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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