The Classical FEM Method and Discretization Methodologyliutiant/slides/rg_tiantian_fall2012.pdf ·...

The Classical FEM Method and Discretization

Methodology SIGGRAPH 2012 Course Note

Re-Presented by Tiantian Liu

• SIGGRAPH 2012 Course • FEM Simulation of 3D Deformable Solids: A practitioner’s guide to theory,

discretization and model reduction • http://www.femdefo.org/ • The classical FEM method and discretization methodology

• Eftychios Sifakis

• Model Reduction • Jernej Barbič

• Eftychios Sifakis • Assistant Professor • Computer Science Department, University of Wisconsin-Madison • sifakis@cs.wisc.edu

Topics • Elasticity in 3D

• Deformation map and deformation gradient • Strain energy and hyperelasticity • Force and traction • Stress tensor

• Strain Measurements Vs. Stress Tensor • Linear elasticity • St. Venant-Kirchhoff model • Corotated linear elasticity • Neohookean elasticity

Basic Concepts

Four Models

Topics (cont’d)

• Discretization • Energy and Force • Linear tetrahedral elements • Explicit integration • Implicit integration

Fundamental Concepts

• What do we need to simulate a deformable body? • For each vertex:

• Position • Velocity • Acceleration / Force

• Concepts we need: • Deformation map and deformation gradient • Strain energy • Force and traction density • Stress tensor

Fundamental Concepts Constitutive Models Discretization

Deformation Map

• We need something describing the deformation

Deformation Map

Example of Deformation Map

• Here’s a deformation map

2116.11500.02067.01866.0

Example of Deformation Map

• Or we can write it another way

116.1500.0067.0866.0

Deformation Gradient Tensor

• Definition • Jacobian Matrix of the Deformation Map

• Notice that F will be spatially varying across Ω, we will notate it as F(X) if such dependence needs to be made explicit.

• Example • Translate

• Example • Scale

• Example • Rotate

Same Potential

Hyperelasticity Hypothesis

• Potential energy associated with a deformed configuration • only depends on the final deformed shape • not depends on the deformation path over time

Strain Energy

• Energy is only a function of current state φ(X) • Based on hyperelasticity

• Define energy density funcition Ψ(φ(X),X)

Strain Energy

• φ(X) is only a function of F(X). (If φ is small)

• Ψ(φ(X),X) can be write as Ψ(F(X))

Strain Energy

• Take a blind guess of Ψ(F(X))?

Strain Energy

• Take a blind guess of Ψ(F(X))?

Strain Energy

• Naive Spring Analogy Model

Energy -> Force

• Example • Potential Energy in Gravity

• Force

Force in non-constant Force Field / Force Density • Aggregated Force = integration of force density

• Problem? • How about zero-volume area, such as surface?

Surface Traction

• Aggregated Force on surface

Ω∂⊂B

The First Piola-Kirchhoff Stress Tensor

• Definition

• Two equally popular (and, in fact, equivalent) ways to describe the material properties of a hyperelastic material

• Explicit formula of Ψ(F(X)); • Explicit formula of P(F(X));

• Force and traction, represented by Piola-Kirchhoff Stress Tensor:

Example from Ψ to P

• Energy density:

• Piola-Kirchhoff Stress Tensor

Example from Ψ to P

• Given Deformation Map

• Calculate P

• Calculate Traction

Rigid body Transformation Invariance

• Energy / Stress Tensor / Force should remain invariant under rigid body transformations (Rotation + Translation)

Problem of Naive Spring-Analogy Model

• Strain Measurement: • F-I

• Stress Tensor • k(F-I)

• Energy density • k/2*(F-I):(F-I)

Constitutive Models of Materials

• Strain Measures • Linear elasticity • St. Venant-Kirchhoff model • Corotated linear elasticity • Isotropic &rotationally invariant • Neohookean elasticity

Strain Measures

• Motivation • Rigid Body Transformation Invariant

• Idea • Since RT = R-1 …

=> FTF = (RS)TRS = STRTRS = STS • Directly Using S => S = RTF

Green Strain Tensor

• Definition

Green Strain Tensor

• Advantage of using the term FTF:

Green Strain Tensor

• Small deformation case:

Small Strain Tensor

• Small deformation case:

Linear elasticity

• Strain energy density defined with small strain tensor

Linear elasticity

• After we have the definition of Ψ(F), we can get the strain tensor by:

Linear elasticity

• The stress P is a linear function of the deformation gradient F. • Problem:

• designed to be accurate exclusively in a small deformation scenario.

Linear elasticity

• http://www.youtube.com/watch?v=e66agRi1R2k

St. Venant-Kirchhoff Model

• Strain energy density defined with Green strain tensor directly

• Similarly, we can get P from the definition of Ψ:

• Potential Problem: • Think about the stress tensor again:

• It’s a cubic function of F, so when F goes

to zero, P does not go to infinity (at least a large value) as wanted.

Fundamental Concepts Constitutive Models Discretization F

• Problem: • When F->0, P = F[2μE+λtr(E)I] -> 0, thus the stress goes to zero! • Experimental critical compression threshold = 58% • If compressed to about 58% of undeformed dimensions, the stress force has a

maximum.

• http://www.youtube.com/watch?v=JelJltJlJc0

Corotated Linear Elasticity

• Use the second idea motioned before, measure strain using: ϵc = S-I = RTF-I • Energy density:

• Still, we can use the definition of Ψ to calculate P:

• Motivation: • mimic what linear elasticity would have been after killing the rotation term

• Problem: • the overhead of corotated vs. linear elasticity includes the cost of the polar

decomposition, and the need to employ nonlinear solvers for certain types of simulation

Rotationally Invariant

Isotropic

• Definition: • resistance to deformation is the same along all possible orientations that such

deformation may be applied

Isotropic

• Definition: • resistance to deformation is the same along all possible orientations that such

deformation may be applied

Isotropic & Rotationally Invariant

• Using the Singular Value Decomposition F = UΣVT, we conclude that rotationally invariant, isotropic materials satisfy:

Isotropic Invariants

• We can define any material with F = UΣVT, but again, the overhead of SVD is high.

• Alternative: • Mimic what we did in St. Venant-Kirchhoff model. • Find some parameters we call isotropic invariant. I(F)

• Define energy function Ψ as a function of I1, I2, I3: Ψ(I1, I2, I3) • We can find the stress tensor P as we did before:

Neohookean Elasticity

• Define energy density Ψ as a function of I1 and I3:

• Then we can get the stress tensor using previous formula:

• Properties: • When F goes to 0, J goes to 0, a*log2(J)-b*log(J) goes to infinity • When F goes to infinity, J goes to infinity, a*log2(J)-b*log(J) goes to infinity

too. (Remember log2(J) goes faster than log(J))

• Properties: (cont’d) • λ value controls the stiffness of the body directly. (λ emphasizes the log2(J)

term which is related to the fraction of volume change) • High λ condition = incompressible

• http://www.youtube.com/watch?v=XYS7STz8iNI

Summary of Models Model Strain Measurement First Piola-Kirchhoff

Stress Tensor Potential Problem

Naive Spring Analogy Model F-I k(F-I) Not rotationally

invariant Linear Elasticity

Model ½*(F+FT)-I 2μϵ+λtr(ϵ)I Not rotationally invariant

St. Venant-Kirchhoff Model ½*(FTF-I) F[2μE+λtr(E)I] Can not compress

extremely Corotated Linear Elasticity Model RTF-I 2μ(F-R)+λtr(RTF-I)R Slow polar

decomposition Neohookean

Elasticity I1, I3 μ(F-μF-T)+λlog(J)F-T -

Discretization

• Energy and Force • Linear tetrahedral elements • Explicit integration • Implicit integration

Deformation Map on Vertices

• We only store the values of the deformation map φ(X ) on a finite number of points X1, X2, . . . , XN, corresponding to the vertices of a discretizated mesh.

• Define an aggregate state of a model:

Discretize Energy

• Original Energy of Deformation

• Discretized Energy on Vertices

Discretize Energy

• Original Energy of Deformation

• Discretized Energy on Vertices

Discretize Energy

• In practice, prior to computing each force, we first separate the energy integral into the contributions of individual elements Ωe

Discretize Force

• From energy to force:

Discretize Force

• Subsequently, the force fi on each node can be computed by adding the contributions of all elements in its immediate neighborhood Ni:

Linear Tetrahedral Elements

• In every tetrahedral Ti, we have:

• Linear Assumption: • Inside one tetrahedron A1 = A2 = A3 = A4 = F b1 = b2 = b3 = b4 = b

Compute F

Energy & Force in One Element

• Since Fi is constant over the tetrahedron Ti, we can redefine the energy function as:

• For one tetrahedron Ti, the forces to the four vertices will be:

Batch Computation of Elastic Forces on a Tetrahedral Mesh (Precomputation)

Batch Computation of Elastic Forces on a Tetrahedral Mesh (Computation)

Explicit Euler integration Scheme

• We have fn=f(xn,vn) already! • an = M-1fn

• vn+1 = vn + dt*an

• xn+1 = xn + dt*vn

• Problem? • Unstable!

Implicit Euler Integration Scheme

• vn+1 = vn + dt*an+1

• xn+1 = xn + dt*vn+1 • Where an+1 = M-1f(xn+1, vn+1) = M-1(fe(xn+1)+fd(xn+1,vn+1)) • M is the mass matrix, lumped to diagonal form. • fe(x*) is the elastic forces at configuration x*, as defined in previous

sections • fd(x*, v*) is the damping force at position x* and velocity v*,

according Rayleigh damping model.

Implicit Euler Integration Scheme

• Problem: • Compute K(x*) is slow.

• Constantly poke a vertex and all its neighbors • Cache Disaster + Construction Cost

• Implicit integration of vn+1 = vn + dt* M-1(fe(xn+1)+fd(xn+1,vn+1)) and xn+1 = xn + dt*vn+1 needs tricky methods

• Solution: • We can compute K(x*)w directly, instead of computing K(X*) first • Define iterative process to compute unknowns Xn+1, Vn+1

Force Differentials

• Define force differential as:

• Write it in H matrix form:

Force Differentials

• Plug in the formula for H:

• 2 More Steps: • construct the deformation gradient increment δF • provide a usable formula for δP(F; δF)

Deformation Gradient Increment δF

Stress Tensor Increment δP(F; δF)

• For St. Venant-Kirchhoff materials

• For Neohookean materials

Batch Computation of Elastic Forces on a Tetrahedral Mesh (Modified)

Iterative Process of Implicit Integration

• Construct sequences of approximations: • Such that:

• Correction variables are defined as

• The linearization around (x(k+1)n+1 , v(k+1)

n+1) yields:

Not pure linearization

• Further Manipulation:

• Update v(k+1)n+1 and x(k+1)

Summary

• Deformation Measurements • Deformation Map, Deformation Gradient

• Strain->Energy->Stress • 5 different models

• Discretization • Linear tetrahedron, explicit/implicit integration

The Classical FEM Method and Discretization Methodologyliutiant/slides/rg_tiantian_fall2012.pdf ·...

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