4MM10 - Advanced computational continuum mechanics
Content
Computational continuum mechanics is nowadays an essential component of advanced computer-aided design. Experimental testing of prototypes is increasingly being replaced by nonlinear finite element simulations (so-called virtual prototyping and digital twinning) because this provides a more rapid and less expensive way to evaluate design concepts. In most companies nonlinear finite element software packages are used as a simulation black box. However, a nonlinear finite element analysis confronts an engineer with many choices and pitfalls. Selections made in the model formulation, mesh discretizations, element types, the solution procedure etc. may have a tremendous effect on the final results. Therefore, an engineer must be aware of these influences in order to provide an adequate interpretation of the results.
This course will focus on the basic aspects of the nonlinear finite element analysis for solids and fluids. The following topics will be discussed and put into practice:
Continuum Mechanics
- Lagrangian versus Eulerian description of continuum
- kinematics
- stresses
- balance laws
- fundamental concepts of constitutive equations
- non-linear elasticity
Non-linear finite element method for solids
- linear finite element method
- geometrically non-linear finite element formulation
- Newton-Raphson method for non-linear equations
- Total Lagrange formulation
- Updated Lagrange formulation
Numerical methods for fluids
- weak formulation for viscous fluids
- velocity-pressure formulation for Navier-Stokes
- Picard and Newton-Raphson method for generalized Newtonian fluids
- Arbitrary Lagrangian-Eulerian (ALE) formulation
- free surfaces鈥�
Objectives
- Distinguish the difference between Eulerian and Lagrangian description of continuum
- Understand the large deformation kinematics that governs large shape changes and large rotations in materials. Properly use, calculate and interpret large deformation tensors (deformation gradient tensor, strain tensors, deformation rate tensor, velocity gradient tensor, spin tensor)
- Understand the different large deformation stress tensors and their physical meaning. Properly use, calculate and interpret the Cauchy stress tensor, first Piola-Kirchhof stress tensor, 2nd Piola-Kirchhoff stress tensor, and the corresponding traction vectors.
- Understand and apply the different balance laws, governing the equilibrium of a continuum: mass balance, balance of momentum, balance of moment of momentum
- Understand and exploit the fundamental concepts of constitutive equations: principle of determinism, principle of local action; objectivity.
- Understand the difference between hyperelastic and hypo-elastic materials; understand the basic concepts underlying hyperelastic materials and their deformation behaviour
- Identify deficiencies of the linear finite element analysis when applied to geometrically non-linear problems
- Distinguish between strong and weak formulations and be able to derive one from another
- Understand, apply and implement in Matlab code Newton-Raphson iterative method for solution of non-linear equations
- Be able to linearize a geometrically non-linear quasi-static elastic problem
- Understand the difference and derive the Total Lagrange and Updated Lagrange formulations for a geometrically non-linear problem
- Implement in Matlab code a finite element program for solution of geometrically non-linear quasi-static elastic problem
- Be able to test a finite element code and adequately interpret the results of a non-linear simulation
- Derive the weak formulation for viscous fluids (Stokes equations).
- Understand and apply Picard and Newton-Raphson method for generalized Newtonian fluids
- Adapt and use a Matlab code for finite elements in fluids.
- Derive and apply the velocity-pressure formulation for the Navier-Stokes equations
- Understand the Arbitrary Lagrangian-Eulerian (ALE) formulation
- Understand treatment of free surfaces