Finete Element

Finite Element Methods (FEM)

Suzanne VogelCOMP 259

Spring, 2002

The finite element method is the formulation of a global model to simulate static or dynamic response to applied forces.• Models: energy, force, volume,…

This differs from a mass spring system, which is a local model.

Definition of FEM

1. Set up a global model in terms of the world coordinates of mass points of the object. These equations will be continuous.2. Discretize the object into a nodal mesh.3. Discretize the equations using finite differences and summations (rather than derivatives and integrals).4. Use (2) and (3) to write the global equations as a stiffness matrix times a vector of (unknown) nodal values.

Top-Down: Steps in FEM

Top-Down: Steps in FEM

6. Solve for the nodal values.•Static – nodal values at equilibrium•Dynamic – nodal values at next time step7. Interpolate values between nodal coordinates.

5

23

14

678

udiscretize interpolate

+global model

object

nodal mesh interpolate values between nodes+

local model

Bottom-Up: Steps in FEMNodes are point masses connected with springs. A continuum equation is solved for the nodes, and intermediate points are interpolated.

A collection of nodes forms an element.

A collection of elements forms the object.

5

23

14

678

u

Elements and Interpolations

Interpolating equations for an element are determined by the number and distribution of nodes within the element.

More nodes mean higher degree, for smoother simulation.

Example: Hermite as 1D CubicInterpolation Equation

1. Assume

u

r

cubic equation

equation using shape (blending) functions

and

Example: Hermite as 1D CubicInterpolation Equation

2. Normalize the element to [0,1] and rewrite

as a matrix equation

or

Example: Hermite as 1D CubicInterpolation Equation

3. Solve for the coefficients Q

4. Plug the coefficients into the cubic equation

5. Rewrite the cubic equation in the form

Example: Hermite as 1D CubicInterpolation Equation

4 + 5. are equivalent to the steps

values at the 4 nodes of the element

shape (blending) functions

Example: Hermite as 1D CubicInterpolation Equation

shape (blending) functions within one elementLet

u

r

1D Elements

(x) (x)

(x)

Example: bungee

2D Elements

(x,y)

(x,y)

(x,y)

Example: cloth

3D Elements

(x,y,z)

(x,y,z)

Example: skin

Static analysis is good for engineering, to find just the end result.

Dynamic analysis is good for simulation, to find all intermediate steps.

Static vs. Dynamic FEM

Types of Global Models[6]

Variational - Find the position function, w(t) that minimizes the some variational integral. This method is valid only if the position computed satisfies the governing differential equations.

Rayleigh-Ritz - Use the variational method assuming some specific form of w(t) and boundary conditions. Find the coefficients and exponents of this assumed form of w(t).

Example of Variational Method[6]Minimizing the variation w.r.t. w of the variational function

under the conditions

satisfies the governing equation, Lagrange’s Equation

Galerkin (weighted residual) - Minimize the residual of the governing differential equation, F(w,w’,w’’,…,t) = 0. The residual is the form of F that results by plugging a specific form of the position function w(t) into F. Find the coefficients and exponents of this assumed form of w(t).

Types of Global Models[6]

We can approximate w(t) using Hooke’s Law

Example of Galerkin Method[6]

If we use that equation to compute the 1st and 2nd time derivatives of w, then we can compute the residual as

Example of Static, Elastic FEMProblem: If you apply the pressure shown, what is the resulting change in length?

Object

First step. Set up a continuum model:

•F = force•P = pressure•A = area•L = initial length•E = Young’s modulus

Entire length:

Infinitessimal length:

Example of Static, Elastic FEM

Since the shape is regular, we can integrate to find the solution analytically. But suppose we want to find the solution numerically.

Next step. Discretize the object.

Example of Static, Elastic FEM

Example of Static, Elastic FEM

Discretization of object intolinear elements bounded by nodes

1 2 3 4n1 n2 n3 n4 n5

Example of Static, Elastic FEM

Next step. Set up a local model.

Stress-Strain Relationship (like Hooke’s Law)

Young’s modulus distance between adjacent nodes

stress (elastic force)

Example of Static, Elastic FEMNext step. Set up a local (element) stiffness matrix.

Rewrite the above as a matrix equation.

Same for the adjacent element.

element stiffness matrix

nodal stresses

nodal coordinates

Example of Static, Elastic FEMNow, all of the element stiffness matrices are as follows.

1 2 3 4n1 n2 n3 n4 n5

ri is the x-coordinate of node ui

Example of Static, Elastic FEM

Next step. Set up a global stiffness matrix.

Pad the element stiffness matrices with zeros and sum them up. Example:

Example of Static, Elastic FEM

Final step. Solve the matrix equation for the nodal coordinates.

Global stiffness matrix.Captures material properties.

Nodal coordinates.Solve for these!

Applied forces

Elastic FEM

A material is elastic if its behavior depends only on its state during the previous time step.•Think: Finite state machine

The conditions under which an “elastic” material behaves elastically are:•Force is small.•Force is applied slowly and steadily.

Inelastic FEM

A material is inelastic if its behavior depends on all of its previous states.

A material may behave inelastically if:•Force is large - fracture, plasticity.•Force is applied suddenly and released, i.e., is transient - viscoelasticity.

Conditions for elastic vs. inelastic depend on the material.

Examples of Elasticity

Elasticity•Springs, rubber, elastic, with small, slowly-applied forces

Examples of InelasticityInelasticity•Viscoelasticity•Silly putty bounces under transient force (but flows like fluid under steady force)

•Plasticity•Taffy pulls apart much more easily under more force (material prop.)

•Fracture•Lever fractures under heavy load

Linear and Nonlinear FEMSimilarly to elasticity vs. inelasticity, there are conditions for linear vs. nonlinear deformation.Often these coincide, as in elastoplastic.

= e

Hooke’s Law

•Describes spring without damping•Linear range of preceding stress vs. strain graph

Elastic Deformation

Elastic vs. Inelastic FEM

e

e

t

loading unloading

orstress strain

Young’s modulus

Elastic vs. Inelastic FEM

Damped Elastic Deformation

e

e

t

loading unloading

viscous linear stress

Rate of deformation is constant.

a1e. a1e

.

Viscoelastic Deformation

Elastic vs. Inelastic FEM

e

e

t

loading unloading

.

viscousnew term!

This graph is actually viscous,but viscoelastic is probably similar

Rate of deformation is greatestimmediately after starting

loading or unloading.

depends on time t

linear stress

Elastoplastic Deformation

Elastic vs. Inelastic FEM

e

This graph is actually plastic,but viscoelastic is probably similar

f

e

x

x

compare

loading

unloading

loading x

elas

tic

plastic

depends on force f

e

Elastic vs. Inelastic FEM

Fracture

•Force response is locally discontinuous•Fracture will propogate if energy release rate is greater than a threshold

e

x

x

loading

unloading

depends on force f

1. World coordinates win inertial frame(a frame with constant velocity)2. Object (material) coordinates rin non-inertial framer(w,t) = rref(w,t) + e(w,t)

Elastic vs. Inelastic FEM4,5

world, orinertial frame

ref

robject, or

non-inertialframe

origin of= center of mass in

Transform•reference component rref

•elastic component e•object frame w.r.t. world frame

r(w,t) = rref(w,t) + e(w,t)

Elastic vs. Inelastic FEM4,5

ref

r

Elastic vs. Inelastic FEMAll these equations are specific for:•Elasticity•Viscosity•Viscoelasticity•Plasticity•Elastoplasticity•Fracture•(not mentioned) “Elastoviscoplasticity”

Ideally: We want a general equation that will fit all these cases.

Elastic vs. Inelastic FEM4,5

A More General ApproachTo simulate dynamics we can use Lagrange’s equation of strain force. At each timestep, the force is calculated and used to update the object’s state (including deformation).

stress componentof force

mass density damping density

elastic potential energyLagrange’s Equation

Elastic vs. Inelastic FEM4,5Given:Mass density and damping density are known.Elastic potential energy derivative w.r.t. r can be approximated using one of various equations.

The current position wt of all nodes of the object are known.Unknown:The new position wt+dt of nodes is solved for at each timestep.

vect

or

vect

or

mat

rice

s

next slide

Lagrange’sEquation

Elastic vs. Inelastic FEM4,5

For both elastic and inelastic deformation, express elastic potential energy as an integral in terms of elastic potential energy density.

elastic potential energy density

elastic potential energy

Elastic vs. Inelastic FEM4,5

Elastic potential energy density can be approximated using one of various equations which incorporate material properties.

•Elastic deformation: Use tensors called metric (1D, 2D, 3D stretch), curvature (1D, 2D bend), and “twist” (1D twist).

•Inelastic deformation: Use controlled-continuity splines.

Elastic FEM4

For elastic potential energy density in 2D, use• metric tensors G (for stretch)• curvature tensors B (for bend)

|| M || = weighted norm of matrix M

Elastic FEM4

Overview of derivation of metric tensor

Since the metric tensor G represents stretch, it incorporates distances between adjacent points.

world coordinatesobject coordinates

Elastic FEM4

Overview of metric and curvature tensors.

From the previous slides, we found:

Similarly:

represents stretch

represents bend

Theorem. G and B together determine shape.

Elastic FEM4

For elastic FEM, elastic potential energy density in 2D incorporates changes in the metric tensor G and the curvature tensor B.

|| M || = weighted norm of matrix Mweights = material properties

Inelastic FEM5

For inelastic FEM, elastic potential energy density is represented as a controlled-continuity spline.

For some degree p, dimensionality d, compute the sum of sums of all combinations of weighted 1st, 2nd,…, mth derivatives of strain e w.r.t. node location r, where m <= p.

weighting function = material property

Inelastic FEM5

Then the elastic potential energy density derivative w.r.t. strain e is:

weighting function = material property

Example: p = 2, d = 3

Elastic vs. Inelastic FEM4,5

InelasticElastic

RecapLagrange’s Eq’ntotal force

(includes stress)

elasticpotential energy

elastic potentialenergy density

4

5

5

material properties

How it has beenexpanded and is continuing

to be expanded...

Elastic FEM4Continuing

elasticpotentialenergy

>0: surface wants to shrink<0: surface wants to expand

>0: surface wants to flatten<0: surface wants to bend

Inelastic FEM5Continuing

Deformation has been modeled by approximating elastic potential energy.

elastic potential energy

elastic potentialenergy density

strain

Inelastic FEM5Continuing

Now rigid-body motion and other aspects of deformation must be computed using physics equations of motion.

In this way, both (in)elastic deformation and rigid-body motion can be modeled, providing a very general framework.

r(w,t) = rref(w,t) + e(w,t)

Inelastic FEM5

Motion of object (non-inertial) frame w.r.t. world (inertial) frame

Combines dynamics ofdeformable and rigid bodies

elastic

rot

trans

Inelastic FEM5

Velocity of node of object (non-inertial) frame w.r.t. world (inertial) frame (radians / sec) x (radius)

Identically, in another coordinate system,r(w,t) = rref(w,t) + e(w,t)w.r.t. object

velocity of reference component

velocity of elastic component

w.r.t. world

Inelastic FEM5

rot

angular momentum

inertia tensor

Angular momentum is conserved in the absense of force. So a time-varying angular momentum indicates the presence of foce.

Inelastic FEM5

rot

indicates changing angle between position and direction of stretch

Inelastic FEM5

elastic

inertial centripetal Coriolis transverse damping

elastic potential energy strain

restoring

If the reference component has no translation or rotation, then

Furthermore, if the elastic component has no acceleration, then

Inelastic FEM5

Recall that non-elastic behavior is characterized by acceleration of the elastic component (strain)...

And elastic behavior is characterized by constant velocity of strain.

loading x

e

Now Lagrange’s equation has been expanded.

Final Steps•Discretize using finite differences (rather than derivatives).•Write as a matrix times a vector of nodal coordinates (rather than a single mass point).•Solve for the object’s new set of positions of all nodes.

Elastic vs. Inelastic FEM4,5

Discretization of FEM4,5

Discretize Lagrange’s equation over all nodes

Procedure described in [4] but not [5]

Discretization of Elastic FEM4

Results of Elastic FEM4

Results of Elastic FEM4

Results of Elastic FEM4

3D plasticine bust of Victor Hugo.180 x 127 mesh; 68,580 equations.

Results of Inelastic FEM5

Results of Inelastic FEM5

Sphere pushing through 2D mesh.23 x 23 mesh; 1,587 equations.

Yield limit is uniform, causing linear tears.

Results of Inelastic FEM5

2D paper tearing by opposing forces.30 x 30 mesh; 2,700 equations.

Yield limit is perturbed stochastically,causing randomly-propogating tears.

References

0. David Baraff. Rigid Body Simulation. Physically Based Modeling, SIGGRAPH Course Notes, August 2001.

1. George Buchanan. Schaum’s Outlines: Finite Element Analysis. McGraw-Hill, 1995.

2. Peter Hunter and Andrew Pullan. FEM/BEM Notes. The University of Auckland, New Zealand, February 21 2001.

References3. Tom Lassanske. [Slides from class lecture]

4. Demetri Terzopoulost, John Platt, Alan Barr, and Kurt Fleischert. Elastically Deformable Models. Computer Graphics, Volume 21, Number 4, July 1987.

5. Demetri Terzopoulos and Kurrt Fleiseher. Modeling Inelastic Deformation: Viscoelasticity, Plasticity, Fracture. Computer Graphics, Volume 22, Number 4, August 1988

Notation