Comp. Part. Mech. (2014) 1:131–145DOI 10.1007/s40571-014-0015-6

A local adaptive discretization algorithm for Smoothed ParticleHydrodynamicsFor the inaugural issue

Fabian Spreng · Dirk Schnabel · Alexandra Mueller ·Peter Eberhard

Received: 22 December 2013 / Revised: 29 March 2014 / Accepted: 9 April 2014 / Published online: 29 April 2014© Springer International Publishing Switzerland 2014

Abstract In this paper, an extension to the Smoothed Parti-cle Hydrodynamics (SPH) method is proposed that allows foran adaptation of the discretization level of a simulated con-tinuum at runtime. By combining a local adaptive refinementtechnique with a newly developed coarsening algorithm, oneis able to improve the accuracy of the simulation results whilereducing the required computational cost at the same time.For this purpose, the number of particles is, on the one hand,adaptively increased in critical areas of a simulation model.Typically, these are areas that show a relatively low parti-cle density and high gradients in stress or temperature. Onthe other hand, the number of SPH particles is decreasedfor domains with a high particle density and low gradients.Besides a brief introduction to the basic principle of the SPHdiscretization method, the extensions to the original formula-tion providing such a local adaptive refinement and coarsen-ing of the modeled structure are presented in this paper. Afterhaving introduced its theoretical background, the applicabil-ity of the enhanced formulation, as well as the benefit gainedfrom the adaptive model discretization, is demonstrated inthe context of four different simulation scenarios focusing onsolid continua. While presenting the results found for theseexamples, several properties of the proposed adaptive tech-nique are discussed, e.g. the conservation of momentum aswell as the existing correlation between the chosen refine-ment and coarsening patterns and the observed quality of theresults.

F. Spreng · D. Schnabel · A. Mueller · P. Eberhard (B)Institute of Engineering and Computational Mechanics,University of Stuttgart, Pfaffenwaldring 9,70569 Stuttgart, Germanye-mail: peter.eberhard@itm.uni-stuttgart.deURL: http://www.itm.uni-stuttgart.de

Keywords Particle simulation · Smoothed ParticleHydrodynamics · Model discretization · Adaptivity ·Refinement · Coarsening

1 Introduction

Due to its meshless nature, typical fields of application ofthe Lagrangian particle method Smoothed Particle Hydro-dynamics (SPH) are simulation scenarios that are character-ized by large deformations of the modeled domain of mater-ial, disintegrating structures, and/or highly dynamic impacts.But although there are no issues coming up that are a resultof a distorted mesh when employing a meshless discretiza-tion method, the model quality is still highly dependent onthe number of interpolation points that are used for the dis-cretization of the investigated spatial domain. For this reason,an important field of research concerning SPH is the develop-ment of discretization strategies that allow for an adaptationof the number of SPH particles being placed in certain areasof a simulation model at runtime. Recent research activitiesdealt with fundamental issues of such adaptive extensions andprovided the basis for some first combined refinement andcoarsening techniques. The approach proposed in [1] givesa good starting point, but relies on an exclusively random-based particle selection strategy for the coarsening process,which is restricted to pairs of adaptive particles. Therefore,we have been working on a more advanced adaptive dis-cretization extension to the SPH method over the last fewyears.

After having developed an enhanced version of the refine-ment procedure presented in [2] and [3], as well as an appro-priate coarsening algorithm, we implemented the combinedadaptive discretization strategy into the SPH plugin for theparticle simulation package Pasimodo [4]. By testing the

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132 Comp. Part. Mech. (2014) 1:131–145

approach for a broad range of different simulation setups, wehave been able to verify its validity. In this paper, the theo-retical basis of this newly developed discretization approach,along with some of the validation examples, is presented.

First, a brief introduction to the SPH spatial discretizationmethod in general as well as to its most important extensionsnecessary to be able to reproduce the behavior found for realsolid matter is provided.

Second, the modifications made to allow for a local adap-tive refinement and coarsening of the modeled structure arepresented. In this context, suitable criteria to decide whetherthe discretization level of an existing particle is to be adaptedor not are discussed. In case of refinement, different pat-terns specifying the positions of the additional particles areintroduced and, in case of coarsening, different strategies toidentify the particles to be merged are presented. Further-more, the way how to determine the state variables of thenew particles are discussed in detail. All of these modifica-tions can also be applied to SPH fluid simulations withoutfurther adjustments.

Third and finally, four of the various simulation scenariosused for the validation of the implemented routine are pre-sented. That way, it is shown that the introduced approachprovides correct results when being applied to simulationsfocusing on solid continua. In a first step, the focus is laidon the basic properties of the presented adaptive technique,e.g. the conservation of kinetic energy. To that end, the resultsfound for the simulations of an ordinary and a notched tensiletest are discussed. For these two rather simple setups, analyt-ical solutions as well as numerical results obtained with theFinite Element Method (FEM) [5] will be compared with theones provided by the adaptive SPH method. In a second step,the applicability of the proposed approach to more typicaland demanding scenarios is demonstrated for two examples;a rigid torus falling onto an elastic membrane and an orthog-onal cutting process.

2 Smoothed Particle Hydrodynamics

Originally developed for the purpose of investigating astro-physical phenomena [6,7], the meshfree Smoothed Parti-cle Hydrodynamics (SPH) method experienced the neces-sary modifications to be also applicable to fluid simulations,i.e., to allow for a discretization of the well-known Navier-Stokes Eq. [8]. In this context, SPH, as a spatial discretizationmethod, is used to replace the considered continuum domainwith a set of particles, which can be interpreted as interpola-tion points. At these points, the underlying differential equa-tions of continuum mechanics are evaluated. In doing so, theinitial system of partial differential equations (PDEs) in timeand space is transformed into one that consists only of ordi-nary differential equations (ODEs) in time [9], which can be

numerically integrated with common integration schemes. Inaddition to fluid simulations, SPH has proven its ability tocorrectly reproduce the physical effects that are observed forreal solid material over the last two decades [10,11].

Next, the basic principle of the SPH discretization methodis presented. After having described its basic idea, the SPHsimulation technique is applied to the Euler equations, whichcan be used to reproduce the behavior of solid matter in simu-lations. Then, the focus is laid on some of the extensions to thebasic SPH solid formulation introduced in [12] that improvethe significance of the obtained results. Namely, these are theJohnson-Cook (JC) plasticity model, the JC damage model,as well as a modified version of the Lennard-Jones potential.

2.1 Basic principle of Smoothed Particle Hydrodynamics

The SPH discretization process to transform a PDE into anODE consists of the following three steps. First, an ensem-ble of abstract, not discrete particles is introduced into theconsidered simulation domain. Each of these SPH particlesrepresents a specific part of the original spatial domain and,according to that, combines all of its properties, such as posi-tion, mass, density, etc. That way, it is possible to eliminatethe dependency on the spatial position and its derivativesin case of any PDE while using the SPH particles, i.e. theintroduced interpolation points, in conjunction with the Diracdelta function.

Second, the Dirac delta function used for the discretiza-tion of a PDE is approximated by a kernel function of finiteextent, e.g. a truncated Gaussian distribution. The contribu-tion of each neighboring particle to a property of the particleof interest is weighted according to the distance betweenthese two SPH particles as well as the actual shape of thechosen kernel function. For this reason, the kernel functionis also referred to as weight function. That way, the level ofsmoothing of the properties of an SPH particle is determined.Commonly, the functions that are used as kernel functionsare cut off at a user-defined distance from the interpolationcenter, i.e. the so-called smoothing length, to save computa-tional effort by not taking into account the relatively minorcontributions from distant particles.

Third, when using a finite number of SPH particles for thediscretization of a spatial domain, the integral over the entiresupport area, which has been introduced in combination withthe Dirac delta function in the first step, has to be replacedwith a sum over all neighboring particles.

2.2 Discretized Euler equations

The basic equations which are used to describe the behaviorof a solid continuum domain are the Euler equations, i.e. theacceleration equation

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Comp. Part. Mech. (2014) 1:131–145 133

dv

dt= 1

ρ∇ · σ + g (1)

and the continuity equation

dρ

dt= −ρ∇ · v , (2)

where v represents the velocity vector, ρ the density, σ theCauchy stress tensor, and g the vector of volume forces perunit mass.

When applying the SPH discretization method to the Eulerequations, as it is shown in [12], the acceleration Eq. (1) canbe written as

dvi

dt=

∑

j

mj

(σ i

ρ2i

+ σ j

ρ2j

)· ∇iW (r i − r j, h)+ g (3)

and the continuity Eq. (2), in a similar manner, as

dρi

dt=

∑

j

mj(vi − vj

) · ∇iW (r i − r j, h), (4)

where m represents the mass, r i and r j denote the positionvectors of the particles i and j , and W (r i−r j, h) is the kernelfunction. The expanse of the support domain and, therefore,the number of neighboring particles involved in the smooth-ing process of particle i is determined by the choice of thesmoothing length h. In many cases, a Gaussian distributionis chosen as kernel function, cutting off the support radius ata distance of 2h, which means that only the particles that arelocated within this section of the kernel function of particlei are taken into account when evaluating (3) and (4).

In addition to the aforementioned Euler equations, anotherequation specifying the relation between stress tensor σ andthe remaining state variables, often referred to as materialstate equation, is required when simulating solid matter. Itconsists of a hydrostatic and a deviatoric part and can bewritten as

σ = −p E︸ ︷︷ ︸hydrostatic part

+ S︸︷︷︸deviatoric part

. (5)

In this equation, p denotes the hydrodynamic pressure, E theidentity matrix, and S the deviatoric stress tensor. The hydro-dynamic pressure can be calculated using the Mie-Grüneisenstate equation [13]

p = c2s (ρ − ρ0) , (6)

where pressure p is a function of sound velocity cs and initialmaterial densityρ0. When assuming, at first, an ideally elasticresponse of the solid material, the Jaumann rate of change ofthe deviatoric stress tensor S can be expressed as

dSdt

= 2 G

(ε − 1

3tr (ε) E

)+ S · ΩT + Ω · S. (7)

In (7), G represents the shear modulus of the consideredmaterial, ε the strain rate tensor given by

ε = 1

2

(∇v + (∇v)T

), (8)

and Ω is the rotation rate tensor defined through

Ω = 1

2

(∇v − (∇v)T

). (9)

The SPH material model described above can be extendedto plastic behavior based on the von Mises equivalent stressσvM. With the von Mises stress and the yield strength σy, aplastic character of the material is identified in the case thatthe von Mises yield criterion

fpl = σy

σvM< 1 (10)

is met. If fpl shows a value lower than 1, the stress statefound for the elastically calculated deviatoric stress tensorSel is located outside the von Mises yield surface aroundthe hydrostatic axis and, as a consequence, that part of thestress tensor σ being responsible for the plastification of thematerial, i.e. the deviatoric part S, is reduced according tothe relation

Spl = fpl Sel (11)

in order to bring it back onto the given yield surface.

2.3 Plasticity model

Plasticity models are used to specify the relationship betweenstress and strain within the plastic region for the simulatedmaterial. To achieve a more realistic representation of theplastic behavior of real matter in the simulations presentedin Sect. 4, the perfectly plastic formulation (yield strengthσy remains constant) is extended to the purely empirical JCflow stress model. In contrast to the perfectly plastic relation-ship between von Mises equivalent stress and correspondingstrain, the JC model also takes the interdependencies betweenthe stress-strain behavior and the material temperature as wellas the strain rate into account [14]. The JC plasticity modelfollows the equation

σy =[

A + Bεnpl

] [1 + C ln

(ε∗pl

)] [1 − T ∗m

], (12)

where εpl represents the equivalent plastic strain, εpl is theequivalent plastic strain rate, and A, B, C , n, and m arematerial dependent constants. Furthermore, the normalizedequivalent plastic strain rate and temperature used in (12) aredefined as

ε∗pl = εpl

εpl,0and T ∗ = T − Tambient

Tmelt − Tambient, (13)

where εpl,0 is the reference equivalent plastic strain rate,Tambient is the ambient temperature, and Tmelt is the

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134 Comp. Part. Mech. (2014) 1:131–145

melting temperature. For all implemented constitutive mod-els, the so-called Bauschinger effect [15], which describesthe phenomenon of a direction-dependent modification ofthe yield strength after previous plastic deformation, is takeninto consideration and the unloading behavior of the modeledmaterial is assumed to be elastic.

2.4 Damage model

As an extension to the plasticity model presented in (12),Johnson and Cook introduced a cumulative-damage frac-ture model [16], which is, just as the already mentionedflow model, not only primarily dependent on the actual plas-tic strain, but also on the strain rate and the temperature ofthe simulated material. To quantify the local damage state,a scalar parameter D is defined for each particle, using aninitial value of 0, which means that the represented domainof material is – still – completely intact. The parameter D isallowed to grow during both tensile and compressive loadinguntil a maximum value of 1 is reached. A value of 1 indicatesthat an SPH particle suffers from total damage and, thus, thatmaterial fracture occurs. The relationship used to describethe loading-dependent evolution of the damage parameter Dreads

D =∑ �εpl

εf, (14)

where �εpl represents the increment of equivalent plasticstrain occurring during one integration cycle and εf denotesthe equivalent strain to fracture relation given by

εf =[

D1 + D2 exp(D3σ

∗) ] [1 + D4 ln

(ε∗pl

)]

×[1 + D5 T ∗] , (15)

where the pressure-stress ratio σ ∗ is defined as the ratio ofthe average of the three normal stresses σm to the von Misesequivalent stress and D1 to D5 are material constants. In alast step, the original stress tensor σ , i.e. the one found in casethat the process of material damage is absent, is replaced bythe damaged stress tensor σD [17], which is defined through

σD = (1 − D) σ , (16)

in order to take into consideration the reduction of the stresslevel resulting from the actual incurred material damage.

2.5 Force model for boundary interaction

Boundaries and nonspherical objects in Pasimodo are rep-resented by triangular surface meshes. To prevent particlesfrom passing through a mesh geometry, a penalty approachis used. According to this, a penalty force is evaluated andapplied for each pair of particle and triangle. The numberof possible penalty functions is large and diversified, but we

settle on one introduced in [18], which is similar to a Lennard-Jones potential, but does not tend to infinity as the distancebetween the particle and the triangle goes to zero. It is definedthrough

F(d) ={ψ(R−d)4−(R−s)2(R−d)2

R2s(2R−s)if d ≤ R

0 otherwise, (17)

where d marks the distance between the particle and the tri-angle and R denotes the maximum distance up to which acontact occurs. The resulting force F points in the direc-tion of the triangle’s normal and changes from repulsion toattraction at distance s, while the scalar ψ determines thestrength of the force in such a way that ψ is its maximumabsolute value and, therefore, F(0) = ψ . Furthermore, thecontact model can include damping and viscosity based onthe difference in velocity.

3 Refinement and coarsening algorithm for SmoothedParticle Hydrodynamics

Many different technical processes of high industrial impor-tance are characterized by a major increase in the level ofplastic deformation, equivalent stress, or temperature thatare restricted to small areas of the processed solid material.This leads to fairly inhomogeneous distributions of the men-tioned state variables over the workpiece structure. There-fore, a finer discretization for these particular parts of theworkpiece model is needed in order to achieve the sameaccuracy of the numerical results as found for the areas thatare influenced only to a certain extent by the actual techni-cal process. But instead of increasing the discretization levelof the whole structure and with it the overall computationalcost, the adaptation shall be restricted to the parts that reallyhave need for it. By developing a combined local adaptiverefinement and coarsening algorithm, we found an SPH for-mulation that allows for such an adaptive increase in the dis-cretization level for specific areas of the model that satisfygiven refinement criteria. At the same time, the number ofSPH particles is decreased in domains where the criteria cho-sen for coarsening are fulfilled.

The adaptive refinement and coarsening approach pre-sented in this section consists of three steps. First, it is decidedif a particle needs to be adapted by checking whether a user-defined criterion is met or not. In the implementation, thiscan be a check of an absolute value, a gradient, and/or arelative geometric position. Second, if the criterion is met,the particle is split into several smaller ones or merged withsome of its neighbors into a bigger one according to a chosenrefinement or coarsening pattern. Third, the state variables ofthe new, adapted particles are calculated either directly fromthe corresponding values of the original particles or by inter-polating them from the values provided by their neighboring

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Comp. Part. Mech. (2014) 1:131–145 135

particles. In the following two subsections, the refinementand coarsening approaches are presented separately, startingwith the refinement algorithm.

3.1 Local adaptive refinement extension

The adaptive refinement approach presented in this sectionfollows the procedure introduced in [2] and [3]. When usingit, the first question to be answered is when to refine an SPHparticle at all. One or several suitable criteria need to be cho-sen for each simulation setup, e.g. single particle values, fieldgradients over the neighborhood, or numerical data. Whiledeveloping and testing the refinement algorithm, it turnedout that for lots of different simulation scenarios, alreadyquite simple refinement criteria that compare a single particlevalue, e.g. one of the particle’s spatial coordinates or its vonMises stress, to a given threshold work very well. Especiallywhen applying the introduced refinement approach to avoida numerical fracture, a criterion that is based on a particle’scoordination number, i.e. the overall number of neighborsthat are located within its smoothing area, proved to be quiteuseful in this context. As we allow for multiple refinement,it is necessary to restrict the particle size or the number oftimes a particle is split up in addition to the user-definedrefinement criteria. Otherwise, the number of particles couldincrease uncontrolled, leading to an enormous computationaleffort, or the refinement process could create particles withalmost zero masses and smoothing lengths. This results invarious numerical problems, such as very small time steps.

Having specified the refinement criteria to be considered,the next step is to define how an SPH particle is actuallyrefined. There are many ways to place the newly created par-ticles. For a two-dimensional simulation, a triangular and ahexagonal pattern as shown in Figs. 1 and 2 are proposedin [2]. At this point, we want to emphasize that, in contrastto [2], we chose the distance with which the new particlesare placed around the original one to be 0.5 ε instead of ε.Therefore, in this work, ε denotes the total dispersion of thepattern, not only its radius. As a cubical grid is often used asan initial setup for arranging the particles in SPH simulations,we implemented an additional refinement pattern retainingthe initial configuration. For this purpose, it splits the originalparticle, in case of a two-dimensional simulation, into foursmaller ones, which are placed on the corners of a cube witha length of the edge of ε as can be seen in Fig. 3. Note thatthe original particle located at the center is removed for thispattern. Correspondingly, the triangular and the hexagonalpattern are enhanced by the possibility to remove the cen-ter particle as well. For three-dimensional simulations, thecubical pattern is extended accordingly.

After having determined the positions of the additionalparticles, their masses have to be specified. It is the simplestway to distribute the mass of the original particle equally

Fig. 1 Two-dimensional triangular refinement pattern and relative par-ticle positions in terms of ε (red: original particle; blue: additional par-ticle). (Color figure online)

Fig. 2 Two-dimensional hexagonal refinement pattern and relativeparticle positions in terms of ε (red: original particle; blue: additionalparticle). (Color figure online)

Fig. 3 Two-dimensional cubical refinement pattern and relative parti-cle positions in terms of ε (red: original particle; blue: additional parti-cle). (Color figure online)

among the new ones. However, this choice pays no regard tothe original density field of the simulated domain of material.Therefore, the use of a mass distribution that introduces theleast disturbance in the smoothed density field is suggestedin [2]. It is defined as

E :=∫ ⎛

⎝miW (xi−x, hi)−∑

j

mjW (xj−x, hj)

⎞

⎠2

dx, (18)

where i denotes the original particle and j the new ones,with the corresponding positions xi and xj and smoothinglengths hi and hj, respectively. If the new particle masses arerewritten as mj = λjmi, with

∑j λj = 1 to give conservation

of mass, (18) yields

E = m2i

⎛

⎝∫

W 2(xi − x, hi) dx

− 2∑

j

λj

∫W (xi − x, hi)W (xj − x, hj) dx

+∑

j,k

λjλk

∫W (xj−x, hj)W (xk −x, hk)dx

⎞

⎠. (19)

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136 Comp. Part. Mech. (2014) 1:131–145

Equation (18) can also be expressed vectorially, i.e.

E = m2i

(C − 2λTb + λT Aλ

), (20)

with

C =∫

W 2(xi − x, hi) dx, (21)

bs =∫

W (xi − x, hi)W (xs − x, hs) dx, and (22)

Ars =∫

W (xr − x, hr)W (xs − x, hs) dx. (23)

With the choice hj = αhi for all particles j , (20) can berewritten as

E = m2i

hqi

(C − 2λT b + λT Aλ

), (24)

with the dimension q and

C =∫

W 2(xi − x, 1) dx, (25)

bs =∫

W (xi − x, 1)W (xs − x, α) dx, and (26)

Ars = 1

αq

∫W (xr − x, α)W (xs − x, α) dx. (27)

For further details, see [2]. With∑

j λj = 1, the optimizationproblem reads

minλ

(C − 2λT b + λT Aλ

). (28)

In general, a solution exists as A is symmetric and positivedefinite, but this solution may also result in unwanted andunphysical negative particle masses. The solution depends onthe choices for the dispersion ε, the smoothing length scaleα, as well as the kernel function. This approach makes thesolution independent from the initial particle distance, mass,and smoothing length. Therefore, it has to be evaluated onlyonce while using the unit mass and length at the beginningof the simulation, which takes little additional computationtime. During the simulation, the pattern can be adapted byscaling it with the actual values of the state variables of theoriginal particle.

Up to now, the positions, masses, and smoothing lengthsof the new particles have been specified. The other properties,such as the densities and the velocities, need to be set, too.While most state variables, as for example the densities, canbe calculated via SPH interpolation from the ones of theoriginal particles without experiencing any issues concerninga further inaccuracy of the simulation results, this is not thecase for the velocities of the additional particles. When theyare set to the ones of the original particles, a conservationof linear and angular momentum as well as kinetic energy is

achieved. Choosing vj = vi for all particles j is the only wayto ensure an exact conservation, which is a must when lookingfor reliable numerical results. Conversely, due to the fact thatthe smaller particles are positioned somewhere around thelocation of the original one without modifying their velocityvalues accordingly when following this approach, this choicemay modify the existing velocity field and, thus, also lead toadditional, unwanted inaccuracies in the simulation results.This dilemma needs to be further investigated in the future.

3.2 Local adaptive coarsening extension

The coarsening approach presented in this paper basicallyfollows the same procedure as the adaptive refinement algo-rithm described above. First, those areas that are of littleinterest are to be identified. For this purpose, coarsening cri-teria need to be defined. These criteria can be chosen similarto the refinement routine, e.g. based on a particle’s velocityor its von Mises stress.

Having marked the particles to be coarsened, the next stepis to find a proper selection of adjacent particles for the coars-ening process. To that end, we added two different selectionstrategies to our SPH code; a simple neighborhood search andan area-based procedure. In case of the neighborhood searchstrategy, it is sufficient to loop over all chosen particles andmerge them with their nearest neighbors. This is an exten-sion to the algorithm introduced in [1], which merges the twonearest neighbors, i.e., only pairs of particles are merged eachtime step. The second approach is a more sophisticated one.For this, a rectangular box is placed around each particle thathas been marked for coarsening. Then, all the boxes over-lapping each other are collected, resulting in a finite numberof linked regions. For each of these regions, an enclosingcuboid is formed and, subsequently, divided into

number of subregions

≈ number of particles in the cuboid

N(29)

smaller ones, where N denotes the desired number of parti-cles to be merged. For the process of dividing the cuboid intosubregions, two different approaches can be employed. Onthe one hand, there is the possibility for a dynamic division,which splits each region into several subregions containingthe same number of SPH particles. On the other hand, a sta-tic division, which divides a region into equally sized sub-regions, can be used. Finally, the particles that belong to thesame subregion are merged.

Originally, the aforementioned procedures were devel-oped for the simulation of fluids. When considering SPHsimulations that have their focus on solid material, the parti-cles often experience a lot less relative motion and reorderingthan in case of fluid simulations. Inspired by the approach

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Comp. Part. Mech. (2014) 1:131–145 137

presented in [19], which keeps the original particles insteadof removing them, we implemented a family-based coarsen-ing strategy, which aims on merging particles that originatefrom the same particle if possible. For this purpose, eachrefined particle is initialized with a variable providing the IDof its origin particle. By adding an additional merging condi-tion that ensures that SPH particles with the same origin arepreferred for merging, one gets a suitable extension to theneighbor-based strategy that takes into account the refine-ment history of an adaptive particle.

After a set of particles to be merged has been found byany of the approaches described above, the properties of thenew, coarsened particles have to be calculated. In order tofulfill the conservation of mass, the mass mj of the coarsenedparticle j needs to be the sum of the merged particles’ massesmi, i.e.

mj =∑

imi. (30)

Furthermore, we ensure a conservation of momentum bypositioning the new particle at the merged particles’ cen-ter of mass and setting its velocity vj to the weighted averageof the particle velocities vi. This yields the two convex com-binations

r j = 1

mj

∑

i

mir i and vj = 1

mj

∑

i

mivi. (31)

Except for the smoothing length hj, all other particle prop-erties that still have to be determined are calculated via SPHinterpolation. The new smoothing length is calculated byreducing the density error in the coarsened particle’s posi-tion, i.e. by searching the root of

mjW (r j − r j, hj)−∑

i

miW (r j − r i, hi). (32)

In order to prevent particles from being coarsened shortlyafter having been refined, for each adaptive SPH particle acounter variable is added. It provides the number of timesteps since the particle’s last refinement. With this counter,it is possible to avoid an alternating coarsening and refine-ment process for an adaptive particle. Besides this counter-based restriction to the coarsening algorithm, a second con-trol mechanism is employed. In order to avoid introducingtoo much inaccuracy when merging particles, we check theerror in the angular momentum against a predefined tolerancelimit for each coarsening process, i.e.

‖∑

i

mi(r i × vi)− mj(r j × vj) ‖2< tolerance limit. (33)

If the condition is unmet, the particles i are not merged inthis step.

4 Examples

Having discussed the theoretical basis of the developed com-bined local adaptive refinement and coarsening algorithm inthe previous section, the focus is laid on its applicabilityto different types of simulation scenarios focusing on solidmatter as well as on the benefit taken from the adaptive for-mulation in the following. First, the SPH simulation of anordinary tensile test setup is presented. In the context of thisquite simple simulation setup, the capability of the intro-duced adaptive discretization technique to avoid a numericalfracture and to ensure a conservation of linear and angularmomentum as well as kinetic energy is illustrated. Second,emphasis is placed on the deviations in the simulation resultsfound for an SPH notched tensile specimen when employ-ing the different refinement patterns introduced in Sect. 3.1.Next, a fully three-dimensional simulation of a rigid torusfalling onto an elastic membrane demonstrates that the adap-tive routine presented in this paper is also suitable to SPHsimulations showing a highly dynamic behavior of the mod-eled structures. As a final simulation setup, which is morerelated to real technical applications than the aforementionedscenarios, the simulation of an orthogonal cutting process ispresented. Unless otherwise stated, the set of parameter val-ues for the combined refinement and coarsening algorithmgiven in Table 1 is used for the SPH solid simulation scenar-ios discussed hereinafter.

4.1 Ordinary tensile test

The uniaxial tensile test is one of the most basic experimentalsetups that are employed to investigate the elastic, the plastic,and the fracture behavior of real solid material and is, for this

Table 1 Parameter values for the combined refinement and coarseningalgorithm

Parameter value

Allowed number of refinement steps per particle 1

Refinement pattern (two- and three-dimensional) Cubical

Dispersion of refinement pattern ε 0.5

Smoothing length multiplication factor α 0.85

Strategy for velocity calculation vj = vi

Allowed number of coarsening steps per particle 1

Coarsening pattern (two- and three-dimensional) Family-based

Desired number of particles to be merged N(two- / three-dimensional)

4 / 8

Tolerance limit for Newton-Raphson method 1 · 10−6

Maximum number of steps for Newton-Raphsonmethod

20

Minimum number of time steps since lastrefinement

5

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138 Comp. Part. Mech. (2014) 1:131–145

Fig. 4 Simulation resultsobtained for the ordinary tensilespecimen. a Stress-strain curvesobtained when employing theoriginal and the adaptive SPHformulation for solids,b Refined SPH model of thetensile specimen showing theparticle’s masses (red: highmass; blue: low mass). (Colorfigure online)

0 0.03 0.06 0.09 0.12 0.150

2

4

6

8x 10

8

strain [−]st

ress

[N

/m2 ]

original SPHadaptive SPH

(a)

(b)

reason, also suitable for a comparison with simulation data.It is the first step in our working process of transferring anordinary flat tensile specimen used in experiments into a sim-ulation model to build up a regular lattice of SPH particlesthat has the same shape and dimensions as the gauge sectionof the real specimen. For building up this lattice, the particlesare positioned according to a structure consisting of quadraticcells. In addition to that, to both ends of the simulated speci-men whose normal vector is parallel to the loading directionadditional rows of SPH particles, which represent the gripsapplied to the real one, are attached. The particles belongingto one of these grip areas are fixed in space, while a constantvelocity is applied to the particles of the other one. Thus,one finds an invariable elongation rate level throughout theentire simulation. The SPH particles that are located withinthe measuring area are not at rest at the beginning of thesimulation, but to them an initial velocity according to theirrelative distance from the grip sections is assigned. That way,a linear velocity field is established in order to avoid havingshock waves running from one end of the specimen to theother resulting in a noisy stress-strain-curve.

When applying the SPH simulation technique in its orig-inal formulation, i.e. without any adaptive extension, to thetensile test simulation or, speaking more generally, to prob-lems showing large deformations of the modeled structure,a possible increase in the distance between adjacent par-ticles can lead to a decrease in the number of neighborsthat are located within the smoothing areas. Sooner or later,this results in a numerically induced fracture. To ensure thatthe damage behavior of the simulated material is controlledexclusively by the fracture model introduced in Sect. 2.4,even in case of large strain values, the presented refinementalgorithm can be used to avoid a numerical fracture by adapt-ing the model discretization in accordance to its deformation.

The relationships between stress and strain found whenemploying both the original and the adaptive SPH formu-lation on the simulation of a flat tensile specimen madeof C45E while using approximately 2 500 particles for thediscretization of its gauge section with an initial length ofl0 = 5 · 10−2 m and an initial width of b = 1.25 · 10−2 m

are shown in Fig. 4a. As can be clearly seen, both the stress-strain curve found for the SPH simulation with a constantnumber of particles and the one obtained with the adaptivemodel take the same elastic and plastic path up to the pointwhen the tensile specimen simulated with the unmodifiedSPH method shows a loss of influence between neighboringparticles and, thus, a numerical fracture can be observed. Atthis point, the SPH particles that are located within the areashowing the largest value of strain have too few neighbors inthe stretching direction and, as a consequence of this, becomedecoupled from each other. Running the identical simulationsetup as done in case of the standard SPH method with theextended one leads to a fairly different, but expected behavior.With an increase in the elongation of the tensile specimen, thedistance between neighboring particles is increased, too. Inthe case that the coordination number of a particle decreases,the discretization level of this particular part of the specimenis adapted accordingly to stop the formation of a numericalfracture already in its early state. As a result, the number ofneighboring particles that are located within the smoothingareas of the refined ones stays above the critical value untilthe end of the simulation and the stress-strain curve of theadaptive SPH simulation follows the perfectly plastic pathup to a strain value of ε = 0.15. As expected from a theoret-ical point of view, a decrease in the coordination number isobserved first for the SPH particles that form the neck of thetensile specimen. According to that, these are also the parti-cles that first experience an adaptation as shown in Fig. 4b.

When using the introduced refinement algorithm to elim-inate the effects leading to a numerical fracture, one has toensure that the linear and angular momentum as well as thekinetic energy is conserved in order to keep the reliabilityof the obtained numerical results. For this purpose, a secondadaptive scenario is set up. It is quite similar to the one ofthe ordinary tensile specimen presented above and shown inFig. 5a. But instead of employing a refinement criterion thatdepends on a particle’s coordination number, two position-based criteria are introduced this time; one used for the refine-ment and the other one used for the coarsening algorithm.Now, the entire tensile specimen is moved in such a way that

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(a)

0 0.2 0.4 0.6 0.8 10

2

4

6

8

x 10−5

normalized simulation time [−]

kine

tic e

nerg

y [J

]

0.01250.02500.0375

(b)

Fig. 5 Modified tensile test simulation setup used to assess the con-servation properties of the presented local adaptive discretization algo-rithm. a Modified SPH tensile specimen showing the particle’s longitu-dinal velocities (red: high longitudinal velocity; blue: low longitudinal

velocity), b Kinetic energy of the original SPH particles, the refinedones, and the coarsened particles for three different relative longitudi-nal positions. (Color figure online)

Fig. 6 Simulation resultsobtained for the modified tensiletest setup when using differentcoarsening patterns (red: highmass; blue: low mass). aNeighbor-based coarseningpattern, b Family-basedcoarsening pattern. (Color figureonline)

(a) (b)

it first passes the position of the refinement criterion, repre-sented by the left of the two gray planes depicted in Fig. 5a,and, after a while, runs through the one of the coarsening cri-terion. Thus, the SPH particles of the adaptive model are firstrefined based on the chosen cubical refinement pattern and,afterwards, coarsened according to the introduced family-based pattern. To make it easier to distinguish between theparticles of different size, the initial overall number of SPHparticles is decreased to approximately 500, of which 100belong to the two grip areas and are not to be refined orcoarsened, and the graphical radius of the coarsened parti-cles is decreased, too. In Fig. 5b, the amount of kinetic energythat is stored in three of the original SPH particles at differentlongitudinal positions of the adaptive tensile specimen, thevalues of accumulated energy for the refined particles intowhich the original ones have been split up, and the kineticenergy values for the merged particles is shown. As can beseen, both the refinement and the coarsening process do notlead to a change in the kinetic energy level when cloning thevelocities of the new SPH particles from the one of the pre-vious particles. Therefore, also the overall amount of kineticenergy being stored in the adaptive tensile model is conservedthroughout the entire simulation.

In Sect. 3.2, both the neighbor-based coarsening patternand its extension to a family-based one have been introduced.Unlike the basic version, the latter one takes into account thehierarchic structure as well as the refinement and coarsen-ing history of an SPH particle for a possible further process

of adapting its discretization level. This modified coarsen-ing pattern is especially useful when simulating solids withthe presented adaptive SPH formulation. While the basicneighbor-based coarsening strategy includes a unification ofrandomly picked neighboring particles, the family-based onetries to merge SPH particles that originate from the same par-ticle. In the case that the simulated solid bodies experience noor only small changes in their topologies while being refinedand, afterwards, coarsened again, the neighbor-based patterndoes not ensure that the original particle distribution over thediscretized structure is restored. This can lead to a significantdegradation in the result quality due to a nonoptimal particledistribution as shown in Fig. 6a, especially in the areas of themodel that are located close to its outer surface. In contrast tothis, the family-based strategy helps to reconstruct the orig-inal discretization of the simulation model, see Fig. 6b, and,therefore, to achieve a quality of the obtained results that isas good as the one of the original distribution.

4.2 Notched tensile test

A more challenging evaluation setup for the introduced adap-tive SPH formulation than the ordinary tensile specimen con-sidered above is the simulation of a notched specimen dueto the fairly inhomogeneous stress distribution found in thiscase. In analogy to the tensile specimen regarded in the pre-vious section, a regular lattice with quadratic cells of SPHparticles serves as a basis for the two-dimensional model of

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Fig. 7 Distribution of the vonMises stress found for the modelof a double-edge notched tensilespecimen when using differentsimulation methods (red: highstress; blue: low stress). aAdaptive SPH, b SPH, c FEM, dDEM [21]. (Color figure online)

(a) (b) (c) (d)

Fig. 8 Different stages of therefinement of the adaptive SPHnotched tensile specimen (red:high stress; blue: low stress).(Color figure online)

a flat double-edge notched tensile specimen with an initialwidth of 1 · 10−2 m and an initial length of 1.34 · 10−2 m.Halfway up the length of the created geometry round notcheswith a radius r = 1.5 · 10−3 m and an offset of 2 · 10−4 mfrom the outer edge are introduced on both long sides of thespecimen. To achieve a satisfactory level of reproduction ofthe high gradients in stress that can be observed next to theroots of the notches, an initial particle spacing of 2.3 ·10−3 mis used for the discretization of the adaptive SPH simula-tion, leading to an overall number of particles of about 2 700at the beginning of the simulation. The cubical refinementpattern as depicted in Fig. 3 with ε = 0.5 and α = 0.85is employed and the chosen refinement criterion is stress-based. For comparison, an unrefined SPH simulation with aninitial particle spacing of 1.15 · 10−3 m is performed, lead-ing to an overall number of particles of about 10 000. Asit is hardly possible to determine the stress distribution thatcan be found for a notched tensile specimen in an analyticalway, two simulation-based references are used for compar-ing the results computed by the SPH method; a simulationof a notched tensile test performed by the Finite Elementprogram ADINA and one employing the Discrete ElementMethod (DEM) [20].

As can be clearly seen when analyzing the distribution ofthe von Mises equivalent stress given in Fig. 7, the resultsobtained when employing the four different simulation tech-niques on the notched tensile scenario are in good agreement.Thus, the adaptive SPH method is able to reproduce the stress

distribution observed for a real notched specimen. Accord-ing to that, the maximum value of the von Mises stress canbe found right at the roots of the two notches, whereas thelevel of equivalent stress decreases when following the pathsfrom the roots to the centerline of the tensile specimen. Fur-thermore, the von Mises stress attains lower and lower valuesalong the notch radii, starting with a high stress level at theroots and showing stress-free areas next to the outer surfaceof the specimen. When examining the three reference solu-tions, an elliptic section of the model that is characterized bya high magnitude of equivalent stress and is aligned with itsminor axis along the ligament line is also obvious in case ofthe adaptive SPH notched tensile test.

To give an impression how the adaptively refined areaevolves with increasing simulation time, Fig. 8 shows threedifferent stages of the refinement process found for the SPHdouble-edge notched tensile specimen. As one can see, theareas of highest stress are also the areas where the simula-tion model first experiences an increase in the discretizationlevel. Accordingly, the refinement process is initiated right atthe roots of the two notches and, subsequently, spreads intodeeper particle layers along the elliptic section of the speci-men that is characterized by a high magnitude of von Misesequivalent stress. In contrast to this, small parts of the notchedtensile specimen, which are located close to the outer surfaceof the particle model, remain unchanged due to the fact thatthe stress level stays below the specified refinement thresholdvalue for these regions. Following this procedure, the adap-

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Fig. 9 Combinations of theparameters ε and α leading topositive and negative massvalues when using differentrefinement patterns (white:negative mass value; blue:positive mass value).a Hexagonal pattern with centerparticle and unequal masses,b Triangular pattern with centerparticle and unequal masses.(Color figure online)

0.2 0.4 0.6 0.8 1 1.2 1.4

0.2

0.4

0.6

0.8

dispersion of pattern ε [−]sm

ooth

ing

leng

th f

acto

r α

[−]

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0.2 0.4 0.6 0.8 1 1.2 1.4

0.2

0.4

0.6

0.8

dispersion of pattern ε [−]

smoo

thin

g le

ngth

fac

tor

α[ −

]

(b)

Fig. 10 Comparison of thetwo-dimensional cubical andhexagonal refinement pattern atearly stage (red: high stress;blue: low stress). a Cubicalpattern, b Hexagonal patternwithout center particle, cHexagonal pattern with centerparticle and unequal masses.(Color figure online)

(a) (b) (c)

tive SPH method provides the same simulation results asfound for the original, unmodified formulation, although theoverall number of SPH particles and, thus, the necessary com-putational effort is much lower for the adaptive simulation.

For a further validation of the adaptive algorithm, theresults found for the three different refinement patterns intro-duced in Sect. 3.1, i.e. the cubical one with ε = 0.5, thehexagonal one with ε = 2/3, and the triangular one withε = 2/3, are compared. For each of these refinement patterns,the value of the dispersion parameter ε is chosen to give anequidistant particle spacing in the refined area between theroots of the two notches. In case of the hexagonal and thetriangular pattern with an additional center particle and anunequal distribution of mass along the particles of the samerefinement cell, the smoothing length multiplication factor αhas to be increased. Otherwise, negative mass values wouldbe used in the optimization process. This is illustrated inFig. 9, which provides the pairs of values of the parameters εand α for which positive particle masses are found. Since thecritical value of the smoothing length multiplication factor αis roughly 0.945 for both patterns when setting ε = 2/3, wechoose α = 0.95 to ensure that the masses are positive.

Results from the simulations with the two-dimensionalcubical and hexagonal pattern at an early stage are shown in

Fig. 10. As can be clearly seen, the hexagonal pattern leads ingeneral to a similar behavior of the notched tensile model asseen before, with the highest stress located next to the rootsof the notches and the formation of an elliptic section of highequivalent stress. There are only small differences betweenthe cubical pattern, which is shown in Fig. 10a, and the hexag-onal one without a centrally located particle, which is shownin Fig. 10b. For instance, the edge areas of the refined part ofthe adaptive model found for the cubical pattern seems to be abit more inhomogeneous, i.e., there are some more unrefinedparticles that are located between the already refined ones inthis case. In contrast to these minor deviations in the simula-tion results, the hexagonal pattern with a center particle andan unequally distributed mass leads to a stress distributionthat is even more concentrated in the notch-near areas as wellas the elliptic section of the specimen that is characterized bya high magnitude of von Mises equivalent stress as shown inFig. 10c. Thus, the simulation employing the hexagonal pat-tern with center particles and an unequal distribution of massshows for the same point in the simulation time a refinedarea that is smaller and fits more tightly around the ellipticsection of high stress than the ones found for the other adap-tive models. According to that, the expansion process of therefined area of the hexagonal pattern placing an additionalparticle in the center of a refinement cell lags behind the one

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Fig. 11 Comparison of thetwo-dimensional cubical andtriangular refinement pattern atlate stage (red: high stress; blue:low stress). a Cubical pattern,b Triangular pattern withoutcenter particle, c Triangularpattern with center particle andunequal masses. (Color figureonline)

(a) (b) (c)

(a) (b) (c)

Fig. 12 Simulation results obtained for the rigid torus falling onto anelastic membrane. a Side view of the simulation setup showing the torusand the membrane (red: fixed particles; blue: free particles), b Top viewof the refined membrane model showing the particle’s masses (red: high

mass; blue: low mass), c Top view of the coarsened membrane modelshowing the particle’s masses (red: high mass; blue: low mass). (Colorfigure online)

of the cubical and the one of the hexagonal pattern withouta central particle.

All the differences between the hexagonal pattern withand without a centrally located particle described above canalso be found for the triangular refinement pattern. In addi-tion to that, also the orientation of the introduced triangularparticle cells has an influence on the obtained solution. Ascan be seen in Fig. 11, the triangular pattern gives a betterapproximation of the borderline between the refined and theunrefined area, but, at the same time, the refined model nolonger shows a perfect symmetry as can be seen when com-paring the discretization of the two notch radii depicted inFig. 11c. For this reason and due to the fact that the cubi-cal refinement pattern provides the same result quality asfound for the hexagonal one while introducing a much lowernumber of SPH particles during the refinement process, thispattern retaining the initial particle configuration is also usedfor the simulations presented below.

4.3 Torus falling onto a membrane

The adaptive SPH formulation proposed in this paper is alsoapplicable to fully three-dimensional simulations that are

characterized by a highly dynamic behavior of the mod-eled solid structures. This is demonstrated in the follow-ing by presenting the results found for the simulation of arigid torus falling onto an elastic membrane. While for therepresentation of the torus a triangular surface mesh alongwith the force model for boundary interactions introduced inSect. 2.5 is deployed, the membrane is discretized using theSPH method. As shown in Fig. 12a, the elastic body consistsof three layers of adaptive particles. The overall number ofparticles at the beginning of the simulation is about 5 300,of which the ones that are located close to the sides of themodel are fixed in space.

In the initial configuration of the simulation setup, thetorus is placed at some distance above the membrane. Afterhaving released the torus from its starting position, the grav-itational force acting on it leads to a free fall motion of therigid body until it hits the elastic membrane. With this con-tact, the process of transferring kinetic energy from the torusto the membrane begins. Subsequently, the rigid body is grad-ually decelerated until its vertical velocity completely van-ishes. Due to the interparticle bonds, the amount of kineticenergy absorbed by the adaptive SPH particles is step-by-step transformed into elastic deformation energy. For this

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reason, the stress level of the particles is increased and theyare refined once the employed stress-based refinement crite-rion is met. As can be seen in Fig. 12b, this first happens forthe particles being directly in contact with the meshed bodyand, as a consequence, the adaptive discretization algorithmprojects the geometry of the torus (dashed lines) onto theuppermost layer of SPH particles, i.e. the surface of the elasticmembrane.

In the second part of the simulation, some of the energybeing stored in the elastic membrane is retransferred to thetorus as part of the rebound effect. Therefore, the torus doesnot stay at rest, but is accelerated in vertical direction andbegins an upward motion. When the torus and the surfacelayer of the membrane are no longer in contact, the stress levelof the uppermost membrane particles falls below the spec-ified coarsening threshold value and, as shown in Fig. 12c,the previously refined particles are coarsened again.

4.4 Orthogonal cutting process

A simulation scenario that is more related to real technicalapplications than the previous examples is the orthogonalcutting process. The orthogonal cutting setup represents anelementary cutting process where the motion of the tool isperpendicular to the cutting edge. When assuming a suffi-ciently large width of cut compared to the chosen depth ofcut, a two-dimensional state of stress can be assumed in theworkpiece and the machining process becomes a planar prob-lem [22].

As part of the evaluation process of the introduced com-bined local adaptive refinement and coarsening algorithm,a model of an orthogonal cutting specimen is created whileusing, as already done in case of the previously discussedvalidation scenarios, a regular lattice with quadratic cells forthe arrangement of the SPH particles and an initial smooth-ing length of the particles of h = 5 · 10−4 m. To preventthe workpiece model from following an unconstrained trans-lational motion, five additional rows of particles, which arefixed in space, are attached to the bottom of the specimen.The material used for the cutting simulation is, again, steelC45E, which is chosen because of its wide application inmechanical engineering. The corresponding material para-meters found for this type of steel at room temperature aregiven in Table 2.

Besides the workpiece material, the geometry of the cut-ting tool can have significant influence on the behavior of thecutting system and, therefore, the obtained results. Due tomanufacturing reasons, the cutting edge of a real tool revealsdeviations in its geometry from an ideal wedge shape [23]. Asa consequence, not a tool with an ideal geometry, i.e. a wedgeshape, but a more sophisticated one taking into account theimperfections resulting from limitations of the manufactur-ing process is employed for the presented machining sim-

Table 2 Material properties for steel C45E at room temperature

Parameter Value

Density ρ 7 700 kg/m3

Young’s modulus E 2.1 · 1011 N/m2

Poisson ratio ν 0.3

Yield strength σy 7.11 · 108 N/m2

Fracture strain εF 0.133

ulation. One model that promises a sufficient imitation ofa real tool is introduced in [24]. For the calculation of theparticle forces resulting from the interaction with the cuttingtool, once more the penalty approach presented in Sect. 2.5is deployed. Further information on the presented orthog-onal cutting setup can be found in [25]. The behavior ofthe cutting model that can be observed when using both theoriginal as well as the adaptive SPH formulation is discussedhereinafter.

When the cutting of material takes place, the distribu-tion of the von Mises stress that is shown in Fig. 13a, withred meaning high stress and blue indicating a low level ofvon Mises equivalent stress, can be found for the originalSPH model. The highest level of stress appears next to thetip of the tool where the separation of material happens.This zone showing a high magnitude of equivalent stressis surrounded by a zone of mid-level stress, which is colorcoded in yellow and green. Both the particle layers that arelocated at some distance from the tool tip as well as the partsof the material that form the chip show low values of vonMises stress, as indicated by the blue color of these SPHparticles. In similarity to experimental results, a continu-ous chip is separated from the remaining workpiece materialin the cutting simulation. In this context, it is important toremember that the particles that are shown in Fig. 13a onlyillustrate the interpolation points where the equations of theunderlying continuum model are evaluated, not domains ofmaterial.

If the introduced adaptive discretization technique isapplied to the orthogonal cutting simulation, one observes arefinement of the load-carrying particles in the different shearzones of the cutting specimen as it is shown in Fig. 13b. Whilethe blue-colored particles still have the initial size, the red-colored ones have been refined in order to be able to resolvethe high gradients in stress found for these regions. Again,a stress-based refinement criterion together with the cubicalrefinement pattern introduced in Sect. 3.1 is used. For theareas of the workpiece model that have already been passedby the tool geometry, the previously refined SPH particles areadaptively coarsened, as indicated by their red color, whenshowing a stress level that is at least ten times lower than thespecified refinement threshold value.

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(a)

(b)

Fig. 13 Simulation results obtained for the orthogonal cutting speci-men when using both the original and the adaptive SPH formulation.a Distribution of the von Mises stress found for the original SPH model(red: high stress; blue: low stress), b Adaptive SPH model showing theparticle’s masses (red: high mass; blue: low mass). (Color figure online)

5 Conclusion

In this paper, a local adaptive discretization algorithm forthe SPH method has been proposed. With the presentedapproach, it is possible to improve the accuracy of the simu-lation results while reducing the required computational costat the same time. To that end, the number of SPH particles isadaptively increased in certain areas of a simulation model ifnecessary, whereas it is decreased for domains of low inter-est. In a first step towards such a combined refinement andcoarsening strategy, an enhanced version of the refinementprocedure has been integrated into the SPH solid formulationthat had been extended to the JC plasticity model, the JC dam-age model, as well as an appropriate boundary force model.In a second step, we developed a coarsening algorithm that iscompatible with the refinement routine implemented before.Both adaptive components follow the same basic procedure,

which consists of the following three steps. First, it is decidedif the discretization level of an already existing SPH particleis to be adapted by checking whether a user-defined criterionis met or not. Second, if the criterion is met, the particle issplit into several smaller ones or, respectively, merged withsome of its neighbors into a bigger one according to a chosenrefinement or coarsening pattern. Third, the state variables ofthe additional particles are calculated either by interpolationor by cloning and subsequent scaling the ones of the originalparticle. In a last step, the validity of the coupled approach,as well as its applicability, was shown by applying it to abroad range of different simulation setups.

Four of the various simulation scenarios used for the val-idation of the implemented adaptive discretization routinewere then presented. In the context of an ordinary tensile testsetup, the capability of the introduced adaptive discretizationtechnique to avoid a numerical fracture as well as to ensurea conservation of linear and angular momentum as well askinetic energy was demonstrated. Next, it was shown for anotched tensile test setup that the combined refinement andcoarsening strategy provides the same results as found withthe FEM or a lot more computational expensive SPH sim-ulations with a fixed number of particles. The fact that theresults quality of the adaptive SPH formulation depends onthe choice which of the introduced refinement and coarseningpatterns are employed was discussed. Besides these two sim-ulation scenarios used to illustrate the basic qualities of theintroduced local adaptive discretization algorithm, it is alsoapplicable to SPH simulations showing a highly dynamicbehavior of the modeled structures and is suitable to solveproblems related to real technical applications. This has beenshown in the context of a simulation of a rigid torus fallingonto an elastic membrane and one of an orthogonal cuttingprocess.

Acknowledgments The research leading to the presented results hasreceived funding from the German Research Foundation (DFG) underthe Priority Program SPP 1480 grant EB 195/12-2. This financial sup-port is highly appreciated.

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