1 1 M Müller – Elasto-Plastic FEM ETH Zurich Interactive Simulation of Elasto-Plastic Materials using the Finite Element Method Matthias Müller Seminar – Wintersemester 02/03 Movie 2 M Müller – Elasto-Plastic FEM ETH Zurich Outline FEM vs. Mass-Spring Stiffness The Stiffness Matrix Static/Dynamic Deformation Continuum Mechanics and FEM Strain and Stress Tensors Continuous PDE’s FEM Discretization Plasticity Plastic Strain Update Rules Fracture Principal Stresses Crack Computation 3 M Müller – Elasto-Plastic FEM ETH Zurich Mass-Spring vs. FEM 1. Discretization of an object into mass points 2. Representation of forces between mass points with springs 3. Computation of the dynamics deformable mass-spring system 1. Discretization of an object into elements (tetrahedra) 2. Discretization of continuous energy equations into algebraic equations for forces acting at vertices 3. Computation of the dynamics deformable FEM system 4 M Müller – Elasto-Plastic FEM ETH Zurich Pros and Cons of FEM Pros: • No individual spring constants needed (only 2 known material parameters E,ν) • No inversion problems (inverted tetrahedra produce forces) • Stress and strain tensors allow • fracture and • plasticity simulations Cons: • (Pre-)compute stiffness matrix • Store stiffness matrix (3x3) per edge • Store original and actual positions of vertices
11
Embed
Finite Element Method Elasto-Plastic Materials using the ... · Finite Element Method Matthias Müller Seminar – Wintersemester 02/03 Movie 2 M Müller – Elasto-Plastic FEM ETH
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
1
1M Müller – Elasto-Plastic FEM
ETH Zurich
Interactive Simulation of Elasto-Plastic Materials using the Finite Element Method
Matthias MüllerSeminar – Wintersemester 02/03
Movie
2M Müller – Elasto-Plastic FEM
ETH ZurichOutline
FEM vs. Mass-SpringStiffness
The Stiffness MatrixStatic/Dynamic Deformation
Continuum Mechanics and FEMStrain and Stress TensorsContinuous PDE’sFEM Discretization
PlasticityPlastic StrainUpdate Rules
FracturePrincipal StressesCrack Computation
3M Müller – Elasto-Plastic FEM
ETH ZurichMass-Spring vs. FEM
1. Discretization of an object into mass points
2. Representation of forces between mass points with springs
3. Computation of the dynamics
deformable mass-spring system
1. Discretization of an object into elements (tetrahedra)
2. Discretization of continuousenergy equations into algebraic equations for forces acting at vertices
3. Computation of the dynamics
deformable FEM system
4M Müller – Elasto-Plastic FEM
ETH ZurichPros and Cons of FEM
Pros:• No individual spring constants needed
(only 2 known material parameters E,ν)• No inversion problems
(inverted tetrahedra produce forces)• Stress and strain tensors allow
• fracture and• plasticity simulations
Cons:• (Pre-)compute stiffness matrix• Store stiffness matrix (3x3) per edge• Store original and actual positions of vertices
2
5M Müller – Elasto-Plastic FEM
ETH ZurichOutline
FEM vs. Mass-SpringStiffness
The Stiffness MatrixStatic/Dynamic Deformation
Continuum Mechanics and FEMStrain and Stress TensorsContinuous PDE’sFEM Discretization
PlasticityPlastic StrainUpdate Rules
FracturePrincipal StressesCrack Computation
6M Müller – Elasto-Plastic FEM
ETH ZurichOne-dimensional Spring
f
∆x
f = k · ∆x
7M Müller – Elasto-Plastic FEM
ETH ZurichThree-dimensional Object
Finite Element Mesh• 903 tetrahedra• 393 vertices• 3 x 393 = 1179 dof.