SBPoperators combining flux-splitting techniques for hyper-bolic systems were proposed via noncentral-differencestencil in the interior which introduced artificial dissipationon a nonstaggered or staggered grid

In seismology and in a number of fields such asacoustics electromagnetics and oceanography a commonlyencountered difficulty is the treatment of unbounded do-mains e local high-order absorbing boundary condition(ABC) [41ndash43] and perfectly matched layer (PML) [44ndash47]are the most commonly used techniques for trimming thesedomains and ldquoabsorbingrdquo outgoing waves as much aspossible in order to simulate elastic wave motion in finitecomputation domains ABC is a boundary condition defined

on an artificial boundary in order to make small spuriousreflections occur when elastic waves propagate at theboundary Absorbing layers were proposed bymodifying theunderlying equations in order to rapidly dampen the wavein a layer Due to the difficulties associated with thetransformation between time and frequency domains en-ergy estimation was not applicable to the PML within thetime domain [48] A local high-order ABC as well as anonreflecting boundary condition (BC) has been widelyused for their feasibility intuitive nature and generaliz-ability when combined with the SATmethod enabling well-proven approximates [49ndash58] We employed the nonre-flecting BC since it provided a simple formulation of theelastic wave problem As an extension a method combiningPML with the SBP-SAT (summation by parts + simulta-neous approximation terms) methodology of the symmetricmatrix form (SMF) of elastic wave equations was developedin a general curvilinear geometry and is presented in thiswork Nonreflecting BCs derived in the Laplace space can beobserved in [59] Additionally a well-proposed SBP-SATmethodology was used on grids with nonconforming in-terfaces [60ndash64]

e aim of this work was to establish a generic repre-sentation of the elastic wave equation using the SMF As aresult the SMF framework was able to formulate SH andP-SV wave equations in a similar manner thereby simpli-fying this calculation while also facilitating the transfor-mation of the extension of dimensions (two-dimensional[2D] and three-dimensional [3D]) Meanwhile using theSMF a general continuous stability analysis is presentedcombined with BCs in a general hyperbolic system Anequivalent form can be derived in a curve domain similar tothe form in a cuboid domain by using curvilinear trans-formation of the elastic wave equationsereafter a genericstable discretization of the SMF on curvilinear coordinateswas developed using SBP-SAT where it was observed thatthe SMF is able to provide an equivalent type of the cuboiddomain In this instance the results of similar stabilityanalyses greatly simplified the calculation and proof processHence this framework is applicable to a wide range of elasticwave equations

e remainder of this paper is presented as follows inSection 2 the SMF of elastic wave equations is establishede well-posed BCs of the SMF of elastic wave equationsare proposed using the flux-splitting form provided inSection 3 Section 4 presents the stability analysis using thecontinuous energy method A semidiscrete and appliedSBP-SAT methodology is presented and stability analysiswithin a discrete norm is discussed in Section 5 Numericalstudies are performed in Section 6 Subsequently con-clusions are drawn and future research is presented inSection 7

2 SMF of Elastic Wave Equations

21 Formulation of SHWave Let the Cartesian coordinatesof a two-dimensional spatial domain defined by (x y) and tdenotes the time variable Consider the first-order elasticwave equation of the SH form

2 Shock and Vibration

ρzvz

zt

zσxz

zx+

zσyz

zy

zσxz

zt G

zvz

zx

zσyz

zt G

zvz

zx

(1)

where vz is the velocity in the z direction of the SH wavedefined by vs

Gρ

1113968 and σxz and σyz are the stresses ρ is

the density of the elastic medium and G is the shearmodulus Equation (1) can then be revised as a symmetricsystem that has a form

zωzt

Ax

zωzx

+ Ay

zωzy

(2)

where

Ax

0 vs 0

vs 0 0

0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Ay

0 0 vs

0 0 0

vs 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ω ρ2

1113969vz

12G

1113969σxz

12G

1113969σyz1113876 1113877

T

(3)

22 Formulation of P-SV Wave e P-SV from the elasticwave equation can be written as

zσxx

zt

zσyy

zt

zσxy

zt

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

C

zvx

zx

zvy

zy

zvx

zy+

zvy

zx

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ρzvx

zt

zσxx

zx+

zσxy

zy

ρzvy

zt

zσxy

zx+

zσyy

zy

(4)

where vx and vy are the particle velocities σij are the stressesρ is density andC CT is the stiffness matrix In the contextof the isotropic case the stiffness matrix is described by twoindependent elastic coefficients the Lame parameters λ andμ Note that C is a symmetric positive definite

C

λ + 2μ λ 0

λ λ + 2μ 0

0 0 μ

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (5)

It will be convenient to use wave velocities vp and vs asmedium properties

vp

λ + 2μ

ρ

1113971

vs

μρ

1113970

(6)

Equation (4) can be formulated as a symmetric system as

1113954A0z1113954ωzt

1113954Ax

z1113954ωzx

+ 1113954Ay

z1113954ωzy

(7)

where 1113954ω σxxρ σyyρ σxyρ vx vy1113960 1113961T

1113954A0

a b 0 0 0

b a 0 0 0

0 0 c 0 0

0 0 0 1 0

0 0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113954Ax

0 0 0 1 0

0 0 0 0 0

0 0 0 0 1

1 0 0 0 0

0 0 1 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113954Ay

0 0 0 0 0

0 0 0 0 1

0 0 0 1 0

0 0 1 0 0

0 1 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(8)

a v2p

4v2p v2p minus v2s1113872 1113873

b minusv2p minus 2v2s

4v2p v2p minus v2s1113872 1113873

c 1v2s

(9)

Shock and Vibration 3

e matrices in (8) are symmetric and 1113954A0 is positivedefinite Equation (7) also can be written as a symmetricfirst-order hyperbolic system as a generic system (2)

zωzt

Ax

zωzx

+ Ay

zωzy

(10)

In order to transform (7) to (10) we shall ansatz1113954A0 PTΛP where

P

12

radic1

2

radic0 0 0

12

radicminus 1

2

radic0 0 0

0 0 1 0 00 0 0 1 00 0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Λ

12 v2p minus v2s1113872 1113873

12v2s

1v2s1

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(11)

P is an orthonormal matrix that satisfies PTP Idenoting

M PTΛminus 12P

Mminus 1 PTΛ12P

(12)

us MT 1113954A0 Mminus 1 and premultiplying M from Equa-tion (10) yields

MT 1113954A01113954ωt MT 1113954Ax 1113954ωx + MT 1113954Ay 1113954ωy

MT 1113954AxMMminus 1 1113954ωx + MT 1113954AyMMminus 1 1113954ωy(13)

as

Mminus 1 1113954ωt MT 1113954AxMMminus 11113954ωx + MT 1113954AyMMminus 11113954ωy (14)

Here we transform (14) into

ωt Axωx + Ayωy

ω Mminus 11113954ω

(15)

Matrix M satisfying the above conditions is

M PTΛminus 12P

v2p minus v2s

1113969+ vs

2

radic

v2p minus v2s

1113969minus vs

2

radic 0 0 0

v2p minus v2s

1113969minus vs

2

radic

v2p minus v2s

1113969+ vs

2

radic 0 0 0

0 0 vs 0 0

0 0 0 1 0

0 0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(16)

23 Curvilinear Coordinates and Coordinate TransformationRegular grids have difficulty in managing numerical simu-lations for geometrically complex models Consider theelastic wave equation in a domain with a curved boundaryshape where a transform from the curvilinear domainΩ isin (x y) to the cuboid domain Ωprime isin (ξ η) was effected(Figure 1) Here we introduced mapping defined by

(x(ξ η) y(ξ η))⟷(ξ(x y) η(x y)) (17)

e Jacobian determinant is

J

zx

zξzx

zη

zy

zξzy

zη

111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

xξyη minus xηyξ (18)

erefore the derivatives in each direction can bewritten as

Jωx yηωξ minus yξωη

Jωy minus xηωξ + xξωη(19)

Using the relations it follows that

Aξ Axyη minus Ayxη

J

Aη minus Axyξ + Ayxξ

J

(20)

us we obtain the same formula as (2) and (10)

ωt Aξωξ + Aηωη (21)

where Aξ andAη are symmetric matrices since Ax and Ay

are symmetric as well

3 Well-Posed Boundary Conditions

A generic representation of the elastic wave equation re-ferred to as the SMF is presented Using this framework aspecific class of the elastic wave equation can be analyzedand approximated Meanwhile the implementation of BCsis also simplified Wave propagation is often extremely largeeven in unbounded spatial domains Considerable physicaldomains must be replaced by suitable-sized computationaldomains by introducing artificial BCs to trim these do-mains Moreover a free-surface BC is necessary for simu-lating elastic wave propagation at the surface or crackstructure e traditional approach for simulating differentBCs is to employ the properties of waves at differentboundaries eg the traction perpendicular to the surface ofthe free boundary is zero Different BCs have differentmathematical formulas which gives rise to difficulties in thederivation and unity of the elastic wave equations frame-work e SMF is not simply a generic representation of theelastic wave equation but also creates a generic system ofdifferent BCs To derive a nonreflecting BC and the free-surface BC a well-proven characteristic type of ABC was


considered where the number of variables propagated intothe domain was equal to the number of variables propagatedout the domain [65 66] Here we denote four boundaries in adomain west east south and north (Figure 1) e free-surface boundary condition at the north boundary wasconsidered A nonreflecting BC was proposed at the westeast and south boundaries More details of characteristicBCs are provided in [67 68] In this paper we first introducethe splitting of the coefficient matrix

Aplusmnξ TTAξ

ΔAξplusmn ΔAξ

11138681113868111386811138681113868

11138681113868111386811138681113868

2⎛⎝ ⎞⎠TT

Aξ (22)

en having

Aplusmnξ T plusmnAξΔ plusmnAξ

T plusmnAξ1113874 1113875

T

Aplusmnη T plusmnAηΔ plusmnAη

T plusmnAη1113874 1113875

T

(23)

Using (23) the characteristic variables associated withpropagating in and out of the domain at the boundary are

pwest TminusAwestξ

1113874 1113875T

ωwest

qwest T+Awestξ

1113874 1113875T

ωwest

peast T+Aeastξ

1113874 1113875T

ωeast

qeast TminusAeastξ

1113874 1113875T

ωeast

psouth Tminus

Asouthη

1113874 1113875T

ωsouth

qsouth T+Asouthη

1113874 1113875T

ωsouth

pnorth T+

Anorthη

1113874 1113875T

ωnorth

qnorth TminusAnorthη

1113874 1113875T

ωnorth

(24)

Well-posed BCs posit that only incoming variables of thecorrect quantity must be prescribed on the boundary us

the well-posed problem can be specified by the incomingcharacteristic variables composed of T which is

p Rq (25)

where R serves as a reflection coefficient ieR isin [minus 1 1]Aimed at different BCs R minus 1 represents a free-surface BCand a nonreflecting BC as defined by R 0

4 Continuous Energy Analysis

In this section an energy analysis is presented for the SMF ofthe elastic wave equation (10) alongside the BC in (25) eenergy method was based on constructing a norm for thegiven problem that did not grow over time To simplifynotation in the forthcoming energy analysis a number ofdefinitions are necessary e result at the grid point wasstacked as a vector where uT

1 [u(1)1 u

(2)1 u

(k)1 ] and

uT2 [u

(1)2 u

(2)2 u

(k)2 ] e inner product was given by

(u1u2)A 1113938xr

xluT1A(x)u2dx and A(x) AT(x)gt 0 in the

domain u1u2 isin L[xl xr] us the corresponding normwas expressed as u2A (u u)A Here we again included theSMF of elastic wave equations in a bounded domain for thesake of convenience


Multiplying (26) by ωT setting data to zero and addingits transpose then integrating over the domain yield

1113946 ωTt ω + ωTωt1113872 1113873dS 1113946 ωTAξωξ + ωTAηωη + ωT

ξ Aξω + ωTηAηω1113872 1113873dS

(27)

Equation (27) is equivalent to

z

ztω

2 BTwest + BTeast + BTsouth + BTnorth (28)

where (zzt)ω2 is the time derivative and (28) gives

BTwest minus 1113946west

ωTAξωdl

BTeast 1113946east

ωTAξωdl

BTsouth minus 1113946south

ωTAηωdl

BTnorth 1113946west

ωTAηωdl

(29)

A bounded energy for (28) required (zzt)ω2 le 0 aswell as BTwestBTeastBTsouthBTnorth le 0 In this instancewe show that the BC terms (29) were nonpositive definiteAs BCs were the same except at the north boundary weonly included the treatment of the south and northboundaries

Mapping

North North

South

East EastWest

South

Curvilinear domain Cuboid domain

West

x

y η

ξ

Figure 1 Coordinate transformation


BTsouth 1113946south

ωT T+AηΔ+

AηT+

Aη1113874 1113875

T

+ TminusAηΔminus

AηTminus

Aη1113874 1113875

T

1113888 1113889ω dl

minus 1113946south

ωT T+AηΔ+

AηT+

Aη1113874 1113875

T

1113888 1113889ω dlle 0

(30)

BTnorth 1113946west

ωT1113888 RTminus

Aη1113874 1113875Δ+

AηRTminus

Aη1113874 1113875

T

+ TminusAηΔminus

AηTminus

Aη1113874 1113875

T

1113889ω dl

ωTTminusAη

RΔ+Aη

RT

+ ΔminusAη

1113874 1113875 TminusAη

1113874 1113875T

ω dlle 0

(31)

Equations (30) and (31) were nonpositive so that (28)satisfied the energy estimate Using the SMF the energyanalysis could be simplified to a large extent as a genericsystem

5 Semidiscrete Approximations

In this section an energy-stable semidiscrete approximationwas derived We used SBP finite-difference operators toapproximate the spatial domain and imposed BCs using theSAT method on an unstaggered grid

51 Spatial Discrete Operators To begin with consider theuniform discretization of the domain x isin (xl xr) withN+ 1 grid points and step hgt 0

xi ih i 0 1 N + 1 h xr minus xl( 1113857

N (32)

e solution is stacked as a vector U (u(xL)

u(Δx) u(xR)) and its discrete form is V (v0 v1

vN+1) An SBP operator approximating zzx has the form

D Hminus 1Q

DV Vx(33)

e matrix D satisfies integration by parts if the fol-lowing properties hold

H HT gt 0

Q + QT EN minus E0

(34)

where E0 erarr

0 erarrT

0 EN erarr

N erarrT

N erarr

0 1 0 middot middot middot 0( 1113857T

and erarr

N 0 0 1( 1113857T In this paper 6th order SBPoperators are used as mentioned in [15]

52 SBP-SATMethodology of SMF of ElasticWave EquationsIn this section an SBP-SATmethodology of SMF of elasticwave equations is proposed and its numerical stability isproved A computation domain was discretized using (Nξ +

1) times (Nη + 1) grids e numerical solution of SH waveequations on the grids is stacked as vectors

ω ω(1)00 ω(2)

00 ω(3)00 ω(1)

NξNηω(2)

NξNηω(3)

NξNη1113876 1113877

T

(35)

For the P-SV wave equations (35) then take the form

ω ω(1)00 ω(2)

00 ω(3)00 ω(4)

00 ω(5)00 ω(1)

NξNηω(2)

NξNηω(3)

NξNηω(4)

NξNηω(5)

NξNη1113876 1113877

T

(36)

e SBP method in higher space dimensions is derivedusing the Kronecker product

BotimesC

b00C middot middot middot b0nC

⋮ ⋱ ⋮

bm0C middot middot middot bmnC

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (37)

where B is an (m + 1) times (n + 1) matrix and C is a (p + 1) times

(q + 1) matrix so that (37) is a [(m + 1)(p + 1)] times

[(n + 1)(q + 1)] matrix (37) fulfills the rules of (AotimesB)

(CotimesD) (AC)otimes (BD) and (AotimesB)T AotimesBT Define I bethe 3 times 3 identity matrix for the SH wave equation and 5 times 5identity matrix for the P-SV wave equation Iξ be the(Nξ + 1) times (Nξ + 1) and Iη be the (Nη + 1) times (Nη + 1)

matrix SBP operators in 2D are

Dξ D1Dξ otimesHη otimes I

Dη Hξ otimesD1Dη otimes I

(38)

e coefficient matrices are determined in 2D yields

Aξ Iξ otimes Iη otimesA1Dξ

Aη Iξ otimes Iη otimesA1Dη

(39)

Moreover the inner product is defined as follows

Hξη Hξ otimesHη otimes I (40)

To combine with boundary conditions we set

Ewest elξ otimes Iη otimes I

Eeast erξ otimes Iη otimes I

Esouth Iξ otimes elη otimes I

Enorth Iξ otimes erη otimes I

(41)

Here the SBP-SAT discretization of (21) with the BCs(24) and (25) can be written as

zωzt

AξDξω + AηDηω + SAT (42)

where


SATwest +Hminus 1ξ EwestT

minusAwestξΔminus

Awestξ

TminusAwestξ

1113874 1113875T

ωwest

SATeast minus Hminus 1ξ EeastT

+AeastξΔ+

Aeastξ

T+Aeastξ

1113874 1113875T

ωeast

SATsouth +Hminus 1η EsouthT

minus

Asouthη

Δminus

Asouthη

Tminus

Asouthη

1113874 1113875T

ωsouth

SATnorth minus Hminus 1η EnorthT

+

Anorthη

Δ+

Anorthη

T+

Anorthη

minus RTminus

Anorthη

1113874 1113875T

ωnorth

(43)

For stability analysis multiplying (42) by ωTHξη andadding the transpose lead to

z

ztω

2ξη BTwest + BTeast + BTsouth + BTnorth (44)

where the boundary conditions are

BTwest minus ωTwestH

westη Awest

ξ ωwest + 2ωTwestH

westη Tminus

AwestξΔminus

AξTminus

Awestξ

1113874 1113875T

ωwest

BTeast ωTeastH

eastη Aeast

ξ ωeast minus 2ωTeastH

eastη T+

AeastξΔ+

AξT+

Aeastξ

1113874 1113875T

ωeast

BTsouth minus ωTsouthH

southξ Asouth

η ωsouth + 2ωTsouthH

southξ Tminus

Asouthη

ΔminusAη

Tminus

Asouthη

1113874 1113875T

ωsouth

BTnorth ωTnorthH

northξ Anorth

η ωnorth minus 2ωTnorthH

northξ T+

Anorthη

Δ+

Anorthη

T+

Anorthη

minus RTminus

Anorthη

1113874 1113875T

ωnorth

(45)

Similar to (30) and (31) we have

BTeast minus ωTeastH

eastη T+

AeastξΔ+

AξT+

Aeastξ

1113874 1113875T

ωeast + ωTeastH

eastη Tminus

AeastξΔminus

AξTminus

Aeastξ

1113874 1113875T

ωeast


westη T+

AwestξΔ+

AξT+

Awestξ

1113874 1113875T

ωwest + ωTwestH

westη Tminus

AwestξΔminus

AξTminus

Awestξ

1113874 1113875T

ωwest


southη T+

Asouthξ

Δ+

Asouthξ

T+

Asouthξ

1113874 1113875T

ωsouth + ωTsouthH

southη Tminus

Asouthξ

Δminus

Asouthξ

Tminus

Asouthξ

1113874 1113875T

ωsouth

BTnorth ωTnorthH

northη Tminus

Anorthξ

RΔ+

Anorthη

RT+ Δminus

Anorthη

1113874 1113875 Tminus

Anorthξ

1113874 1113875T

ωnorth minus ωTnorthH

northξ T+

Anorthη

minus RTminus

Anorthη

1113874 1113875Δ+

Anorthη

T+

Anorthη

minus RTminus

Anorthη

1113874 1113875T

ωnorth

(46)

Since (46) are nonpositive definite matrices (zzt)ω2ξηle 0 verifies the robustness

For large scale as well as complex structural spatialdomains a computation block is not a preferable choice fornumerical simulation We divided the computation spaceinto several blocks with grids conforming at the blockinterface when interfaces were continuous which had aform matching that in (47) Meanwhile the multiblockSBP-SATmethodology of elastic wave equations was able tomanage discontinuous structures such as cracks by usingthe interface definition e form of two blocks is shown inFigure 2

SAT(1)east minus Hminus 1(1)

ξ E(1)eastT

+(1)

AeastξΔ+(1)

Aeastξ

T+(1)

Aeastξ

1113874 1113875T

ω(1)east minus ω(2)

west1113872 1113873

SAT(2)west +Hminus 1(2)

ξ E(2)westT

minus (2)

AwestξΔminus (2)

Awestξ

Tminus (2)

Awestξ

1113874 1113875T

ω(2)west minus ω(1)

east1113872 1113873

(47)

when the interface conditions were applied for simulatingcracks that had

SAT(1)east minus Hminus 1

ξ EeastT+AeastξΔ+

Aeastξ

T+Aeastξ

1113874 1113875T

ωeast

SAT(2)west +Hminus 1

ξ EwestTminusAwestξΔminus

Awestξ

TminusAwestξ

1113874 1113875T

ωwest

(48)

6 Numerical Experiments

61 Multiblock Condition of P-SV Wave In this section weconsider a multiblock condition that can divide the com-putation domain into appropriate scale blocks Each blockwas able to transform data through an adjacent boundarywhich had the ability to effect computations in parallel Todemonstrate the potential of this multiblock treatment weimposed nonreflecting BCs at boundaries and continuousinterface conditions for blockse properties in blocks werevp 1800ms vs 1000ms and ρ 1000 kgm3 e


computation domain was bounded by [0 3000m] times

[0 3000m] which involved nine blocks with Nξ 100 andNη 100 as shown in Figure 3 e time step is set todt 10minus 3 s and the initial data are

g(t) expminus (x minus 1500)2 +(y minus 1500)2

80001113888 1113889 (49)

Using the SBP-SATmethodology for the SMF of elasticwave equations we obtained amplitude diagrams for dif-ferent times e x- and y-direction wave fields showeddifferent waveforms in the P-SV mode since the initial datahad been applied in the x direction P-wave propagated fromblock five to other blocks at time 03 s (Figure 4) isindicated that the P-SV wave field motivated in block fiveshowed little change when propagating to other blocks

P-wave propagated to the nonreflection boundary attime 09 s (Figure 5) and SV-wave propagated to thenonreflection boundary at time 15 s (Figure 6) isindicated the feasibility of unbounded domainsimulation

e treatment of BCs is always more complex whenapplying the higher-order difference method Generally theaccuracy of BCs is lower than that for the inner mediumerefore the stability problem often occurs at the boundaryposition is problem becomes more prominent whenmultiple computational domains are combined When do-ing so we divided the domain into nine subblocks and 36boundaries which posed a significant challenge to thestability and impact of the numerical simulation Stability isdemonstrated in Section 5 as part of this numerical ex-periment Meanwhile when the elastic wave propagated tothe inner boundary of the medium there was no disturbancefield that is the block form had little influence on the wavefield which indicated that the multiblock simulation of thewave equation was feasible e SBP-SAT methodologyapplied in this work clearly had a positive promoting effecton numerical simulations which can significantly expandthe simulation scale

To investigate the connection between the multiblockand propagation of the elastic wave we next present theamplitude of velocity in the x direction at time 06 s inblock two independently (Figure 7) We verified that con-tinuous interface conditions had little impact on the overallwave field when waves propagated to the interface bound-aries Meanwhile the values of the boundary nodes of thetwo adjacent computational domains coincided with oneanother (Figure 8) It is shown that a multiblock domain canenlarge the simulation scale and significantly improvecomputational efficiency under reasonable settings

62 Crack Structure for SH Wave When the inner com-putational domain was considered as a whole the boundarytypes in the subcomputational region could also be diver-sified which provided convenience for simulating complexstructures For example the traditional finite-differencemethod is required to perform complex operations or ap-proximations in order to realize a discontinuous mediumfor example filling intermittent parts with low-speed mediaIn order to visually demonstrate the propagation process ofelastic waves in cracked media SH waves were utilized sincewaveform transformation was not involved e propertiesof the medium were vs 100ms ρ 2 kgm3 and G

104 Pa e computational domain was bounded by[0 300m] times [0 300m] which involved nine blocks withNξ 100 andNη 100e time step was set to dt 10minus 3 sand the initial data were similar to that in (49)

When an SH wave propagated to the crack structure itgenerated reflected and diffracted waves which causedcontinuous disturbance of the wave field in the mediume superposition and mutual influence of the wave fieldhad a significant impact on wave propagation character-istics Meanwhile the amplitude of diffracted waves de-creased in an obvious manner (Figure 9) is helped us toexplain that in the case of real earthquakes seismic waveattenuation factors must be considered based on the impactof cracks Moreover the method proposed in this paper ismore convenient for realizing a crack structure that is ableto avoid the amplification of errors by low-speed mediafilling

63 Curve Crack Structure in Curvilinear Domain In thisnumerical experiment we considered curve crack structurein a curvilinear domain e interface BC was set betweenblock five and block eight and a crack with no contactbetween block two and block five was employed to show theadvantages of this method in modeling and numerical

S(1)

E(1)

N(1)

S(2)

W(2)W(1) E(2)

N(2)

1 2η

ξ

Figure 2 Multiblock model

0

500

1000

1500

2000

2500

3000

Y

500 1000 1500 2000 2500 30000X

3

2

1 4

5

6

7

8

9

Figure 3 Illustrative model containing multiblock


0

008

006

004

002

ndash002

ndash004

ndash006

014

012

01

(a)

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

(b)

Figure 4 Velocity distribution in x and z directions at time 03 s (a) x direction (b) y direction

01

005

0

ndash005

(a)

006

004

002

0

ndash002

ndash004

ndash006

(b)

Figure 5 Velocity distribution in x and y directions at time 09 s (a) x direction (b) y direction

008

006

004

002

0

ndash002

ndash004

(a)

004

003

002

001

0

ndash001

ndash002

ndash003

ndash004

(b)



simulations (Figure 10) e remaining settings were thesame as in the previous example

In this test the wave field was more complex whichcaused persistent disturbances by reflected and diffractedwaves but without spurious waves being generated by curvemesh conditions (Figure 11) We observed long-time dis-turbances and a reduced maximum of amplitude whichprovided references for the seismic and postseismic analysisof the structure is test also characterized the applicabilityand feasibility of SBP-SAT of the SMF of elastic waveequations to simulate structures with complex shapes

64 Large-Scale Simulation for Complex Terrain To dem-onstrate the full potential of the SBP-SATmethodology ofthe SMF in seismology we next considered specificterrain in Lushan China Elevation data were derivedfrom a profile of Baoxing seismic station as shown in

Figure 12 for an area 798 km in length Since multiblocktreatment involving complex terrain was the main focusof the numerical simulation we employed homogeneousand isotropic parameters where vp 6400ms andvs 3695 ms and 19 times 6 blocks with 201 times 101 gridsdiscretized in each

A method was developed combining PML [48] with theSBP-SATmethodology of the SMF of elastic wave equationsand a general curvilinear geometry was presented by thistest We focused on elastic wave propagation in the mediumand the stability of the simulation e initial data were setsimilar to that in (49)

e wave field at 5 s (Figure 13) showed that hypo-central P-wave propagated to the free surface whileFigure 14 showed the wave field at 25 s once elastic waveshad been reflected and scattered by the irregular surfacecausing the wave field to be extremely complex e wavefield at 50 s and 70 s is shown in Figures 15 and 16

ndash004

ndash002

0

002

004

006

008

01

012

014

016

V x (m

s)

02 04 06 08 1 12 14 16 18 20Time (s)

(Nx = 101 Ny = 51) in block 2(Nx = 1 Ny = 51) in block 5

Figure 8 Velocity amplitude in the x direction

20001800

Y (m)1600

120010001400

800

X (m)6001200

4002001000

0

2200ndash005

0

005

01

V x (m

s)

Figure 7 Velocity distribution in the x direction at time 15 s


01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(a)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash01

ndash008

(b)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(c)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(d)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(e)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(f )

Figure 9 Velocity distribution in the z direction (a) at time 07 s (b) at time 09 s (c) at time 12 s (d) at time 14 s (e) at time 15 sand (f) at time 18 s


0

50

100

150

200

250

300

Y

50 100 150 200 250 3000X

2

3

1 4

5

6

7

8

9

Figure 10 Illustrative model containing curve cracks

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(a)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(b)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(c)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(d)

Figure 11 Continued


respectively As can be seen the numerical simulation ledto a converging result Since the PML methodology wasused for the curve mesh the absorption effect at ABCs was

additionally strengthened and did not undermine therobust nature of the system

Figures 15 and 16 show the wave field at 50 s and 70 seterrain was generally mountainous on the left and relatively

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(e)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(f )


ndash10000ndash9000ndash8000ndash7000ndash6000ndash5000ndash4000ndash3000ndash2000ndash1000

010002000300040005000

Elev

atio

n (m

)

200000 400000 600000 8000000Distance (m)

Figure 12 Velocity distribution in the z direction

ndash60 ndash40 ndash20 0 20 40 60 80 100 120


ndash80 ndash60 ndash40 ndash20 0 20 40 60 80






ndash15 ndash10 ndash5 0 5 10 15





flat on the right so that the wave field was more complexon the left due to the scattered wave Figures 17 and 18 showthe wave field at 100 s and 175 s A long term and large-scalesimulation with a stable wave field highlights the superiornature of using a multiblock approach in order to divide thelarge model into appropriate subblocks which can improvesimulation efficiency Meanwhile the continuous interfaceBCs had little impact on the wave field

7 Conclusions and Future Work

eprimarymotivation of conducting this work was to derivea high-order accurate SBP-SATapproximation of the SMF ofelastic wave equations which can realize elastic wave prop-agation in a medium with complex cracks and boundariesBased on the SMF the stability of the algorithm can be provenmore concisely through the energy method which infers thatthe formula calculation and implementation of the programcompilation for discretizing of the elastic wave equation willbe more convenient in addition to being able to expand thedimension of elastic waves e SBP-SATdiscretization of theSMF of elastic wave equations appears to be robust despitethe fact that an area is divided into multiple blocks and thenumber of BCs is increased e method developed in thepresent study was successfully applied to several simulationsinvolving cracks and multiple blocks It was also successfullyapplied to actual elevation data in Lushan China indicatingthat our method has the potential for simulating actualearthquakes

In follow-up work we aim to build on the current modelwith more accurate nonreflecting BCs without underminingthe modelrsquos stability Meanwhile we also hope to includeother SBP operators such as upwind or staggered andupwind operators Numerical simulation indicates that themethod has wide application prospects It can be challengingto model discretize and simulate actual elevation andgeological structure and strict requirements are necessary toensure the stability of such a method

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (Grant no 11872156) National KeyResearch and Development Program of China (Grant no2017YFC1500801) and the program for Innovative ResearchTeam in China Earthquake Administration

References

[1] Z Alterman and F Karal ldquoPropagation of elastic waves inlayered media by finite difference methodsrdquo Bulletin of the

Seismological Society of America vol 58 no 1 pp 367ndash3981968

[2] R M Alford K R Kelly and D M Boore ldquoAccuracy offinite-difference modeling of the acoustic wave equationrdquoGeophysics vol 39 no 6 pp 834ndash842 1974

[3] K R Kelly R W Ward S Treitel and R M AlfordldquoSynthetic seismograms a finite-difference approachrdquo Geo-physics vol 41 no 1 pp 2ndash27 1976

[4] W D Smith ldquoe application of finite element analysis tobody wave propagation problemsrdquo Geophysical Journal of theRoyal Astronomical Society vol 42 no 2 pp 747ndash768 2007

[5] K J Marfurt ldquoAccuracy of finite-difference and finite-ele-ment modeling of the scalar and elastic wave equationsrdquoGeophysics vol 49 no 5 pp 533ndash549 1984

[6] D Komatitsch and J Tromp ldquoIntroduction to the spectralelement method for three-dimensional seismic wave propa-gationrdquo Geophysical Journal International vol 139 no 3pp 806ndash822 1999

[7] J Kim and A S Papageorgiou ldquoDiscrete wave-numberboundary-element method for 3-D scattering problemsrdquoJournal of Engineering Mechanics vol 119 no 3 pp 603ndash6241993

[8] A S Papageorgiou and D Pei ldquoA discrete wavenumberboundary element method for study of the 3-D response 2-Dscatterersrdquo Earthquake Engineering amp Structural Dynamicsvol 27 no 6 pp 619ndash638 1998

[9] Z Ba and X Gao ldquoSoil-structure interaction in transverselyisotropic layered media subjected to incident plane SHwavesrdquo Shock and Vibration vol 2017 Article ID 283427413 pages 2017

[10] M Dumbser M Kaser and J de La Puente ldquoArbitrary high-order finite volume schemes for seismic wave propagation onunstructured meshes in 2D and 3Drdquo Geophysical JournalInternational vol 171 no 2 pp 665ndash694 2007

[11] B Fornberg ldquoe pseudospectral method comparisons withfinite differences for the elastic wave equationrdquo Geophysicsvol 52 no 4 pp 483ndash501 1987

[12] B Fornberg ldquoHigh-order finite differences and the pseudo-spectral method on staggered gridsrdquo SIAM Journal on Nu-merical Analysis vol 27 no 4 pp 904ndash918 1990

[13] R Madariaga ldquoDynamics of an expanding circular faultrdquoBulletin of the Seismological Society of America vol 66 no 3pp 639ndash666 1976

[14] B Gustafsson High Order Difference Methods for Time De-pendent PDE Springer Berlin Germany 2007

[15] B Strand ldquoSummation by parts for finite difference ap-proximations for ddxrdquo Journal of Computational Physicsvol 110 no 1 pp 47ndash67 1994

[16] N Albin and J Klarmann ldquoAn algorithmic exploration of theexistence of high-order summation by parts operators withdiagonal normrdquo Journal of Scientific Computing vol 69 no 2pp 633ndash650 2016

[17] K Mattsson M Almquist and E van der Weide ldquoBoundaryoptimized diagonal-norm SBP operatorsrdquo Journal of Com-putational Physics vol 374 pp 1261ndash1266 2018

[18] K Mattsson ldquoSummation by parts operators for finite dif-ference approximations of second-derivatives with variablecoefficientsrdquo Journal of Scientific Computing vol 51 no 3pp 650ndash682 2012

[19] H O Kreiss and G Scherer ldquoFinite element and finite dif-ference methods for hyperbolic partial differential equationsrdquoin Mathematical Aspects of Finite Elements in Partial Dif-ferential Equations pp 195ndash212 Elsevier AmsterdamNetherlands 1974


[20] M H Carpenter D Gottlieb and S Abarbanel ldquoTime-stableboundary conditions for finite-difference schemes solvinghyperbolic systems methodology and application to high-order compact schemesrdquo Journal of Computational Physicsvol 111 no 2 pp 220ndash236 1994

[21] M H Carpenter J Nordstrom and D Gottlieb ldquoRevisitingand extending interface penalties for multi-domain sum-mation-by-parts operatorsrdquo Journal of Scientific Computingvol 45 no 1ndash3 pp 118ndash150 2010

[22] K Mattsson and P Olsson ldquoAn improved projectionmethodrdquo Journal of Computational Physics vol 372pp 349ndash372 2018

[23] P Olsson ldquoSupplement to summation by parts projectionsand stability IrdquoMathematics of Computation vol 64 no 211p S23 1995

[24] P Olsson ldquoSummation by parts projections and stability IIrdquoMathematics of Computation vol 64 no 212 p 1473 1995

[25] D C Del Rey Fernandez J E Hicken and D W ZinggldquoReview of summation-by-parts operators with simultaneousapproximation terms for the numerical solution of partialdifferential equationsrdquo Computers amp Fluids vol 95pp 171ndash196 2014

[26] M Svard and J Nordstrom ldquoReview of summation-by-partsschemes for initial-boundary-value problemsrdquo Journal ofComputational Physics vol 268 pp 17ndash38 2014

[27] K Mattsson ldquoBoundary procedures for summation-by-partsoperatorsrdquo Journal of Scientific Computing vol 18 no 1pp 133ndash153 2003

[28] M Svard M H Carpenter and J Nordstrom ldquoA stable high-order finite difference scheme for the compressible Navier-Stokes equations far-field boundary conditionsrdquo Journal ofComputational Physics vol 225 no 1 pp 1020ndash1038 2007

[29] M Svard and J Nordstrom ldquoA stable high-order finite dif-ference scheme for the compressible Navier-Stokes equa-tionsrdquo Journal of Computational Physics vol 227 no 10pp 4805ndash4824 2008

[30] G Ludvigsson K R Steffen S Sticko et al ldquoHigh-ordernumerical methods for 2D parabolic problems in single andcomposite domainsrdquo Journal of Scientific Computing vol 76no 2 pp 812ndash847 2018

[31] B Gustafsson H O Kreiss and J Oliger Time DependentProblems and Difference Methods Wiley Hoboken NJ USA1995

[32] P Olsson and J Oliger Energy andMaximumNorm Estimatesfor Nonlinear Conservation Laws National Aeronautics andSpace Administration Washington DC USA 1994

[33] J E Hicken andDW Zingg ldquoSummation-by-parts operatorsand high-order quadraturerdquo Journal of Computational andApplied Mathematics vol 237 no 1 pp 111ndash125 2013

[34] D C Del Rey Fernandez P D Boom and D W Zingg ldquoAgeneralized framework for nodal first derivative summation-by-parts operatorsrdquo Journal of Computational Physicsvol 266 pp 214ndash239 2014

[35] H Ranocha ldquoGeneralised summation-by-parts operators andvariable coefficientsrdquo Journal of Computational Physicsvol 362 pp 20ndash48 2018

[36] J E Hicken D C Del Rey Fernandez and D W ZinggldquoMultidimensional summation-by-parts operators generaltheory and application to simplex elementsrdquo SIAM Journal onScientific Computing vol 38 no 4 pp A1935ndashA1958 2016

[37] D C Del Rey Fernandez J E Hicken and D W ZinggldquoSimultaneous approximation terms for multi-dimensionalsummation-by-parts operatorsrdquo Journal of Scientific Com-puting vol 75 no 1 pp 83ndash110 2018

[38] L Dovgilovich and I Sofronov ldquoHigh-accuracy finite-dif-ference schemes for solving elastodynamic problems incurvilinear coordinates within multiblock approachrdquo AppliedNumerical Mathematics vol 93 pp 176ndash194 2015

[39] K Mattsson ldquoDiagonal-norm upwind SBP operatorsrdquoJournal of Computational Physics vol 335 pp 283ndash310 2017

[40] K Mattsson and O OrsquoReilly ldquoCompatible diagonal-normstaggered and upwind SBP operatorsrdquo Journal of Computa-tional Physics vol 352 pp 52ndash75 2018

[41] R L Higdon ldquoAbsorbing boundary conditions for elasticwavesrdquo Geophysics vol 56 no 2 pp 231ndash241 1991

[42] D Givoli and B Neta ldquoHigh-order non-reflecting boundaryscheme for time-dependent wavesrdquo Journal of ComputationalPhysics vol 186 no 1 pp 24ndash46 2003

[43] D Baffet J Bielak D Givoli T Hagstrom andD RabinovichldquoLong-time stable high-order absorbing boundary conditionsfor elastodynamicsrdquo Computer Methods in Applied Mechanicsand Engineering vol 241ndash244 pp 20ndash37 2012

[44] J-P Berenger ldquoA perfectly matched layer for the absorptionof electromagnetic wavesrdquo Journal of Computational Physicsvol 114 no 2 pp 185ndash200 1994

[45] D Appelo and G Kreiss ldquoA new absorbing layer for elasticwavesrdquo Journal of Computational Physics vol 215 no 2pp 642ndash660 2006

[46] Z Zhang W Zhang and X Chen ldquoComplex frequency-shifted multi-axial perfectly matched layer for elastic wavemodelling on curvilinear gridsrdquo Geophysical Journal Inter-national vol 198 no 1 pp 140ndash153 2014

[47] K Duru and G Kreiss ldquoEfficient and stable perfectly matchedlayer for CEMrdquo Applied Numerical Mathematics vol 76pp 34ndash47 2014

[48] K Duru J E Kozdon and G Kreiss ldquoBoundary conditionsand stability of a perfectly matched layer for the elastic waveequation in first order formrdquo Journal of ComputationalPhysics vol 303 pp 372ndash395 2015

[49] K Mattsson F Ham and G Iaccarino ldquoStable boundarytreatment for the wave equation on second-order formrdquoJournal of Scientific Computing vol 41 no 3 pp 366ndash3832009

[50] S Nilsson N A Petersson B Sjogreen and H-O KreissldquoStable difference approximations for the elastic wave equa-tion in second order formulationrdquo SIAM Journal on Nu-merical Analysis vol 45 no 5 pp 1902ndash1936 2007

[51] Y Rydin K Mattsson and J Werpers ldquoHigh-fidelity soundpropagation in a varying 3D atmosphererdquo Journal of ScientificComputing vol 77 no 2 pp 1278ndash1302 2018

[52] J E Kozdon E M Dunham and J Nordstrom ldquoInteractionof waves with frictional interfaces using summation-by-partsdifference operators weak enforcement of nonlinearboundary conditionsrdquo Journal of Scientific Computing vol 50no 2 pp 341ndash367 2012

[53] J E Kozdon E M Dunham and J Nordstrom ldquoSimulationof dynamic earthquake ruptures in complex geometries usinghigh-order finite difference methodsrdquo Journal of ScientificComputing vol 55 no 1 pp 92ndash124 2013

[54] K Duru and E M Dunham ldquoDynamic earthquake rupturesimulations on nonplanar faults embedded in 3D geometri-cally complex heterogeneous elastic solidsrdquo Journal ofComputational Physics vol 305 pp 185ndash207 2016

[55] O OrsquoReilly J Nordstrom J E Kozdon and E M DunhamldquoSimulation of earthquake rupture dynamics in complexgeometries using coupled finite difference and finite volumemethodsrdquo Communications in Computational Physics vol 17no 2 pp 337ndash370 2015


[56] B A Erickson E M Dunham and A Khosravifar ldquoA finitedifference method for off-fault plasticity throughout theearthquake cyclerdquo Journal of the Mechanics and Physics ofSolids vol 109 pp 50ndash77 2017

[57] K Torberntsson V Stiernstrom K Mattsson andE M Dunham ldquoA finite difference method for earthquakesequences in poroelastic solidsrdquo Computational Geosciencesvol 22 no 5 pp 1351ndash1370 2018

[58] L Gao D C Del Rey Fernandez M Carpenter and D KeyesldquoSBP-SAT finite difference discretization of acoustic waveequations on staggered block-wise uniform gridsrdquo Journal ofComputational and Applied Mathematics vol 348 pp 421ndash444 2019

[59] S Eriksson and J Nordstrom ldquoExact non-reflecting boundaryconditions revisited well-posedness and stabilityrdquo Founda-tions of Computational Mathematics vol 17 no 4 pp 957ndash986 2017

[60] K Mattsson and M H Carpenter ldquoStable and accurate in-terpolation operators for high-order multiblock finite dif-ference methodsrdquo SIAM Journal on Scientific Computingvol 32 no 4 pp 2298ndash2320 2010

[61] J E Kozdon and L C Wilcox ldquoStable coupling of non-conforming high-order finite difference methodsrdquo SIAMJournal on Scientific Computing vol 38 no 2 pp A923ndashA952 2016

[62] L Friedrich D C Del Rey Fernandez A R WintersG J Gassner D W Zingg and J Hicken ldquoConservative andstable degree preserving SBP operators for non-conformingmeshesrdquo Journal of Scientific Computing vol 75 no 2pp 657ndash686 2018

[63] S Wang K Virta and G Kreiss ldquoHigh order finite differencemethods for the wave equation with non-conforming gridinterfacesrdquo Journal of Scientific Computing vol 68 no 3pp 1002ndash1028 2016

[64] S Wang ldquoAn improved high order finite difference methodfor non-conforming grid interfaces for the wave equationrdquoJournal of Scientific Computing vol 77 no 2 pp 775ndash7922018

[65] H-O Kreiss ldquoInitial boundary value problems for hyperbolicsystemsrdquo Communications on Pure and Applied Mathematicsvol 23 no 3 pp 277ndash298 1970

[66] H-O Kreiss and J Lorenz Initial-Boundary Value Problemsand the Navier-Stokes Equations Society for Industrial andApplied Mathematics Philadelphia PA USA 2004

[67] P Secchi ldquoWell-posedness of characteristic symmetric hy-perbolic systemsrdquo Archive for Rational Mechanics andAnalysis vol 134 no 2 pp 155ndash197 1996

[68] T H Pulliam and D W Zingg Fundamental Algorithms inComputational Fluid Dynamics Springer International Pub-lishing Cham Switzerland 2014


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approximation of boundary conditions and the establish-ment and simulation of complex structures such as cracksFurthermore a larger amount of computation and longercomputation time are generally required leading to com-putation limitations

In large-scale simulation cases particularly the com-putational load and memory utilization of traditionalmethods will face significant challenges Multiblock simu-lation methods proposed for composite domains can saveprocessing memory by partitioning the region representedin a large-scale model thereby greatly improving computingefficiency by enabling parallel calculations between blocks

A relatively appropriate HOFDM that combines initialboundary value conditions involves summation-by-parts(SBP) operators [15ndash19] is approach is employed forapproximating the derivatives in a spatial domain togetherwith the weak enforcement of boundaries and interfaceconditions using either the simultaneous-approximation-term (SAT) method [20 21] or the projection method[22ndash24] e SBP-SAT method can develop strictly stableproof using the energy method leading to robustness inspatial and time domains Some review papers of SBP op-erators can be found in [25 26] and examples of SBP-SATcan be found in [27ndash30]

e essential characteristics of SBP operators are thatthey mimic the integration-by-parts (IBP) property so that itis possible to mimic the energy dissipation of a continuousproblem e basic theory of first-order SBP operators wasproposed by Kreiss and Scherer [19] in 1974 whereby theenergy method is applied for stability proof leading to theHOFDM known as strict stability [31 32]

e SBP operators introduced above were derived in auniform grid with central-difference stencils in the interiorof the area and using one-sided difference stencils near theboundaries thereby maintaining the properties of SBPoperators ese operators can be referred to as classical SBPoperators Other types of SBP operators are generalized SBP[25 33ndash35] multidimensional SBP [36 37] upwind SBP[38 39] and staggered and upwind SBP operators [40]Generalized SBP operators have one or more characteristicsof nonrepeating interior operators nonuniform nodal dis-tribution and exclude one or both boundary nodes Mul-tidimensional SBP operators remove the limitations ofclassical SBP operators for applying the tensor-productformulation to multidimensional conditions because mostclassical SBP operators are one-dimensional Upwind SBPoperators combining flux-splitting techniques for hyper-bolic systems were proposed via noncentral-differencestencil in the interior which introduced artificial dissipationon a nonstaggered or staggered grid

In seismology and in a number of fields such asacoustics electromagnetics and oceanography a commonlyencountered difficulty is the treatment of unbounded do-mains e local high-order absorbing boundary condition(ABC) [41ndash43] and perfectly matched layer (PML) [44ndash47]are the most commonly used techniques for trimming thesedomains and ldquoabsorbingrdquo outgoing waves as much aspossible in order to simulate elastic wave motion in finitecomputation domains ABC is a boundary condition defined

on an artificial boundary in order to make small spuriousreflections occur when elastic waves propagate at theboundary Absorbing layers were proposed bymodifying theunderlying equations in order to rapidly dampen the wavein a layer Due to the difficulties associated with thetransformation between time and frequency domains en-ergy estimation was not applicable to the PML within thetime domain [48] A local high-order ABC as well as anonreflecting boundary condition (BC) has been widelyused for their feasibility intuitive nature and generaliz-ability when combined with the SATmethod enabling well-proven approximates [49ndash58] We employed the nonre-flecting BC since it provided a simple formulation of theelastic wave problem As an extension a method combiningPML with the SBP-SAT (summation by parts + simulta-neous approximation terms) methodology of the symmetricmatrix form (SMF) of elastic wave equations was developedin a general curvilinear geometry and is presented in thiswork Nonreflecting BCs derived in the Laplace space can beobserved in [59] Additionally a well-proposed SBP-SATmethodology was used on grids with nonconforming in-terfaces [60ndash64]

e aim of this work was to establish a generic repre-sentation of the elastic wave equation using the SMF As aresult the SMF framework was able to formulate SH andP-SV wave equations in a similar manner thereby simpli-fying this calculation while also facilitating the transfor-mation of the extension of dimensions (two-dimensional[2D] and three-dimensional [3D]) Meanwhile using theSMF a general continuous stability analysis is presentedcombined with BCs in a general hyperbolic system Anequivalent form can be derived in a curve domain similar tothe form in a cuboid domain by using curvilinear trans-formation of the elastic wave equationsereafter a genericstable discretization of the SMF on curvilinear coordinateswas developed using SBP-SAT where it was observed thatthe SMF is able to provide an equivalent type of the cuboiddomain In this instance the results of similar stabilityanalyses greatly simplified the calculation and proof processHence this framework is applicable to a wide range of elasticwave equations

e remainder of this paper is presented as follows inSection 2 the SMF of elastic wave equations is establishede well-posed BCs of the SMF of elastic wave equationsare proposed using the flux-splitting form provided inSection 3 Section 4 presents the stability analysis using thecontinuous energy method A semidiscrete and appliedSBP-SAT methodology is presented and stability analysiswithin a discrete norm is discussed in Section 5 Numericalstudies are performed in Section 6 Subsequently con-clusions are drawn and future research is presented inSection 7

2 SMF of Elastic Wave Equations

21 Formulation of SHWave Let the Cartesian coordinatesof a two-dimensional spatial domain defined by (x y) and tdenotes the time variable Consider the first-order elasticwave equation of the SH form


ρzvz

zt

zσxz

zx+

zσyz

zy

zσxz

zt G

zvz

zx

zσyz

zt G

zvz

zx

(1)


Gρ



zωzt

Ax

zωzx

+ Ay

zωzy

(2)

where

Ax

0 vs 0

vs 0 0

0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Ay

0 0 vs

0 0 0

vs 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ω ρ2

1113969vz

12G

1113969σxz

12G

1113969σyz1113876 1113877

T

(3)


zσxx

zt

zσyy

zt

zσxy

zt

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

C

zvx

zx

zvy

zy

zvx

zy+

zvy

zx

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ρzvx

zt

zσxx

zx+

zσxy

zy

ρzvy

zt

zσxy

zx+

zσyy

zy

(4)


C

λ + 2μ λ 0

λ λ + 2μ 0

0 0 μ

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (5)


vp

λ + 2μ

ρ

1113971

vs

μρ

1113970

(6)


1113954A0z1113954ωzt

1113954Ax

z1113954ωzx

+ 1113954Ay

z1113954ωzy

(7)


1113954A0

a b 0 0 0

b a 0 0 0

0 0 c 0 0

0 0 0 1 0

0 0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113954Ax

0 0 0 1 0

0 0 0 0 0

0 0 0 0 1

1 0 0 0 0

0 0 1 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113954Ay

0 0 0 0 0

0 0 0 0 1

0 0 0 1 0

0 0 1 0 0

0 1 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(8)

a v2p

4v2p v2p minus v2s1113872 1113873


4v2p v2p minus v2s1113872 1113873

c 1v2s

(9)



zωzt

Ax

zωzx

+ Ay

zωzy

(10)


P

12

radic1

2

radic0 0 0

12

radicminus 1

2

radic0 0 0

0 0 1 0 00 0 0 1 00 0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Λ

12 v2p minus v2s1113872 1113873

12v2s

1v2s1

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(11)


M PTΛminus 12P

Mminus 1 PTΛ12P

(12)




as



ωt Axωx + Ayωy


(15)


M PTΛminus 12P

v2p minus v2s

1113969+ vs

2

radic

v2p minus v2s

1113969minus vs

2

radic 0 0 0

v2p minus v2s

1113969minus vs

2

radic

v2p minus v2s

1113969+ vs

2

radic 0 0 0

0 0 vs 0 0

0 0 0 1 0

0 0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(16)


(x(ξ η) y(ξ η))⟷(ξ(x y) η(x y)) (17)


J

zx

zξzx

zη

zy

zξzy

zη

111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868







J


J

(20)









Aplusmnξ TTAξ

ΔAξplusmn ΔAξ

11138681113868111386811138681113868

11138681113868111386811138681113868

2⎛⎝ ⎞⎠TT

Aξ (22)

en having


T plusmnAξ1113874 1113875

T


T plusmnAη1113874 1113875

T

(23)


pwest TminusAwestξ

1113874 1113875T

ωwest

qwest T+Awestξ

1113874 1113875T

ωwest

peast T+Aeastξ

1113874 1113875T

ωeast

qeast TminusAeastξ

1113874 1113875T

ωeast

psouth Tminus

Asouthη

1113874 1113875T

ωsouth

qsouth T+Asouthη

1113874 1113875T

ωsouth

pnorth T+

Anorthη

1113874 1113875T

ωnorth


1113874 1113875T

ωnorth

(24)



p Rq (25)




1 [u(1)1 u

(2)1 u

(k)1 ] and

uT2 [u

(1)2 u

(2)2 u


(u1u2)A 1113938xr






ξ Aξω + ωTηAηω1113872 1113873dS

(27)


z

ztω




ωTAξωdl

BTeast 1113946east

ωTAξωdl


ωTAηωdl

BTnorth 1113946west

ωTAηωdl

(29)


Mapping

North North

South

East EastWest

South


West

x

y η

ξ




ωT T+AηΔ+

AηT+

Aη1113874 1113875

T

+ TminusAηΔminus

AηTminus

Aη1113874 1113875

T

1113888 1113889ω dl

minus 1113946south

ωT T+AηΔ+

AηT+

Aη1113874 1113875

T

1113888 1113889ω dlle 0

(30)

BTnorth 1113946west

ωT1113888 RTminus

Aη1113874 1113875Δ+

AηRTminus

Aη1113874 1113875

T

+ TminusAηΔminus

AηTminus

Aη1113874 1113875

T

1113889ω dl

ωTTminusAη

RΔ+Aη

RT

+ ΔminusAη

1113874 1113875 TminusAη

1113874 1113875T

ω dlle 0

(31)






N (32)




D Hminus 1Q

DV Vx(33)


H HT gt 0

Q + QT EN minus E0

(34)

where E0 erarr

0 erarrT

0 EN erarr

N erarrT

N erarr


and erarr




ω ω(1)00 ω(2)

00 ω(3)00 ω(1)

NξNηω(2)

NξNηω(3)

NξNη1113876 1113877

T

(35)


ω ω(1)00 ω(2)

00 ω(3)00 ω(4)

00 ω(5)00 ω(1)

NξNηω(2)

NξNηω(3)

NξNηω(4)

NξNηω(5)

NξNη1113876 1113877

T

(36)


BotimesC


⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (37)








(38)




(39)








(41)


zωzt


where



minusAwestξΔminus

Awestξ

TminusAwestξ

1113874 1113875T

ωwest


+AeastξΔ+

Aeastξ

T+Aeastξ

1113874 1113875T

ωeast


minus

Asouthη

Δminus

Asouthη

Tminus

Asouthη

1113874 1113875T

ωsouth


+

Anorthη

Δ+

Anorthη

T+

Anorthη

minus RTminus

Anorthη

1113874 1113875T

ωnorth

(43)


z

ztω




westη Awest


westη Tminus

AwestξΔminus

AξTminus

Awestξ

1113874 1113875T

ωwest

BTeast ωTeastH

eastη Aeast


eastη T+

AeastξΔ+

AξT+

Aeastξ

1113874 1113875T

ωeast


southξ Asouth


southξ Tminus

Asouthη

ΔminusAη

Tminus

Asouthη

1113874 1113875T

ωsouth

BTnorth ωTnorthH

northξ Anorth


northξ T+

Anorthη

Δ+

Anorthη

T+

Anorthη

minus RTminus

Anorthη

1113874 1113875T

ωnorth

(45)



eastη T+

AeastξΔ+

AξT+

Aeastξ

1113874 1113875T

ωeast + ωTeastH

eastη Tminus

AeastξΔminus

AξTminus

Aeastξ

1113874 1113875T

ωeast


westη T+

AwestξΔ+

AξT+

Awestξ

1113874 1113875T

ωwest + ωTwestH

westη Tminus

AwestξΔminus

AξTminus

Awestξ

1113874 1113875T

ωwest


southη T+

Asouthξ

Δ+

Asouthξ

T+

Asouthξ

1113874 1113875T

ωsouth + ωTsouthH

southη Tminus

Asouthξ

Δminus

Asouthξ

Tminus

Asouthξ

1113874 1113875T

ωsouth

BTnorth ωTnorthH

northη Tminus

Anorthξ

RΔ+

Anorthη

RT+ Δminus

Anorthη

1113874 1113875 Tminus

Anorthξ

1113874 1113875T


northξ T+

Anorthη

minus RTminus

Anorthη

1113874 1113875Δ+

Anorthη

T+

Anorthη

minus RTminus

Anorthη

1113874 1113875T

ωnorth

(46)




ξ E(1)eastT

+(1)

AeastξΔ+(1)

Aeastξ

T+(1)

Aeastξ

1113874 1113875T


west1113872 1113873


ξ E(2)westT

minus (2)

AwestξΔminus (2)

Awestξ

Tminus (2)

Awestξ

1113874 1113875T


east1113872 1113873

(47)




Aeastξ

T+Aeastξ

1113874 1113875T

ωeast



Awestξ

TminusAwestξ

1113874 1113875T

ωwest

(48)







80001113888 1113889 (49)









S(1)

E(1)

N(1)

S(2)

W(2)W(1) E(2)

N(2)

1 2η

ξ


0

500

1000

1500

2000

2500

3000

Y

500 1000 1500 2000 2500 30000X

3

2

1 4

5

6

7

8

9



0

008

006

004

002

ndash002

ndash004

ndash006

014

012

01

(a)

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

(b)


01

005

0

ndash005

(a)

006

004

002

0

ndash002

ndash004

ndash006

(b)


008

006

004

002

0

ndash002

ndash004

(a)

004

003

002

001

0

ndash001

ndash002

ndash003

ndash004

(b)









ndash004

ndash002

0

002

004

006

008

01

012

014

016

V x (m

s)

02 04 06 08 1 12 14 16 18 20Time (s)



20001800

Y (m)1600

120010001400

800

X (m)6001200

4002001000

0

2200ndash005

0

005

01

V x (m

s)



01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(a)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash01

ndash008

(b)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(c)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(d)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(e)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(f )



0

50

100

150

200

250

300

Y

50 100 150 200 250 3000X

2

3

1 4

5

6

7

8

9


01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(a)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(b)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(c)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(d)

Figure 11 Continued





01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(e)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(f )



010002000300040005000

Elev

atio

n (m

)

200000 400000 600000 8000000Distance (m)



















Data Availability




Acknowledgments


References











































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


ρzvz

zt

zσxz

zx+

zσyz

zy

zσxz

zt G

zvz

zx

zσyz

zt G

zvz

zx

(1)


Gρ



zωzt

Ax

zωzx

+ Ay

zωzy

(2)

where

Ax

0 vs 0

vs 0 0

0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Ay

0 0 vs

0 0 0

vs 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ω ρ2

1113969vz

12G

1113969σxz

12G

1113969σyz1113876 1113877

T

(3)


zσxx

zt

zσyy

zt

zσxy

zt

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

C

zvx

zx

zvy

zy

zvx

zy+

zvy

zx

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ρzvx

zt

zσxx

zx+

zσxy

zy

ρzvy

zt

zσxy

zx+

zσyy

zy

(4)


C

λ + 2μ λ 0

λ λ + 2μ 0

0 0 μ

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (5)


vp

λ + 2μ

ρ

1113971

vs

μρ

1113970

(6)


1113954A0z1113954ωzt

1113954Ax

z1113954ωzx

+ 1113954Ay

z1113954ωzy

(7)


1113954A0

a b 0 0 0

b a 0 0 0

0 0 c 0 0

0 0 0 1 0

0 0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113954Ax

0 0 0 1 0

0 0 0 0 0

0 0 0 0 1

1 0 0 0 0

0 0 1 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113954Ay

0 0 0 0 0

0 0 0 0 1

0 0 0 1 0

0 0 1 0 0

0 1 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(8)

a v2p

4v2p v2p minus v2s1113872 1113873


4v2p v2p minus v2s1113872 1113873

c 1v2s

(9)



zωzt

Ax

zωzx

+ Ay

zωzy

(10)


P

12

radic1

2

radic0 0 0

12

radicminus 1

2

radic0 0 0

0 0 1 0 00 0 0 1 00 0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Λ

12 v2p minus v2s1113872 1113873

12v2s

1v2s1

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(11)


M PTΛminus 12P

Mminus 1 PTΛ12P

(12)




as



ωt Axωx + Ayωy


(15)


M PTΛminus 12P

v2p minus v2s

1113969+ vs

2

radic

v2p minus v2s

1113969minus vs

2

radic 0 0 0

v2p minus v2s

1113969minus vs

2

radic

v2p minus v2s

1113969+ vs

2

radic 0 0 0

0 0 vs 0 0

0 0 0 1 0

0 0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(16)


(x(ξ η) y(ξ η))⟷(ξ(x y) η(x y)) (17)


J

zx

zξzx

zη

zy

zξzy

zη

111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868







J


J

(20)









Aplusmnξ TTAξ

ΔAξplusmn ΔAξ

11138681113868111386811138681113868

11138681113868111386811138681113868

2⎛⎝ ⎞⎠TT

Aξ (22)

en having


T plusmnAξ1113874 1113875

T


T plusmnAη1113874 1113875

T

(23)


pwest TminusAwestξ

1113874 1113875T

ωwest

qwest T+Awestξ

1113874 1113875T

ωwest

peast T+Aeastξ

1113874 1113875T

ωeast

qeast TminusAeastξ

1113874 1113875T

ωeast

psouth Tminus

Asouthη

1113874 1113875T

ωsouth

qsouth T+Asouthη

1113874 1113875T

ωsouth

pnorth T+

Anorthη

1113874 1113875T

ωnorth


1113874 1113875T

ωnorth

(24)



p Rq (25)




1 [u(1)1 u

(2)1 u

(k)1 ] and

uT2 [u

(1)2 u

(2)2 u


(u1u2)A 1113938xr






ξ Aξω + ωTηAηω1113872 1113873dS

(27)


z

ztω




ωTAξωdl

BTeast 1113946east

ωTAξωdl


ωTAηωdl

BTnorth 1113946west

ωTAηωdl

(29)


Mapping

North North

South

East EastWest

South


West

x

y η

ξ




ωT T+AηΔ+

AηT+

Aη1113874 1113875

T

+ TminusAηΔminus

AηTminus

Aη1113874 1113875

T

1113888 1113889ω dl

minus 1113946south

ωT T+AηΔ+

AηT+

Aη1113874 1113875

T

1113888 1113889ω dlle 0

(30)

BTnorth 1113946west

ωT1113888 RTminus

Aη1113874 1113875Δ+

AηRTminus

Aη1113874 1113875

T

+ TminusAηΔminus

AηTminus

Aη1113874 1113875

T

1113889ω dl

ωTTminusAη

RΔ+Aη

RT

+ ΔminusAη

1113874 1113875 TminusAη

1113874 1113875T

ω dlle 0

(31)






N (32)




D Hminus 1Q

DV Vx(33)


H HT gt 0

Q + QT EN minus E0

(34)

where E0 erarr

0 erarrT

0 EN erarr

N erarrT

N erarr


and erarr




ω ω(1)00 ω(2)

00 ω(3)00 ω(1)

NξNηω(2)

NξNηω(3)

NξNη1113876 1113877

T

(35)


ω ω(1)00 ω(2)

00 ω(3)00 ω(4)

00 ω(5)00 ω(1)

NξNηω(2)

NξNηω(3)

NξNηω(4)

NξNηω(5)

NξNη1113876 1113877

T

(36)


BotimesC


⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (37)








(38)




(39)








(41)


zωzt


where



minusAwestξΔminus

Awestξ

TminusAwestξ

1113874 1113875T

ωwest


+AeastξΔ+

Aeastξ

T+Aeastξ

1113874 1113875T

ωeast


minus

Asouthη

Δminus

Asouthη

Tminus

Asouthη

1113874 1113875T

ωsouth


+

Anorthη

Δ+

Anorthη

T+

Anorthη

minus RTminus

Anorthη

1113874 1113875T

ωnorth

(43)


z

ztω




westη Awest


westη Tminus

AwestξΔminus

AξTminus

Awestξ

1113874 1113875T

ωwest

BTeast ωTeastH

eastη Aeast


eastη T+

AeastξΔ+

AξT+

Aeastξ

1113874 1113875T

ωeast


southξ Asouth


southξ Tminus

Asouthη

ΔminusAη

Tminus

Asouthη

1113874 1113875T

ωsouth

BTnorth ωTnorthH

northξ Anorth


northξ T+

Anorthη

Δ+

Anorthη

T+

Anorthη

minus RTminus

Anorthη

1113874 1113875T

ωnorth

(45)



eastη T+

AeastξΔ+

AξT+

Aeastξ

1113874 1113875T

ωeast + ωTeastH

eastη Tminus

AeastξΔminus

AξTminus

Aeastξ

1113874 1113875T

ωeast


westη T+

AwestξΔ+

AξT+

Awestξ

1113874 1113875T

ωwest + ωTwestH

westη Tminus

AwestξΔminus

AξTminus

Awestξ

1113874 1113875T

ωwest


southη T+

Asouthξ

Δ+

Asouthξ

T+

Asouthξ

1113874 1113875T

ωsouth + ωTsouthH

southη Tminus

Asouthξ

Δminus

Asouthξ

Tminus

Asouthξ

1113874 1113875T

ωsouth

BTnorth ωTnorthH

northη Tminus

Anorthξ

RΔ+

Anorthη

RT+ Δminus

Anorthη

1113874 1113875 Tminus

Anorthξ

1113874 1113875T


northξ T+

Anorthη

minus RTminus

Anorthη

1113874 1113875Δ+

Anorthη

T+

Anorthη

minus RTminus

Anorthη

1113874 1113875T

ωnorth

(46)




ξ E(1)eastT

+(1)

AeastξΔ+(1)

Aeastξ

T+(1)

Aeastξ

1113874 1113875T


west1113872 1113873


ξ E(2)westT

minus (2)

AwestξΔminus (2)

Awestξ

Tminus (2)

Awestξ

1113874 1113875T


east1113872 1113873

(47)




Aeastξ

T+Aeastξ

1113874 1113875T

ωeast



Awestξ

TminusAwestξ

1113874 1113875T

ωwest

(48)







80001113888 1113889 (49)









S(1)

E(1)

N(1)

S(2)

W(2)W(1) E(2)

N(2)

1 2η

ξ


0

500

1000

1500

2000

2500

3000

Y

500 1000 1500 2000 2500 30000X

3

2

1 4

5

6

7

8

9



0

008

006

004

002

ndash002

ndash004

ndash006

014

012

01

(a)

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

(b)


01

005

0

ndash005

(a)

006

004

002

0

ndash002

ndash004

ndash006

(b)


008

006

004

002

0

ndash002

ndash004

(a)

004

003

002

001

0

ndash001

ndash002

ndash003

ndash004

(b)









ndash004

ndash002

0

002

004

006

008

01

012

014

016

V x (m

s)

02 04 06 08 1 12 14 16 18 20Time (s)



20001800

Y (m)1600

120010001400

800

X (m)6001200

4002001000

0

2200ndash005

0

005

01

V x (m

s)



01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(a)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash01

ndash008

(b)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(c)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(d)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(e)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(f )



0

50

100

150

200

250

300

Y

50 100 150 200 250 3000X

2

3

1 4

5

6

7

8

9


01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(a)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(b)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(c)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(d)

Figure 11 Continued





01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(e)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(f )



010002000300040005000

Elev

atio

n (m

)

200000 400000 600000 8000000Distance (m)



















Data Availability




Acknowledgments


References











































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



zωzt

Ax

zωzx

+ Ay

zωzy

(10)


P

12

radic1

2

radic0 0 0

12

radicminus 1

2

radic0 0 0

0 0 1 0 00 0 0 1 00 0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Λ

12 v2p minus v2s1113872 1113873

12v2s

1v2s1

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(11)


M PTΛminus 12P

Mminus 1 PTΛ12P

(12)




as



ωt Axωx + Ayωy


(15)


M PTΛminus 12P

v2p minus v2s

1113969+ vs

2

radic

v2p minus v2s

1113969minus vs

2

radic 0 0 0

v2p minus v2s

1113969minus vs

2

radic

v2p minus v2s

1113969+ vs

2

radic 0 0 0

0 0 vs 0 0

0 0 0 1 0

0 0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(16)


(x(ξ η) y(ξ η))⟷(ξ(x y) η(x y)) (17)


J

zx

zξzx

zη

zy

zξzy

zη

111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868







J


J

(20)









Aplusmnξ TTAξ

ΔAξplusmn ΔAξ

11138681113868111386811138681113868

11138681113868111386811138681113868

2⎛⎝ ⎞⎠TT

Aξ (22)

en having


T plusmnAξ1113874 1113875

T


T plusmnAη1113874 1113875

T

(23)


pwest TminusAwestξ

1113874 1113875T

ωwest

qwest T+Awestξ

1113874 1113875T

ωwest

peast T+Aeastξ

1113874 1113875T

ωeast

qeast TminusAeastξ

1113874 1113875T

ωeast

psouth Tminus

Asouthη

1113874 1113875T

ωsouth

qsouth T+Asouthη

1113874 1113875T

ωsouth

pnorth T+

Anorthη

1113874 1113875T

ωnorth


1113874 1113875T

ωnorth

(24)



p Rq (25)




1 [u(1)1 u

(2)1 u

(k)1 ] and

uT2 [u

(1)2 u

(2)2 u


(u1u2)A 1113938xr






ξ Aξω + ωTηAηω1113872 1113873dS

(27)


z

ztω




ωTAξωdl

BTeast 1113946east

ωTAξωdl


ωTAηωdl

BTnorth 1113946west

ωTAηωdl

(29)


Mapping

North North

South

East EastWest

South


West

x

y η

ξ




ωT T+AηΔ+

AηT+

Aη1113874 1113875

T

+ TminusAηΔminus

AηTminus

Aη1113874 1113875

T

1113888 1113889ω dl

minus 1113946south

ωT T+AηΔ+

AηT+

Aη1113874 1113875

T

1113888 1113889ω dlle 0

(30)

BTnorth 1113946west

ωT1113888 RTminus

Aη1113874 1113875Δ+

AηRTminus

Aη1113874 1113875

T

+ TminusAηΔminus

AηTminus

Aη1113874 1113875

T

1113889ω dl

ωTTminusAη

RΔ+Aη

RT

+ ΔminusAη

1113874 1113875 TminusAη

1113874 1113875T

ω dlle 0

(31)






N (32)




D Hminus 1Q

DV Vx(33)


H HT gt 0

Q + QT EN minus E0

(34)

where E0 erarr

0 erarrT

0 EN erarr

N erarrT

N erarr


and erarr




ω ω(1)00 ω(2)

00 ω(3)00 ω(1)

NξNηω(2)

NξNηω(3)

NξNη1113876 1113877

T

(35)


ω ω(1)00 ω(2)

00 ω(3)00 ω(4)

00 ω(5)00 ω(1)

NξNηω(2)

NξNηω(3)

NξNηω(4)

NξNηω(5)

NξNη1113876 1113877

T

(36)


BotimesC


⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (37)








(38)




(39)








(41)


zωzt


where



minusAwestξΔminus

Awestξ

TminusAwestξ

1113874 1113875T

ωwest


+AeastξΔ+

Aeastξ

T+Aeastξ

1113874 1113875T

ωeast


minus

Asouthη

Δminus

Asouthη

Tminus

Asouthη

1113874 1113875T

ωsouth


+

Anorthη

Δ+

Anorthη

T+

Anorthη

minus RTminus

Anorthη

1113874 1113875T

ωnorth

(43)


z

ztω




westη Awest


westη Tminus

AwestξΔminus

AξTminus

Awestξ

1113874 1113875T

ωwest

BTeast ωTeastH

eastη Aeast


eastη T+

AeastξΔ+

AξT+

Aeastξ

1113874 1113875T

ωeast


southξ Asouth


southξ Tminus

Asouthη

ΔminusAη

Tminus

Asouthη

1113874 1113875T

ωsouth

BTnorth ωTnorthH

northξ Anorth


northξ T+

Anorthη

Δ+

Anorthη

T+

Anorthη

minus RTminus

Anorthη

1113874 1113875T

ωnorth

(45)



eastη T+

AeastξΔ+

AξT+

Aeastξ

1113874 1113875T

ωeast + ωTeastH

eastη Tminus

AeastξΔminus

AξTminus

Aeastξ

1113874 1113875T

ωeast


westη T+

AwestξΔ+

AξT+

Awestξ

1113874 1113875T

ωwest + ωTwestH

westη Tminus

AwestξΔminus

AξTminus

Awestξ

1113874 1113875T

ωwest


southη T+

Asouthξ

Δ+

Asouthξ

T+

Asouthξ

1113874 1113875T

ωsouth + ωTsouthH

southη Tminus

Asouthξ

Δminus

Asouthξ

Tminus

Asouthξ

1113874 1113875T

ωsouth

BTnorth ωTnorthH

northη Tminus

Anorthξ

RΔ+

Anorthη

RT+ Δminus

Anorthη

1113874 1113875 Tminus

Anorthξ

1113874 1113875T


northξ T+

Anorthη

minus RTminus

Anorthη

1113874 1113875Δ+

Anorthη

T+

Anorthη

minus RTminus

Anorthη

1113874 1113875T

ωnorth

(46)




ξ E(1)eastT

+(1)

AeastξΔ+(1)

Aeastξ

T+(1)

Aeastξ

1113874 1113875T


west1113872 1113873


ξ E(2)westT

minus (2)

AwestξΔminus (2)

Awestξ

Tminus (2)

Awestξ

1113874 1113875T


east1113872 1113873

(47)




Aeastξ

T+Aeastξ

1113874 1113875T

ωeast



Awestξ

TminusAwestξ

1113874 1113875T

ωwest

(48)







80001113888 1113889 (49)









S(1)

E(1)

N(1)

S(2)

W(2)W(1) E(2)

N(2)

1 2η

ξ


0

500

1000

1500

2000

2500

3000

Y

500 1000 1500 2000 2500 30000X

3

2

1 4

5

6

7

8

9



0

008

006

004

002

ndash002

ndash004

ndash006

014

012

01

(a)

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

(b)


01

005

0

ndash005

(a)

006

004

002

0

ndash002

ndash004

ndash006

(b)


008

006

004

002

0

ndash002

ndash004

(a)

004

003

002

001

0

ndash001

ndash002

ndash003

ndash004

(b)









ndash004

ndash002

0

002

004

006

008

01

012

014

016

V x (m

s)

02 04 06 08 1 12 14 16 18 20Time (s)



20001800

Y (m)1600

120010001400

800

X (m)6001200

4002001000

0

2200ndash005

0

005

01

V x (m

s)



01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(a)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash01

ndash008

(b)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(c)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(d)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(e)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(f )



0

50

100

150

200

250

300

Y

50 100 150 200 250 3000X

2

3

1 4

5

6

7

8

9


01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(a)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(b)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(c)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(d)

Figure 11 Continued





01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(e)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(f )



010002000300040005000

Elev

atio

n (m

)

200000 400000 600000 8000000Distance (m)



















Data Availability




Acknowledgments


References











































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



Aplusmnξ TTAξ

ΔAξplusmn ΔAξ

11138681113868111386811138681113868

11138681113868111386811138681113868

2⎛⎝ ⎞⎠TT

Aξ (22)

en having


T plusmnAξ1113874 1113875

T


T plusmnAη1113874 1113875

T

(23)


pwest TminusAwestξ

1113874 1113875T

ωwest

qwest T+Awestξ

1113874 1113875T

ωwest

peast T+Aeastξ

1113874 1113875T

ωeast

qeast TminusAeastξ

1113874 1113875T

ωeast

psouth Tminus

Asouthη

1113874 1113875T

ωsouth

qsouth T+Asouthη

1113874 1113875T

ωsouth

pnorth T+

Anorthη

1113874 1113875T

ωnorth


1113874 1113875T

ωnorth

(24)



p Rq (25)




1 [u(1)1 u

(2)1 u

(k)1 ] and

uT2 [u

(1)2 u

(2)2 u


(u1u2)A 1113938xr






ξ Aξω + ωTηAηω1113872 1113873dS

(27)


z

ztω




ωTAξωdl

BTeast 1113946east

ωTAξωdl


ωTAηωdl

BTnorth 1113946west

ωTAηωdl

(29)


Mapping

North North

South

East EastWest

South


West

x

y η

ξ




ωT T+AηΔ+

AηT+

Aη1113874 1113875

T

+ TminusAηΔminus

AηTminus

Aη1113874 1113875

T

1113888 1113889ω dl

minus 1113946south

ωT T+AηΔ+

AηT+

Aη1113874 1113875

T

1113888 1113889ω dlle 0

(30)

BTnorth 1113946west

ωT1113888 RTminus

Aη1113874 1113875Δ+

AηRTminus

Aη1113874 1113875

T

+ TminusAηΔminus

AηTminus

Aη1113874 1113875

T

1113889ω dl

ωTTminusAη

RΔ+Aη

RT

+ ΔminusAη

1113874 1113875 TminusAη

1113874 1113875T

ω dlle 0

(31)






N (32)




D Hminus 1Q

DV Vx(33)


H HT gt 0

Q + QT EN minus E0

(34)

where E0 erarr

0 erarrT

0 EN erarr

N erarrT

N erarr


and erarr




ω ω(1)00 ω(2)

00 ω(3)00 ω(1)

NξNηω(2)

NξNηω(3)

NξNη1113876 1113877

T

(35)


ω ω(1)00 ω(2)

00 ω(3)00 ω(4)

00 ω(5)00 ω(1)

NξNηω(2)

NξNηω(3)

NξNηω(4)

NξNηω(5)

NξNη1113876 1113877

T

(36)


BotimesC


⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (37)








(38)




(39)








(41)


zωzt


where



minusAwestξΔminus

Awestξ

TminusAwestξ

1113874 1113875T

ωwest


+AeastξΔ+

Aeastξ

T+Aeastξ

1113874 1113875T

ωeast


minus

Asouthη

Δminus

Asouthη

Tminus

Asouthη

1113874 1113875T

ωsouth


+

Anorthη

Δ+

Anorthη

T+

Anorthη

minus RTminus

Anorthη

1113874 1113875T

ωnorth

(43)


z

ztω




westη Awest


westη Tminus

AwestξΔminus

AξTminus

Awestξ

1113874 1113875T

ωwest

BTeast ωTeastH

eastη Aeast


eastη T+

AeastξΔ+

AξT+

Aeastξ

1113874 1113875T

ωeast


southξ Asouth


southξ Tminus

Asouthη

ΔminusAη

Tminus

Asouthη

1113874 1113875T

ωsouth

BTnorth ωTnorthH

northξ Anorth


northξ T+

Anorthη

Δ+

Anorthη

T+

Anorthη

minus RTminus

Anorthη

1113874 1113875T

ωnorth

(45)



eastη T+

AeastξΔ+

AξT+

Aeastξ

1113874 1113875T

ωeast + ωTeastH

eastη Tminus

AeastξΔminus

AξTminus

Aeastξ

1113874 1113875T

ωeast


westη T+

AwestξΔ+

AξT+

Awestξ

1113874 1113875T

ωwest + ωTwestH

westη Tminus

AwestξΔminus

AξTminus

Awestξ

1113874 1113875T

ωwest


southη T+

Asouthξ

Δ+

Asouthξ

T+

Asouthξ

1113874 1113875T

ωsouth + ωTsouthH

southη Tminus

Asouthξ

Δminus

Asouthξ

Tminus

Asouthξ

1113874 1113875T

ωsouth

BTnorth ωTnorthH

northη Tminus

Anorthξ

RΔ+

Anorthη

RT+ Δminus

Anorthη

1113874 1113875 Tminus

Anorthξ

1113874 1113875T


northξ T+

Anorthη

minus RTminus

Anorthη

1113874 1113875Δ+

Anorthη

T+

Anorthη

minus RTminus

Anorthη

1113874 1113875T

ωnorth

(46)




ξ E(1)eastT

+(1)

AeastξΔ+(1)

Aeastξ

T+(1)

Aeastξ

1113874 1113875T


west1113872 1113873


ξ E(2)westT

minus (2)

AwestξΔminus (2)

Awestξ

Tminus (2)

Awestξ

1113874 1113875T


east1113872 1113873

(47)




Aeastξ

T+Aeastξ

1113874 1113875T

ωeast



Awestξ

TminusAwestξ

1113874 1113875T

ωwest

(48)







80001113888 1113889 (49)









S(1)

E(1)

N(1)

S(2)

W(2)W(1) E(2)

N(2)

1 2η

ξ


0

500

1000

1500

2000

2500

3000

Y

500 1000 1500 2000 2500 30000X

3

2

1 4

5

6

7

8

9



0

008

006

004

002

ndash002

ndash004

ndash006

014

012

01

(a)

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

(b)


01

005

0

ndash005

(a)

006

004

002

0

ndash002

ndash004

ndash006

(b)


008

006

004

002

0

ndash002

ndash004

(a)

004

003

002

001

0

ndash001

ndash002

ndash003

ndash004

(b)









ndash004

ndash002

0

002

004

006

008

01

012

014

016

V x (m

s)

02 04 06 08 1 12 14 16 18 20Time (s)



20001800

Y (m)1600

120010001400

800

X (m)6001200

4002001000

0

2200ndash005

0

005

01

V x (m

s)



01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(a)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash01

ndash008

(b)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(c)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(d)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(e)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(f )



0

50

100

150

200

250

300

Y

50 100 150 200 250 3000X

2

3

1 4

5

6

7

8

9


01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(a)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(b)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(c)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(d)

Figure 11 Continued





01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(e)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(f )



010002000300040005000

Elev

atio

n (m

)

200000 400000 600000 8000000Distance (m)



















Data Availability




Acknowledgments


References











































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



ωT T+AηΔ+

AηT+

Aη1113874 1113875

T

+ TminusAηΔminus

AηTminus

Aη1113874 1113875

T

1113888 1113889ω dl

minus 1113946south

ωT T+AηΔ+

AηT+

Aη1113874 1113875

T

1113888 1113889ω dlle 0

(30)

BTnorth 1113946west

ωT1113888 RTminus

Aη1113874 1113875Δ+

AηRTminus

Aη1113874 1113875

T

+ TminusAηΔminus

AηTminus

Aη1113874 1113875

T

1113889ω dl

ωTTminusAη

RΔ+Aη

RT

+ ΔminusAη

1113874 1113875 TminusAη

1113874 1113875T

ω dlle 0

(31)






N (32)




D Hminus 1Q

DV Vx(33)


H HT gt 0

Q + QT EN minus E0

(34)

where E0 erarr

0 erarrT

0 EN erarr

N erarrT

N erarr


and erarr




ω ω(1)00 ω(2)

00 ω(3)00 ω(1)

NξNηω(2)

NξNηω(3)

NξNη1113876 1113877

T

(35)


ω ω(1)00 ω(2)

00 ω(3)00 ω(4)

00 ω(5)00 ω(1)

NξNηω(2)

NξNηω(3)

NξNηω(4)

NξNηω(5)

NξNη1113876 1113877

T

(36)


BotimesC


⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (37)








(38)




(39)








(41)


zωzt


where



minusAwestξΔminus

Awestξ

TminusAwestξ

1113874 1113875T

ωwest


+AeastξΔ+

Aeastξ

T+Aeastξ

1113874 1113875T

ωeast


minus

Asouthη

Δminus

Asouthη

Tminus

Asouthη

1113874 1113875T

ωsouth


+

Anorthη

Δ+

Anorthη

T+

Anorthη

minus RTminus

Anorthη

1113874 1113875T

ωnorth

(43)


z

ztω




westη Awest


westη Tminus

AwestξΔminus

AξTminus

Awestξ

1113874 1113875T

ωwest

BTeast ωTeastH

eastη Aeast


eastη T+

AeastξΔ+

AξT+

Aeastξ

1113874 1113875T

ωeast


southξ Asouth


southξ Tminus

Asouthη

ΔminusAη

Tminus

Asouthη

1113874 1113875T

ωsouth

BTnorth ωTnorthH

northξ Anorth


northξ T+

Anorthη

Δ+

Anorthη

T+

Anorthη

minus RTminus

Anorthη

1113874 1113875T

ωnorth

(45)



eastη T+

AeastξΔ+

AξT+

Aeastξ

1113874 1113875T

ωeast + ωTeastH

eastη Tminus

AeastξΔminus

AξTminus

Aeastξ

1113874 1113875T

ωeast


westη T+

AwestξΔ+

AξT+

Awestξ

1113874 1113875T

ωwest + ωTwestH

westη Tminus

AwestξΔminus

AξTminus

Awestξ

1113874 1113875T

ωwest


southη T+

Asouthξ

Δ+

Asouthξ

T+

Asouthξ

1113874 1113875T

ωsouth + ωTsouthH

southη Tminus

Asouthξ

Δminus

Asouthξ

Tminus

Asouthξ

1113874 1113875T

ωsouth

BTnorth ωTnorthH

northη Tminus

Anorthξ

RΔ+

Anorthη

RT+ Δminus

Anorthη

1113874 1113875 Tminus

Anorthξ

1113874 1113875T


northξ T+

Anorthη

minus RTminus

Anorthη

1113874 1113875Δ+

Anorthη

T+

Anorthη

minus RTminus

Anorthη

1113874 1113875T

ωnorth

(46)




ξ E(1)eastT

+(1)

AeastξΔ+(1)

Aeastξ

T+(1)

Aeastξ

1113874 1113875T


west1113872 1113873


ξ E(2)westT

minus (2)

AwestξΔminus (2)

Awestξ

Tminus (2)

Awestξ

1113874 1113875T


east1113872 1113873

(47)




Aeastξ

T+Aeastξ

1113874 1113875T

ωeast



Awestξ

TminusAwestξ

1113874 1113875T

ωwest

(48)







80001113888 1113889 (49)









S(1)

E(1)

N(1)

S(2)

W(2)W(1) E(2)

N(2)

1 2η

ξ


0

500

1000

1500

2000

2500

3000

Y

500 1000 1500 2000 2500 30000X

3

2

1 4

5

6

7

8

9



0

008

006

004

002

ndash002

ndash004

ndash006

014

012

01

(a)

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

(b)


01

005

0

ndash005

(a)

006

004

002

0

ndash002

ndash004

ndash006

(b)


008

006

004

002

0

ndash002

ndash004

(a)

004

003

002

001

0

ndash001

ndash002

ndash003

ndash004

(b)









ndash004

ndash002

0

002

004

006

008

01

012

014

016

V x (m

s)

02 04 06 08 1 12 14 16 18 20Time (s)



20001800

Y (m)1600

120010001400

800

X (m)6001200

4002001000

0

2200ndash005

0

005

01

V x (m

s)



01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(a)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash01

ndash008

(b)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(c)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(d)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(e)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(f )



0

50

100

150

200

250

300

Y

50 100 150 200 250 3000X

2

3

1 4

5

6

7

8

9


01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(a)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(b)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(c)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(d)

Figure 11 Continued





01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(e)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(f )



010002000300040005000

Elev

atio

n (m

)

200000 400000 600000 8000000Distance (m)



















Data Availability




Acknowledgments


References











































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



minusAwestξΔminus

Awestξ

TminusAwestξ

1113874 1113875T

ωwest


+AeastξΔ+

Aeastξ

T+Aeastξ

1113874 1113875T

ωeast


minus

Asouthη

Δminus

Asouthη

Tminus

Asouthη

1113874 1113875T

ωsouth


+

Anorthη

Δ+

Anorthη

T+

Anorthη

minus RTminus

Anorthη

1113874 1113875T

ωnorth

(43)


z

ztω




westη Awest


westη Tminus

AwestξΔminus

AξTminus

Awestξ

1113874 1113875T

ωwest

BTeast ωTeastH

eastη Aeast


eastη T+

AeastξΔ+

AξT+

Aeastξ

1113874 1113875T

ωeast


southξ Asouth


southξ Tminus

Asouthη

ΔminusAη

Tminus

Asouthη

1113874 1113875T

ωsouth

BTnorth ωTnorthH

northξ Anorth


northξ T+

Anorthη

Δ+

Anorthη

T+

Anorthη

minus RTminus

Anorthη

1113874 1113875T

ωnorth

(45)



eastη T+

AeastξΔ+

AξT+

Aeastξ

1113874 1113875T

ωeast + ωTeastH

eastη Tminus

AeastξΔminus

AξTminus

Aeastξ

1113874 1113875T

ωeast


westη T+

AwestξΔ+

AξT+

Awestξ

1113874 1113875T

ωwest + ωTwestH

westη Tminus

AwestξΔminus

AξTminus

Awestξ

1113874 1113875T

ωwest


southη T+

Asouthξ

Δ+

Asouthξ

T+

Asouthξ

1113874 1113875T

ωsouth + ωTsouthH

southη Tminus

Asouthξ

Δminus

Asouthξ

Tminus

Asouthξ

1113874 1113875T

ωsouth

BTnorth ωTnorthH

northη Tminus

Anorthξ

RΔ+

Anorthη

RT+ Δminus

Anorthη

1113874 1113875 Tminus

Anorthξ

1113874 1113875T


northξ T+

Anorthη

minus RTminus

Anorthη

1113874 1113875Δ+

Anorthη

T+

Anorthη

minus RTminus

Anorthη

1113874 1113875T

ωnorth

(46)




ξ E(1)eastT

+(1)

AeastξΔ+(1)

Aeastξ

T+(1)

Aeastξ

1113874 1113875T


west1113872 1113873


ξ E(2)westT

minus (2)

AwestξΔminus (2)

Awestξ

Tminus (2)

Awestξ

1113874 1113875T


east1113872 1113873

(47)




Aeastξ

T+Aeastξ

1113874 1113875T

ωeast



Awestξ

TminusAwestξ

1113874 1113875T

ωwest

(48)







80001113888 1113889 (49)









S(1)

E(1)

N(1)

S(2)

W(2)W(1) E(2)

N(2)

1 2η

ξ


0

500

1000

1500

2000

2500

3000

Y

500 1000 1500 2000 2500 30000X

3

2

1 4

5

6

7

8

9



0

008

006

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002

ndash002

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(a)

008

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002

0

ndash002

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ndash006

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(b)


01

005

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ndash005

(a)

006

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ndash006

(b)


008

006

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(a)

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ndash001

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(b)









ndash004

ndash002

0

002

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V x (m

s)

02 04 06 08 1 12 14 16 18 20Time (s)



20001800

Y (m)1600

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2200ndash005

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005

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V x (m

s)



01

008

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ndash002

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ndash01

(a)

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ndash01

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(b)

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(c)

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(d)

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(e)

01

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ndash002

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(f )



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01

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ndash002

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(a)

01

008

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002

0

ndash002

ndash004

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ndash008

ndash01

(b)

01

008

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ndash008

ndash01

(c)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(d)

Figure 11 Continued





01

008

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002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(e)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(f )



010002000300040005000

Elev

atio

n (m

)

200000 400000 600000 8000000Distance (m)



















Data Availability




Acknowledgments


References











































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





80001113888 1113889 (49)









S(1)

E(1)

N(1)

S(2)

W(2)W(1) E(2)

N(2)

1 2η

ξ


0

500

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Y

500 1000 1500 2000 2500 30000X

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0

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(a)

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(b)


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(a)

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(b)


008

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(a)

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(b)









ndash004

ndash002

0

002

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006

008

01

012

014

016

V x (m

s)

02 04 06 08 1 12 14 16 18 20Time (s)



20001800

Y (m)1600

120010001400

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X (m)6001200

4002001000

0

2200ndash005

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005

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V x (m

s)



01

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ndash006

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ndash01

(a)

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(b)

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ndash008

ndash01

(c)

01

008

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0

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ndash004

ndash006

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ndash01

(d)

01

008

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002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(e)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(f )



0

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2

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01

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ndash01

(a)

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008

006

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002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(b)

01

008

006

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002

0

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ndash004

ndash006

ndash008

ndash01

(c)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(d)

Figure 11 Continued





01

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006

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002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(e)

01

008

006

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002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(f )



010002000300040005000

Elev

atio

n (m

)

200000 400000 600000 8000000Distance (m)



















Data Availability




Acknowledgments


References











































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


0

008

006

004

002

ndash002

ndash004

ndash006

014

012

01

(a)

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

(b)


01

005

0

ndash005

(a)

006

004

002

0

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ndash006

(b)


008

006

004

002

0

ndash002

ndash004

(a)

004

003

002

001

0

ndash001

ndash002

ndash003

ndash004

(b)









ndash004

ndash002

0

002

004

006

008

01

012

014

016

V x (m

s)

02 04 06 08 1 12 14 16 18 20Time (s)



20001800

Y (m)1600

120010001400

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X (m)6001200

4002001000

0

2200ndash005

0

005

01

V x (m

s)



01

008

006

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002

0

ndash002

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ndash006

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(a)

01

008

006

004

002

0

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ndash004

ndash006

ndash01

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(b)

01

008

006

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0

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ndash01

(c)

01

008

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0

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ndash008

ndash01

(d)

01

008

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ndash004

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ndash008

ndash01

(e)

01

008

006

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002

0

ndash002

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ndash006

ndash008

ndash01

(f )



0

50

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Y

50 100 150 200 250 3000X

2

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1 4

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01

008

006

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0

ndash002

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(a)

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008

006

004

002

0

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ndash004

ndash006

ndash008

ndash01

(b)

01

008

006

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002

0

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ndash004

ndash006

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ndash01

(c)

01

008

006

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002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(d)

Figure 11 Continued





01

008

006

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002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(e)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(f )



010002000300040005000

Elev

atio

n (m

)

200000 400000 600000 8000000Distance (m)



















Data Availability




Acknowledgments


References











































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia








ndash004

ndash002

0

002

004

006

008

01

012

014

016

V x (m

s)

02 04 06 08 1 12 14 16 18 20Time (s)



20001800

Y (m)1600

120010001400

800

X (m)6001200

4002001000

0

2200ndash005

0

005

01

V x (m

s)



01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(a)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash01

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(b)

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ndash008

ndash01

(c)

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(d)

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(e)

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ndash006

ndash008

ndash01

(f )



0

50

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Y

50 100 150 200 250 3000X

2

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1 4

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01

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ndash006

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ndash01

(a)

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008

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002

0

ndash002

ndash004

ndash006

ndash008

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(b)

01

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ndash002

ndash004

ndash006

ndash008

ndash01

(c)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(d)

Figure 11 Continued





01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(e)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(f )



010002000300040005000

Elev

atio

n (m

)

200000 400000 600000 8000000Distance (m)



















Data Availability




Acknowledgments


References











































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(a)

01

008

006

004

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0

ndash002

ndash004

ndash006

ndash01

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(b)

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(c)

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(d)

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(e)

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(f )



0

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2

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1 4

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01

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(a)

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008

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0

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(b)

01

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002

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ndash008

ndash01

(c)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(d)

Figure 11 Continued





01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(e)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(f )



010002000300040005000

Elev

atio

n (m

)

200000 400000 600000 8000000Distance (m)



















Data Availability




Acknowledgments


References











































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


0

50

100

150

200

250

300

Y

50 100 150 200 250 3000X

2

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1 4

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8

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01

008

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(a)

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(b)

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ndash008

ndash01

(c)

01

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ndash006

ndash008

ndash01

(d)

Figure 11 Continued





01

008

006

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0

ndash002

ndash004

ndash006

ndash008

ndash01

(e)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(f )



010002000300040005000

Elev

atio

n (m

)

200000 400000 600000 8000000Distance (m)



















Data Availability




Acknowledgments


References











































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(e)

01

008

006

004

002

0

ndash002

ndash004

ndash006

ndash008

ndash01

(f )



010002000300040005000

Elev

atio

n (m

)

200000 400000 600000 8000000Distance (m)



















Data Availability




Acknowledgments


References











































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






Data Availability




Acknowledgments


References











































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


Documents

MultiblockSBP-SATMethodologyofSymmetricMatrixFormof ...timedomain[48].Alocalhigh-orderABC,aswellasa nonreﬂecting boundary condition (BC), has been widely used for their feasibility,