Shephard Short Course

8/8/2019 Shephard Short Course

http://slidepdf.com/reader/full/shephard-short-course 1/47

1

Mark S. ShephardScientific Computation Research Center, Rensselaer Polytechnic Institute

Contributions from:

N. Chevaugeon, D. Datta, J.E. Flaherty, X. Li, X. Luo, J. Wan; Rensselaer

J.F. Remacle; Université Catholique de Louvain

M.W. Beall, R.M. O’bara, J. Walsh and B.E. Webster; Simmetrix Inc.

T.P. Gielda; Visteon

Outline" Reliable automatic mesh

generation from CAD

" Mesh control for adaptiveanalysis

" Automated adaptive analysisin simulation-based design

Automated Simulation in Engineering Design

2

Domain Definition" Geometric Models

# CAD systems

" Image data# Use representation consistent with

image data (e.g., marching cubes)

# Convert the image data to solid model

" Discrete (Mesh) Models# Infer model from mesh

Attributes - Information past domain" Generalized structures,

operators and dependencies

" Spatial and temporal variations

Fields" Distribution of tensor fields

over discrete models

Geometry-Based Problem Definition

CAD

medicalimage

data



3

CAD System Information

Most models stored as feature data plus resulting boundary representationCAD system boundary representation" Topology - entities defining the boundary and their adjacencies" Shape parameters" Tolerances" Methods for evaluating tolerances & validity

Often based on modeling kernels (ACIS, Granite, Parasolid…)

4

Attributes

Attributes are everything other than geometry needed to fullydefine the problem" problem definition attributes (describe the problem to be solved)

material properties, loads, b.c.’s

Attributes are applied togeometric model only

Attributes may be generalfunctional distributions,functions of: space, time,

fields, modeling operations



5

Fields

Tensor quantities defined in terms of the two level discretization A field describes tensor over entities in a geometric model" Field defined in terms of

# interpolations over a discrete model entities

# dof associated with those interpolants

" Fields can vary in both time and space

Operations defined for the case dofs being known or unknown" If known: evaluation results in a tensor value for the field at that point

" If unknown: evaluation results in coefficient matrix related to the dofs -supports element level calculations to be independent of interpolation

field

6

CAD System Boundary Representations

Often uses relatively large and variable tolerances" eliminates small features" makes algorithms more robust

Tolerances & methods for evaluating tolerances" The methods used in the CAD system modeling

engines are written to deal with tolerances ina consistent manner

" The methods used in the modeling engines are notavailable outside of the modeling engines, therefore,any form of translation introduces “dirty geometry”

" Tolerances and tangencies

line beingintersected

geometrictolerance

geometrictolerance

line beingintersected

Issues of Tangency

a hard case a harder case



7

CAD System Boundary Representations

Models well understood with in the CAD system" Model fits together to within tolerances understood by the system" Geometry is not “dirty”

Differences between modeling sources" Each CAD system modeling engine uses different representations of

geometric entities - most use similar queries for basic interrogation" Each CAD system modeling engine uses different representations of

topology data structure" Modeling kernels (ACIS, Granite, Parasolid) help to reduce the scope of

this problem

8

Techniques for CAD Boundary Representation Access

Translation & Healing" Translation may use direct translators or standards." IGES does not address issues with representations, global tolerances, features,

tolerancing or tolerance methods and typically results in “dirty” geometry" Standards such as VDAFS and STEP

# address representations and global tolerances# do not address tolerancing or tolerance methods# often results in “dirty” geometry (typically cleaner than IGES).

" Companies have invested millions in healing technologies# ITI, Elysium, Spatial,TransMagic,CAD-CAMe,TTI,TTF, …# Progress has been made but process is still not reliable or robust.# Fundamental issue of tolerance methods has not been addressed

" “Dirty” geometry due to differences representations, tolerances and methods" Defeaturing is difficult since feature data and information is lost in translation

# Feature-based translators attempt to reproduce models from featurerepresentations but do not address tolerance methods

# Feature suppression with non feature-based translators requires featurerecognition algorithms

" Robustness varies dramatically with thecombinations of tools selected



9


Discrete Representations" Facet representation from the CAD system

# Designed for visualization and may not close

# Often done on a surface by surface basisand may not match across surfaceboundaries

# Dependent on CAD system facetter" Feature, shape and topology information is lost

# “Small” features typically represented with“small” facets

# Can not recover real domain shapeinformation

" Still can have “dirty” geometry that is not closed

10


Direct Functional Access" Directly use the functionality of the CAD systems to determine all

geometry-based information

" Available through CAD system toolkits# CATIA CAA

# Open IDEAS

# Pro/Toolkit

# UG Open

" Also available through modeling kernels# ACIS

# Granite

# Parasolid

" Uses native system tolerances & methods# Model fits together and is understood

# Lack of uniformity in topological representations - appears to force systemdependent interfaces

" Interrogations easily supported# Can support meshing and other simulation functions

# Does require use of existing model with no simplifications



11


Direct functional access only viable means" Highly robust and no ambiguity" Only method that properly support all needs

Need to address deficiencies" Lack of uniformity of topological models" Support of various features to account for geometric simplification, attribute

geometry, linkage with feature models, etc.

Direct Geometry Access with Unified Topology Model" Abstraction of topology can support a generalized interface B-Rep

# Topological entities and adjacencies$ Unique and independent of any shape or tolerance information$ There are well qualified structures to support general domain definitions$ CAD systems provide the key information to load appropriate structure

$ Storage for topological model is small (compared to other structures)$ Appears to allow effective support of geometric simplification, linkage to

feature models, supporting attribute specification, etc.

" Approach being used commercially to interface to all major CAD Systems# >99% reliability (compared to average success rates of 60-70%)# Supports automatic mesh generation (and much more)

12

Direct Geometry Access with Unified Topological Model

" Unified Topology Model# Need general combination of solids, surfaces and curves

$ Can be represented by non-manifold topology

# Follow Radial Edge Data Structure (Weiler)

# Topology built from topology in modeling source

# Geometric queries passed through to modeling source



13

Geometric Interrogations for Mesh Generation

Operators used during surface mesh generation to define entitiesand ensuring mesh validity

Once a valid surface mesh (properly separates regions and isnon-self intersecting) can use as fine a tolerance as needed (withappropriate care)

Typical Operators" Topological adjacencies

" Point classification

" Surface normals

" Closest point

" Parametric values

" Line/surface intersection

Some geometric operations are quite expensive - minimize use

14

Mesh Representation

Use of topology (primary entities only) for mesh" Effectively supports needs of all mesh generation and modification functions

Typical mesh topological structures" One-Level

# all relations are easy/fast to obtain

# not minimum storage

" Circular# less storage than one-level

# upward adjacencies are more costly to obtain than from one-level

" Reduced Interior

# interior faces and edges not explicitly represented

# ordered region-vertex relation implies interior entities

# less storage than other representations

# more complex to work with



15

Mesh/Model Relationship

Critical to all aspects of the simulation process Relationship is termed “classification”" Mesh Classification: Unique

association of a mesh entity, M id i,

to a geometric model entity, G j

d j,where d i<d j is denoted by

M id i G j

d j

indicates the left-hand entity(or set) represents a portion of theright-hand entity in the discretization

Multiple M id i classified on a G j

d j

" Boundary mesh entities are identifiedin terms of their classifications

" Classification critical to supporting adaptivesimulations and high level problem definitions

mesh region

mesh face

mesh edge

mesh vertex

region

region or face

region, face oredge

region, face,edge, or vertex

GEOMETRIC DOMAINENTITIES

MESHADJACENCIES

16

Algorithm Oriented Mesh Database

Mesh adjacencies" Work with minimal set, complete or not

" Change to any representation as needed by application

Methodology" Mesh entity description and comparison

" Downward adjacencies ordering, Templates and Inverse Templates

" Orientation and Mesh entity iD

Any mesh entity described by a set of lower dimension entities

Direct consequence

" Need at least one downward representation

" All vertices required" Vertices are atomic mesh entities - differentiated using iD’s

Two entities equal if their set of vertices are equal" Not absolutely general but key to practical implementation

" Allows to compare mesh entities (<,>,=)



17

Minimum information AOMD

Functionally complete representation" All vertices and all equally-classified entities:

# All mesh vertices classified on model vertices

# All mesh edges classified on model edges

# All mesh faces classified on model faces

# All mesh regions classified in model regions

" This sufficient minimum can be extended tomeet specific needs (e.g., represent partitionboundaries, non-conforming meshes)

" Starting from that and using templates,AOMD is able to build up the rest

Implementation

" C++, STL and STL like containers, generic programming" Algorithms act on mesh entity containers a level above data storage

18

Dual mesh

" The four topological entity sets {M } on the primal mesh are paired to

dual mesh entity sets { }" Tonti diagram indicates the relationships

A class of finite volume procedures computes on the dual" Most common dual mesh is a Delaunay triangulation and Voronoi diagram

" Easily extended for simplex meshes - dual vertices and centroids ofregions in primal mesh

" Common in CFD and Electromagnetics codes

" Calculations on both primal and dual meshes common

Dual Meshes in Computational Mechanics

is a boundaryadjacency operator∂

d -3i

d i



19

Dual Meshes In AOMD

Dual mesh regions" Simply connected with variable number of faces that can be large

" Can not be represented by simple ordered vertex, edge of face templates

" Can avoid representation of dual entities and adjacencies since dualadjacencies can be constructed form the primal mesh (e.g., the

adjacencies of the dual mesh can be constructed form M {M }adjacencies of the primal mesh).

Dual mesh in AOMD" Inferred from the primal mesh

" Requires specification of regions and verticesin the primal mesh - are the dual of the verticesregions in the dual mesh

" Downward adjacencies are ordered in Mand unordered in

" Upward adjacencies are unordered in M and ordered in

{ }03

i0i

3

20

Mesh Validity Assurance

Definition of a valid mesh with respect to curved domainsrequires more than distance calculations

One definition of a valid mesh of a n-dimensional domain is a

Geometric Triangulation, T

n

, in which a set of mesh entities Tn = { M 1

d 1 , M 2d 2 , M 3

d 3 , … , M N

d N }

with such that" For each (non-overlapping)" Tn is topologically compatible" Tn is geometric similar

A AA

(a) (b) (c)

0 ≤ ≤d ni

i j≠ ,



21

Topological Compatibility

Given a non-self-intersecting mesh with all the mesh vertices,, classified, and the remaining mesh entitieswith boundary entity sets , consider a modelentity with boundary entities . If each

is used by two , and each is used by one, the mesh is topologically compatible with

A mesh is topologically compatible if it is compatible with allmodel entities.

M 1

3

M 1

2 M

1

1

M 1

4

M 1

1

M 1

2

M 1

3

{ } M 0 { , } M d n

d 1 ≤ ≤

{ ( ), }∂ M d nd 0 1≤ ≤ −

G j

d ∂ ( )G j

d ∂ ( ) M i

d G j

d

M k

d G j

d ∂ ( ) M id

∂ ( )G j

d

M k

d G j

d G j

d

compatible hole redundancy

22

A definition: A set of mesh entities { M d } is geometrically similarto a model entity when { M d } consists of N mesh entities

Which are classified on the model entity “cover theparametric space” and the “parametric intersection of anytwo mesh entities is null - Important for edges and faces

M 1

1

M 1

2

M 1

4

M 1

3

M 1

5

M 1

6

M

1

1

M 1

2

M 1

4

M 13

M 1

5

M 1

6

a) violating geometricsimilarity

b) satisfying geometricsimilarity

Geometric Similarity

Gi

d

Gi

d



23

ζ

ζ

2

1

Algorithm for Ensuring Mesh Validity

Process of converting a triangulation to a geometric triangulation Algorithms have been given that begin from a mesh of “at least”the “convex hull” of the curved domain

Procedures that generate a surface mesh first make this processa bit simpler (but not trivial)

Geometric similarity can be tricky:" Must account for parametric space trimming

" Need alternative definition if there is no parametricspace or if you are given a “poor one”

Note: most volume meshers also requiredelimination of all intersection of the discretesurface mesh (happens between mesh entitieson different faces or, in the case of a vertextouch can be the same face)

24

Mesh Improvement Strategies for Adaptive Methods

Two related ingredients" Mesh correction indication to decide how the mesh is to be improved

" Mesh “enrichment method”

Most common procedure: employ the element contributions tothe error with a single enrichment strategy" Strategy is to equilibrate the errors in each element - proven optimal for

important classes of equations (Babuska)

More optimal strategy would be use combination informationsources and more flexible enrichment methods" Theory indicates higher rates of convergence can be obtained

"

Correction indication for combined procedures is complex - some heuristicmethods have been developed for hp-adaptive - try to estimate if local p-refinement will yield high convergence - If yes use it, if no use h-refinement

" Accounting for directional nature of solution - most error norms only scalars



25

Mesh Enrichment Strategies

" Using mesh of elements of the same order# Relocating nodes within a given mesh topology (r-refinement)

# Nested refinement templates (h-refinement)

$ Non-conforming

$ Conforming

# Remeshing (generalized h-refinement)

# General local mesh modifications (generalized h-refinement)

" Altering the order on a fixed mesh# p-version finite elements

# Spectral elements

" Addition of special functions# Elements with required jumps

# Elements with proper order singular field

" Combinations of procedures# h- and r-refinement

# hp-refinement

# Etc.

26

r-Refinement

Correction indication" Include positions of vertices in functional - too expensive

" Add nodal velocities as unknowns with penalty term to maintain meshvalidity - integrate mesh moving into the time matching algorithm

Strategy" Move mesh vertices to reduce error while ensuring mesh remains valid

Advantages" Often get large improvements for little effort

" No need to deal with mesh topology changes

Disadvantages

" Fixed limit on level ofimprovement possible

" Difficult to control on2-D and 3-D meshes

Good option in combinationwith other methods

solution field

uniform mesh

r-refined mesh



27

Nested Refinement

Correction indication" Elements with large error subdivided with goal of equidistributing error

Strategies" Bisection of elements yielding non-conforming meshes

" Bisection of elements with temporary splits of neighbors

" Application of strategy to maintain control of shapes such as split longestedge, of alternating edges split

Advantages" Straight forward for non-conforming case

" a-priori control of element shapes changes

" Allows effective solution transfer processes

" Can obtain level of accuracy desired Disadvantages" Dealing with constraint equation in non-conforming case

" Can not coarsen past the initial mesh sizes

" Cannot account properly for curved domains

bisection bisection withtemporary split

28

Remeshing

Correction Indication" Need to use the a posteriori information (error estimates, error indicators,

or correction indicator) to construct new mesh size field

" Mesh size field typically defined discretely over previous mesh or somebackground grid

Strategy" Employ automatic mesh generator that can function from a general mesh

size field

Advantages" Supports general changes in mesh size including construction of

anisotropic meshes

" Can deal with any level of geometric domain complexity" Can obtain level of accuracy desired

Disadvantages" Cost of complete mesh generation

" Solution field transfer expensive and can be inaccurate



29

General Local Mesh Modification

Goal is to approach the flexibility of remeshing while reducingsome of the disadvantages

Correction Indication" a posteriori information (error estimates, error indicators, or correction

indicator) to construct new mesh size field

Strategy" Employ a “complete set” of mesh modification operations to alter the given

mesh into the desired

Advantages" Supports general changes in mesh size including construction of

anisotropic meshes

" Can deal with any level of geometric domain complexity" Can obtain level of accuracy desired

" Solution transfer can be applied incrementally - may have more control

Disadvantages" Nearly as complex as complete mesh generation

30

Altering the Order on a Fixed Mesh

Correction Indication" a posteriori information (error estimates, error indicators, or correction

indicator) determine how to alter element basis functions

Strategy" Employ hierarchical spectral or p-version finite element basis to easily

change order

Advantages" For specific classes of problems method

has improved orders of convergence

" Can support limited anisotropy

" Can obtain level of accuracy desired

Disadvantages" Need analysis code that effectively

supports variable order elements

" Dealing with curved element meshes

p=1

p=3

p=2

p=4

p=3

p=4

p=5

p=6



31

Addition of Special Functions

Correction Indication" Indicators to detect and isolate

features like jumps and singularities

" Procedures to detect order ofsingularity can be required

Strategy" Add appropriate analytic functions for the jumps and singularities

" Tool like partition of unity functions and level sets being used to effectivelyadd the desired functions

Advantages" Can be quite effective when such features are present

Disadvantages" Need specialized methods not commonly supported by available codes

" Cannot not ensure given level of accuracy

Good option in combination with others whenappropriate features are present

elements withcrack tip singularity

32

Combination of Methods

" hr-for can get desired level ofaccuracy and gain advantagesof r-refinement

" h-refinement and special elements -isolate singularity and refine tocontrol the remaining error

" hp-adaptive method - gain improvedrates of convergence possible

" Local mesh modification withp-refinement or special elements -can deal with evolving domains

crack propagation



33

Adaptive Mesh Control with Mesh Modification

Reasons for mesh modification" Provides control of adaptive mesh updates

" Should reduce the accuracy loss in solution field transfer over remeshing

" Should prove effective for cases of frequent application (e.g. adaptivetransient simulations)

Complexities" Adaptive anisotropic mesh modification requires general

mesh control methods

" Curved domains introduce complexities into the applicationof mesh modification

34

Simplex Element Mesh Modification Operators

Single step operators" Face swap

" Edge swap

" Edge collapse

" Region collapse

" Edge split

" Vertex motion

" Curving interior mesh entity

Compound operators modify two or more entities to allow themodification of the problem entity

Examples of two step compound operators

" Swap entity A first, and then swap entity B" Collapse entity A first, and then collapse entity B

" Swap entity A first, and then collapse entity B

" Swap entity A first, and then curve new entity B

" Swap entity A first, and then reposition vertex B

" Etc.



35

Mesh Modification

“Standard approach”" Independent application of refinement/coarsening and shape control

" Adaptive refinement based on templates

" Some procedures limit coarsening to undoing refinements

" Shape control often limited to vertex repositioning

Current approach focused on examination of local situation" Mesh modification selected from “full” set of operations with the goal of

satisfaction of general mesh size field

Example: Refinement of element with poor shape caused by a short edge -Split only the long edges

36

Element Shape

For isotropic meshes several efficient measures possible(Operations for anisotropic meshes done in transformed space)" Can normalize to 1 for equilateral tetrahedron and 0 for zero volume

" Use such a measure to identify elements of unsatisfactory shape

Consideration of operations to improve shape" Requires more careful analysis

# Dihedral angles

# Edge lengths

" Best operators to improve based on situation

Poor shape dueto short edge

Poor shape due tolarge dihedral angles



37

Implementation of Mesh Modification Procedure

Given the “mesh size field”:" Drive the mesh modification loop at the element level

# Look at element edge lengths and shape (in transformed space)

# If both satisfactory continue to the next element

# If not satisfied select “best” modification

# Elements with edges that are too long must have edges split or swapped out

# Short edges eliminated so long and other criteria remain satisfied

" Continue until size and shape is satisfied or no more improvement possible

Determination of “best” mesh modification" Selection of mesh modifications based on satisfaction of the element

requirements

" Appropriate considerations of neighboring elements (not fully resolved in

the anisotropic case)" Ranking mesh modifications and choosing the “best” mesh modification

38

Eliminate Short Edges

The sequence to determinate a desired modification" First collapse the short edge by removing either its bounding vertex if possible

" Use compound operator: “swap(s)/collapse(s) + collapsing the short edge”! The pre-swap(s) or collapse(s) is determined by analyzing the tetrahedra that

become invalid after applying the desired collapsing

For example:

" After a collapsing, always check the number of edges connected to retainedvertex, swap(s) from the longest edge to reduce that number to previous level if

possible. For example:



39

Sequence to determinate a desired modification

" For small volume due to two opposite edges# In case on boundary, simple remove it if possible

# Swap either of the two opposite edges if possible

# Split the two opposite edges, followed by collapsing the new diagonal edgeif possible

" For small volume due to a vertex/face pair# In case on the boundary, simple remove it if possible

# Split the face and collapse the new interior edge if possible

# Swap one of the three bounding edges of the face if possible

# Collapse the vertex to any of the other three vertices

Eliminate Small Volumes

(1) two almost intersected opposite edges (2) a vertex almost sitting in its opposite face

40

Refinement

" Use edge split templates# Forty-two possible configurations to triangulate a tetrahedron

# Always collapse interior vertices that have to be inserted in four of the forty-two configuration through swap + collapse operations

# Always create the shortest diagonal edge in case ambiguous

" Adding new vertices at the middle point of edges in transformed spaceyields better position with respect to mesh size field in physical space

A

B

P

B

A

P

Physical spaceTransformed space

Middle point of thetransformed space

transform



41

Refinement

" Refinement of edges# Allow several steps to reduce element size to desired level# At each iteration, only split the longest edge and those edges close to the

longest with respect to the mesh size field

" Collapse the short edges refinement produced before starting the nextiteration

Blue circles indicate refinement edges in each iteration

42

Placing Vertices on Curved Boundaries

Input: list of refinement mesh vertices classified on curved b’drys Two parts of the procedure:" Part I: Process all that move directly or require local modification only

While any mesh vertex in the list still can be moved aheaddetermine target move for next vertex in list

if the current vertex can move to target location without a problem then

move to that location and remove vertex from list

else

determine the first problem preventing motion

define a move that ensures the first problem is “passed”

“analyze” current situation to determine “best” local modification

if there is an acceptable local modification then

perform selected local modification and remove from list if at target

end if end if

end while



43

Placing Vertices on Curved Boundaries

"

Part II: Ensure moving remaining vertices in the list using cavity re-triangulation

Traverse remaining vertices in the list once

perform local cavity construction apply cavity mesher, remove form list & add any new ones created into list

end traverse

New boundary mesh vertices may be created during cavity re-triangulation. But, this is not common, and they are typically“closer” to boundary

Part I If list is empty Part II

If any createdvertices need to be

snapped

Yes

No No

Yes

Vertex list tobe snapped

Done

Done

44

Definition of first problem plane" The plane containing a mesh face opposite to current snapping vertex

" The first plane the snapping vertex runs into while moving toward target

Other information includes:" Key mesh entities to cause mesh invalid

" Problem mesh faces on the first problem plane

Analyze Current Situation

: the mesh vertex classified on the modelboundary not snapped

M 00

B-C-D: a mesh face opposite to M 00



45

Vertex may be snapped incrementally along different path

Any intermediate target must at least pass first problem plane

Stop before reachingfirst problem plane

Need to Move Past First Problem Plane

Pass first problem plane

(1) original mesh

(2) create poorly-shapedelement not desirable

(3) move to vertex 2, whichgets onto the first plane and

typically a better solution

Green line: modelBlack line: meshRed arrow: snap directionCyan arrow: possible snap path

46

Cavity Creation and Meshing to Place Vertices on B’dry

Green line: model edgeBlack line: mesh edgeRed dash line:Blue dot: P M ( )0

0T P M ( ( )0

0



47

This procedure may create new vertices that need to be snapped – not common and typically “close” to its classified modelboundary

These new vertices will be added into the vertex list to beprocessed by the overall procedure

Handling of New Boundary Vertices Created

Blue dot: A new boundary vertex introducedduring re-triangulation

The new vertex is easier to be snapped sinceit is “close” to classified model boundary

48

" The model, a coarse initial mesh and the mesh after multiplerefinement/coarsening

" The spatial field of desired size:

" # of vertices snapped in refinement/coarsening iterations

Examples - sleeve of ball bearing

x

y z

Iterations of adaptations

# of vertices to be snapped

# of vertices snapped by a

reposition

# of vertices snapped by local

modifications

# of vertices snapped requiring local re-

triangulations

1 342 204 136 2

2 485 369 110 6

3 340 286 52 2

4 74 34 40 -

5 26 3 23 -



49

Mesh Adaptivity Adhering to Geometry

Adaptive simulation to determine electric losses in an accelerator cavity

solid

model

3rd

refine

2nd

refine

initial

mesh

50

Example - Adaptive Resolution of a Helical Particle Path

Result Mesh Initial

Mesh Template-based Enrichment

Max dihedral angle 144.9 174.3 152.1

Min dihedral angle 18.9 3.2 15.0Min tL of mesh 0.951 0.119 0.268Max tL of mesh 5.539 1.579 1.795Average tL of mesh 2.253 0.826 0.929# of mesh vertices 131 6823 4135# of mesh regions 508 34530 21348

" The problem - helical particle path inside a cubic domain, the initialmesh and the enriched mesh

" Compare with template-based procedure

Obtained local edge lengthrequested local edge length



51

Application to Metal Forming Application

An evolving geometry problem" Element shapes degrade and become

invalid if not corrected

" Domain shape and contact surfacesevolve during the simulation

Mesh modification to" Control element shapes degradation

" Provide geometric approximation control

" Provide discretization error control

52

Define size/shape distribution over the domain

One construction - directional length variation at a point

" Transformation matrix field T(x,y,z)

" Ellipsoidal in physical space transformed to normalized sphere

" Volume relation between physical space and the transformed space:

Construction of Anisotropic Mesh Size Field

:,, 321 eee Unit vectors associated withthree principle directions

:,,321

hhhDesired mesh edge lengthsin these directions

transformation

Physical spacetransformed space

V T x y z V transformed physical= ⋅( , , )

e1

e 2

e 3

e1 ,e 2 ,e 3



53

Construction of Anisotropic Mesh Size Field

Two specific forms of anisotropic mesh size field consideredto date" Analytical formulations

# Transformation matrix can be obtained any point inthe mesh

# or is computed using Guassquadrature over mesh edges

" Piecewise linear definition over the current mesh:# Transformation field is attached to vertices of mesh# or is approximated using the

matrices, , attached to the bounding vertices ofthe edge or element

d transforme L d transformeV

d transforme L d transformeV 2D visualization of discrete

anisotropic field

L L h h with he T e T

itransformed i

i iT

= ⋅ =

⋅ ⋅ ⋅=1 2

11 2

( ) ( )( , )

r r

V T V itransformed i= ⋅ =min( ) ( , , , )1 2 3 4

iT

Elements satisfy the transformation field if all edges satisfy the field andvolumes are “consistent” with unit regular element in the transformed field

If an element is not satisfactory, the mesh modification procedure isapplied to the element in the transformed space

e e

54

Examples 1 – Cylindrical Feature in Cubic Domain

Initial mesh The conformed meshwith maximum aspect ratio 100

x

y

z

" Anisotropic mesh size field specified

T(x,y,z)

1/h 0 0

0 1/h 0

0 0 1 /h

r

z

=

⋅ −

= − +

= =

= + = =

− −

θ

θ

α α

α α

α α

cos sin

sin cos

:

. ( ) .

.

, cos / sin /

.

0

0

0 0 1

0 14 1 0 0014

0 14

2 0 25

2 2

2

where

h e

h h

with r x y x r and y r

r

r

z



55

Examples 2 – Anisotropy Around Curved Domain

" The anisotropic mesh size field specified

Geometry modelInitial mesh Adapted mesh with

boundary vertices snapped

T(x,y,z)

1/h 0 0

0 1/h 0

0 0 1/h

r

z

=

⋅ −

= −

= = −

= + =

− −

− −

θ

θ

α α

α α

α α

cos sin

sin cos

:

. ( )

. ( )

, cos / sin

.

.

0

0

0 0 1

0 001 150 149

0 033 4 3

4 0 1

100 1

2 2

where

h e

h h e

with r x y x r and

r r

zr

== y r /

x

y

z

56

Anisotropic Mesh Correction Indication

Key Components to constructing a good anisotropic field" Smoothness indicator

" Construction of anisotropic mesh metric tensor

Smoothness indicator to isolate disconitunities (e.g., shocks)" Need to define field “smoothers” on only one side of discontinuities

" For DG formulation use superconvergence on downwind side to buildsmoothness indicator (see Flaherty, et al.)

" Theory ensures convergence when jumps are resolved - Defining ananisotropic Hessian based on smooth field approximation theory produceshuge errors at these location and 180 degree changes in directions

Construction of anisotropic mesh metric tensor" Mesh metric between jumps and smooth portions need to be matched

" At discontinuities set smallest size normal to discontinuity and determineaspect ratio using gradients along discontinuity

" In smooth portions of the domain away from the discontinuities calculate aHessian metric based on approximation theory



57

Adaptive Specification of Anisotropic Size Field

The new transformation field defined as

Where the scale factor, , is constructed based on 1-Dinterpolation error estimate arguments

Given an element edge, , and associated edge direction, ,the desired length of that edge can be determined as

Where and are the vertices of edge

If the current edge is too long it is marked for refinement

If the current edge is too short it is marked for elimination Technical issues associated with this process include" Determination of the Hessian" Controlling the smoothness of the new mesh size field

ei

M j0

M i1

L M e TT e dt i iT

it

M

M

j

k ( )1

0

0

= ∫ r r

M k 0

M i1

φ T Q= φ

ei

eiT ei

58

Example of Anisotropic Adaptive

" Mesh adaptation through mesh modification (not remesh)

" Works from anisotropic mesh size field - Library needed to support this ismuch more complex that some basic mesh modification library

" Limited experience indicates meshes have many fewer dof so that evenwith the complex operations required total time cut by a factor of 5 - Morework needed to confirm



59

Mach 3 Flow - Meshes at Two Different Times

after 20 mesh adaptation steps

after 120 mesh adaptation steps

60

Colliding Explosions - 250 refinement steps

Problem dominatedby shock/contactinteractions



61

3D Flow Past a Wedge

62

Adaptive Solution Procedures

Why is Adaptive Simulation Not Commonly Used?" Adaptive methods not effectively supported by the program structures

of current codes which were designed based on fixed grids

" Reluctance to devote needed computational resources even though itis cheap compared to the $400,000 to train an expert (D.H. Brown)

" Simulation is not sufficiently integrated into the design process - Thegreatest benefits gained when design engineers (not drafts persons)use reliable automated simulation to develop optimal products

" Technical reasons - although more work is needed, there aresubstantial capabilities

Adaptive Solution Procedures"

Can be constructed by adding a correction indication and meshimprovement strategy to an existing code - organizations that havedone using this capabilities

" Knowledge exists on how to do better - question is when it will becomecommonly available

" Technical areas like better error estimators and effectiveimplicit solves on adaptively evolving meshes need work



63

Program Structures for Parallel Adaptive Analysis

Most current analysis codes" Have a mesh based problem definition only -

need higher level definition to support mesh adaptation

" Data structures and algorithms designed for astatic mesh - not an evolving mesh

" Parallel computations supported on fixed mesh partition

Functions needed to support parallel adaptive" A mesh independent high level definition of the domain

and physical attributes (including the solution fieldstransferred in multiphysics analysis)

" Effective support of adaptive high order methods(e.g. p-version methods)

" Effective mesh modification methods" Effective support of dynamic load balancing

64

Parallel Automated Adaptive Analysis

Flow examples



65

"

p-version capability of exponential rate of convergence" Can produce a sequence of solutions by increasing polynomial order

" Gaining popularity in industry

p-version meshes" Coarse mesh needed to maintain computational efficiency

" Mesh layout needs strong control and gradation

" Mesh must maintain appropriate level of geometric approximation

Curvilinear Mesh Generation" Approaches

# Directly generate curved meshes for curved geometry domains

# Start with straight-sized, planar faces meshes and curve mesh entities oncurved boundaries and, as needed, mesh interior

" Key issues# Shape of mesh entities classified on curved geometry model boundary

# Shape validity verification of curved mesh entities

# Abilities to construct and modify curved mesh entities as needed to obtain validand acceptable curved meshes

Meshes for p-Version Finite Element Method

66

Geometric Approximation Representations

Mesh representation by Lagrange interpolants" Lagrange interpolation to curve initially straight-sided meshes.

# Worked out for quadratic Lagrange elements

# Higher order that quadratic too messy - both geometric control andcomputations required

# results will show quadratic is not sufficient if p-value is raised

Mesh representation by Bezier polynomial" Effectively increase the geometric approximation

order to any order desired

" Focus of current efforts

Mesh representation by Spline methods" More numerical stable

" Most geometry is piecewise (NURB or B-spline)



67

Application of Bezier Polynomial in Curved Meshing

" Advantageous properties of Bezier polynomials# Can be as high a degree as desired

# Convex hull provides smoother and more controllable approximation

# Better properties to allow more efficient intersection checks

# Derivatives and products of Beziers are also Beziers

# Efficient algorithms for degree elevation and subdivision

" Technical issues to define mesh entity shape# Interpolating and/or approximating model geometry

# Accounting for geometric modeling systems face parametric coordinates – periodicity, degeneracy and distortion…

# Chord length parameterization method for mesh edge on model face

$ Cord length, d , for n+1 interpolation points {Qk}, k=0,…,n defined as

$ Edge parameters associated with locations are: t 0

= 0, t n

= 1 and

d Q Qk k

k

n= − −

=

∑ 1

1

t t Q Q

d k k

k k = +

−

−

−

1

1

, k=1,…,n-1

68

Generic Hierarchic Structure of Mesh Entity Shape

" Mesh entity shape has been designed as hierarchic structure# All of the inherited classes share the same interface as the base class

# Effective representation and definition of mixed order curved meshes

# Mesh entity needs to inflate shape based on entities they bound in order tosupport different polynomial orders

Need to inflate face closurebased on bounding edges

Inflating a face fromquadratic to cubic



69

Mesh Region Shape Validity Determination

" Traditional validation methods test the Jacobian at integration points" Goal - provide a general validation for Bezier Regions: Relate Jacobian to the

region control points and determine its minimum bound

The Jacobian of a Bezier Region

" Partial derivatives of the region are themselves Bezier functions

" Jacobian determinant is defined by box-product of partial derivatives

" Since the product of 2 Bezier functions is also a Bezier function, the Jacobian

determinant is also a Bezier function

" In the case of a tetrahedron the function is of order 3 (p- 1), where p is the order of

the original shape

" Since a Bezier function is bounded by its convex hull, the Jacobian determinant

function inside the region is bounded by the convex hull of its control points" A region is valid globally if the minimum control point of the Jacobian determinant

function is > 0

70

Invalid Region

" The box product terms that compose the Jacobian determinant functioncan be used to determine how a region should be corrected

(0,0,0,1)

(1,0,0,0) (0,0,1,0)

(0,1,0,0)

(0,0,0,1) a

b

c

P0 P1

P2

P3

Invalid Tet caused by moving P1 -note that a • (b x c) < 0

" Jacobian Control point, J l

for a tet region of degree d is equal to:

J d

l

a Pi

P j

Pk l ijk

i j k l

=−

• ×

+ + =

∑1

3 11 2 3

( )( )

ζ ζ ζ

ad

i

d

j

d

k ijk =

−

−

−

1 1 1



71

Correcting an Invalid Entity by Local Modification

# Determine the existence of “small” edge length, face area or region volume inthe neighboring of key mesh entity

$ Exists – apply edge, face, region collapse to produce more space

# Appropriate apply swap, split or compound operation

Region collapse

split

72

Quality of Curved Mesh for p-Version Method

" Quality of curved mesh is still an open issue,some possible components# Minimum determinant of Jacobian

# Determinant of Jacobian variation inside one element

# Normalized Minimum determinant of Jacobian

# Geometric approximation error

" Two main difficulties# Lack of mathematical proof

# Hard to relate quality to finite element

solution accuracy

" Example# Compare two valid curved meshes

based on the same geometric mode# Quadratic mesh geometry

# Case (a) focusedon maximizingthe min. Jacobian - leads to stronginterior meshentity curving

# Clear there areopen questions

(a)

(b)



73

Mesh Curving Examples

74

Examples of mesh curving

Model Geometry Original Linear Mesh

QuadraticBeziers

CubicBeziers



75

Simulation-Based Design

Goal" Physics-based simulation a regular part of engineering design processes

for products of all types

Physics-Based Simulation Capabilities Available" Generalized finite element and finite volume procedures available that are

capable of effectively solving many physics-based models

" Some ability to automatically go from detail models (solid models) tomeshes for simulation

Current Engineering Design Practice" Computer-aided design technologies commonly used in industrial design

" Engineering simulation used regularly in design only when either:#

The simulation tools are readily available and directly linked with the CAD data- geometric interference analysis

# The cost (dollars and/or time) of physical prototypes is simply too high -automotive crashworthiness

" Design decisions require complex interaction of several “experts”

76

Background

Needed Capabilities" Ability to perform reliable engineering simulations from design data

" Ability to link simulation results back into the design decision process

Missing Technical Components" Automated adaptive simulation

# Control of discretization errors

# Model selection

# Capture of engineering knowledge

" Automatic construction of simulation models from basic design data

" General methods to link information between simulations and manipulateto answer design questions

Missing Corporate Components" Willingness to make needed organizational changes

" Support for the creation of the Simulation-Based Design Systemsappropriate to their products



77

Background

Missing technical components under development" Builds on company investments in

# Product data management systems (PDM)

# Computer aided design systems (CAD)

# Computer aided engineering tools (CAE)

" Specific new technologies needed for# Simulation model management

# Simulation data management

# Simulation model generation

# Adaptive simulation control

Missing corporate components will follow" Some forward looking companies have made sizable investments

# Example system being used in practice will be overviewed

Environment for the simulation-based design being developed" Needed technological components being developed

" Components integrated with PDM, CAD and CAE

78

Simulation-Based Design Environment



79

Simulation Environment for Engineering Design

Simulation Model Management: Interactions between the product design asdefined in the product management system and the simulation technologies

Simulation Data Management: Structures and methods to define thediscretized simulation information and relate it to the product definition

Simulation Model Generation Tools: Construct models and the appropriatespatial discretizations for a simulation from the product definition

Adaptive Control Tools: Determine appropriate mathematical models, selectdiscretization technologies, evaluate the accuracy of the predictions, anddetermine improvements needed to obtain the desired accuracy

Geometry-Based Simulation Engine: Responsible for executing thenumerical aspects of the simulations building on available CAE tools

80

Example of Simulation-Based Design System

Visteon’s Unified Parametric Vehicle (UPV) has simulation driving design" Integrated engineering design/analysis process

" Took five years to evolve and develop - will be much easier in the future

" Develop products and systems without having to build prototypes

Key UPV features" Parametric CAD Model based in Pro/E parametric modeling tools

" Feature based design model with simulation attributes attached to the designmodel entities - invariant to topology changes

" Meshing is 100% automatic with direct CAD interface

" Adaptive mesh refinement

" Finite element CFD solver

" Coupled wall-to-wall radiation

"

Embedded 1D refrigerant, coolant,and oil system performance software

" Applied to engine cooling, occupantcomfort, and air handling performance



81

UPV Overview

Users can morph design" UPV parameterized design model" CAE models created from design model" Physical/Simulation attributes attached

to design model

Software Tools" CAD – Pro/E" Simulation Middleware – Simmetrix" CAE/CFD engine – AcuSolve" Visualization – Fieldview

Engineering Simulation" Coupled multiphysics" Range from 1-D “engineering models”

to adaptive FEM" All simulation steps automated" Simulation parameters associated with

UPV CAD Model

82

UPV-Engine Cooling

Requirement: Accurate thermal managementprediction to support climate control system design

" Analyze climate control and powertrain coolingsystem performance

" 24 hour turn-around to support design decisions

Design parameters needed

" Radiator top tank coolant temperature

" Flow distribution through heat exchangers

" Fan performance

" Engine compartment air flow

" Heat protection

SBD system performance" Within 5% of wind tunnel air flows for engine

(within the accuracy of wind tunnel testing)

" Top water temperature predictions within 2oC of

test results for 20 vehicles (100s of simulations)



83

UPV–Interior Comfort

Standard Applications" Cool Down and Warm Up Performance

" Defrost Performance

" Hot Climate Soak

Advanced Applications" City Drive Cycle

" Demist Performance

" Automatic Temperature Control Performance& Prediction

Example: Footwell study" Inputs: Vehicle interior, registers, seats and

mannequin geometry, and blower curve

" Outputs: View flow direction in footwell,

temperature distribution and visualize designimprovements

" Benefits: Optimal floor duct design providescomfort and avoids hot foot

Flow pathline by velocity magnitude

84

UPV Summary

UPV used for predictive design of various automotive systems at Visteon

Significant cost and time savings have been realized

" Analysis of one particular engineering attribute went from 3 months ofbuilding and testing to 2-3 weeks of simulation

Significantly more of the design space to be explored during bid process

" Gives greater confidence in installed performance



85

Design Model for Simulation Model Management

Design Model" A high level representation of the design components

" Focuses on functional components and their interactions

Key Assumption" The design model has a sufficient component decomposition and

component relations to support the alternative viewpoints needed

Design Model structures definition efforts" Testing the key assumption with multiple industries

# Assumption appears to hold, but most test are with people looking for SBD

" Design and implementation of an abstract representation that can supportmultiple functional interactions defining simulation idealizations, and

integrates with CAD and other SBD components

86

Consider the UPV example:" Design model consist of functional components

# Design parameters associated with functional components

# Component interactions (relationships) defined by “common geometry” and/orfunctional interaction - desired simulation relationships possible from this set

# Simulation models/idealizations can be associated with functional components

# All attributes can be associated with functional components

# Relating results parameters to the functional components allows effectiveintegration into the company product data management system

" Design model supports automated simulation for multiple levels ofengineering modeling# Parameters for low order engineering models linked directly to components

#

Design model can drive automatic solid model construction to support “fullgeometry” discretization methods

" Design models needs to support multiphysics coupling for multiple levels ofengineering modeling# Requires appropriate linkages to different levels of geometric model

# Requires linkage with fields of simulation results

Role of Design Model



87

Design Models and CAD System Capabilities

" Feature/assembly modelers provided by CAD vendors support# Basic feature/assembly trees and interactions

# Parameterization of geometric quantities and interactions

# Ability to associate generic attributes to features

# Ability to drive solid model construction from feature/assembly model

# User interface for feature/assembly model construction and manipulation

" Feature/assembly modelers provided by CAD vendors do not support# Multiple levels of interactions, particularly functional interactions not defined in

terms of geometric interactions

# Effective means to support full range of simulation attribute interactions

# Multiple levels of geometric representation for different analysis idealizations

# Effective interactions with coordinated attribute sets as needed to drive asimulation

" Not cost effective to attempt to redo what CAD vendors are providing# Must build on the feature/assemble modeling capabilities of CAD systems

# Must continue to take advantage of CAD geometry functions

88

Design Model “Design”

" Functional component representation supporting component relationships

" Component hierarchies supported

" Support easy integration with CAD feature/assembly modelers forcomponent definition

" CAD system to support geometry-based interactions# Geometric definition of components

# Initial association of geometry defined relationships with feature/assemblymodel

# Geometry sizing parameters

# Continue to use the non-manifold topology to support “automatic meshing”, etc.

" Analysis idealizations defined at the design model component level

" Definition of design model must be able to evolve

# Initial definition typically before detailed geometric model - You want simulationto help drive definition of geometry!

# Additional component levels defined during process

# Relationship of components and simulation idealizations defined and adaptedduring process



89

Design Model

Tree structure for the design model components" Components are nodes in tree

" A directed acyclic graph is one representation

" Load components from CAD feature/assembly model

Components of automotive HVAC system

90

Design Model

Support of functional relationships" Design model must support relationships between components

# Relationships represented at any level

# Higher level relationships must be consistent with lower level relationships

" Geometric relationships inherited from CAD system# Feature/assembly model needs to provide this information in terms of

components of the non-manifold geometric model (as it is known at the currentlevel of the design)

" Relationships past the sharing of common boundary entities needed# Can be the only relationships before geometry defined

# Are required by various design and simulation processes



91

Design Model and Analysis Idealizations

Design model and analysis idealizations" Multiple levels of analysis idealization - idealization selection depends on

# Level of design detail available

# Simulation information and accuracy needed

" Analysis idealizations and design model# Analysis idealizations associated with the design model components

# Design component relationships used to match idealizations for a simulation

# Interactions can be as varied as dictated by level of geometric approximation,mathematical model selection, component level idealization, etc.

92

Design Model and Other SBD Structures

Key structures used in simulation model management" Design model components with their relationships and idealizations

# Coordinated with CAD system feature/assembly model

" Appropriate instances of non-manifold topology and functional links toappropriate solid models in CAD system

" Simulation control and simulation attribute cases

" Simulation field structures# Since fields are in terms of meshes - this links through the meshes



Documents

Shephard Short Course