MMS: 2次元弾性体
Theory and Physics
Overview
Professor! Today's topic is MMS: 2D Elastic Body, right? What is it about?
MMS for the Navier's equations of a 2D elastic body. Derive the body force term from the manufactured solution and verify the convergence order of stress and displacement.
Governing Equations
Discretization Method
How do you actually solve this equation on a computer?
We use spatial discretization by the Finite Element Method (FEM). We assemble the element stiffness matrix and construct the global stiffness equation.
We transform it into the weak form (variational form) and use the Galerkin method formulation with test functions and shape functions. The choice of element type (low-order elements vs. higher-order elements, full integration vs. reduced integration) directly affects the trade-off between solution accuracy and computational cost.
Matrix Solution Algorithm
What exactly do you mean by matrix solution algorithm?
Solve the simultaneous equations using direct methods (LU decomposition, Cholesky decomposition) or iterative methods (CG method, GMRES method). Preconditioned iterative methods are effective for large-scale problems.
| Solver | Classification | Memory Usage | Applicable Scale |
|---|---|---|---|
| LU decomposition | Direct Method | O(n²) | Small to Medium Scale |
| Cholesky decomposition | Direct Method (Symmetric Positive Definite) | O(n²) | Small to Medium Scale |
| PCG Method | Iterative Method | O(n) | Large Scale |
| GMRES method | Iterative Method | O(n·m) | Large Scale / Non-symmetric |
| AMG Preconditioner | Preprocessing | O(n) | Very Large Scale |
So, if you cut corners in the finite element method part, you'll pay for it later. I'll keep that in mind!
Implementation in Commercial Tools
So, what software can be used to do MMS: 2D Elastic Body?
| Tool Name | Developer/Current Status | Main File Formats |
|---|---|---|
| MSC Nastran / NX Nastran | MSC Nastran (Hexagon), NX Nastran (Siemens Digital Industries Software) | .bdf, .dat, .f06, .op2, .pch |
| Abaqus FEA (SIMULIA) | Dassault Systèmes SIMULIA | .inp, .odb, .cae, .sta, .msg |
| Ansys Mechanical (formerly ANSYS Structural) | Ansys Inc. | .cdb, .rst, .db, .ans, .mac |
| Ansys Fluent | Ansys Inc. | .cas, .dat, .msh, .jou |
| Simcenter STAR-CCM+ | Siemens Digital Industries Software | .sim, .java, .csv |
| COMSOL Multiphysics | COMSOL AB | .mph |
| OpenFOAM | Open Source (OpenCFD/ESI, OpenFOAM Foundation) | Dictionary files (blockMeshDict, etc.), .foam |
Vendor Lineage and Product Integration History
Is the background of each software quite dramatic?
MSC Nastran / NX Nastran
Next is about MSC Nastran. What's the story?
Developed in the 1960s as NASA Structural Analysis (NASTRAN). Commercialized by MSC Software, later UGS (now Siemens) derived NX Nastran. MSC was acquired by Hexagon AB in 2017.
Current Affiliation: MSC Nastran (Hexagon), NX Nastran (Siemens Digital Industries Software)
Abaqus FEA (SIMULIA)
What exactly is Abaqus FEA about?
Developed in 1978 by HKS (Hibbitt, Karlsson & Sorensen). Acquired by Dassault Systèmes in 2005 and integrated into the SIMULIA brand.
Current Affiliation: Dassault Systèmes SIMULIA
Wait, wait, structural analysis... so can it also be used for cases like this?
Ansys Mechanical (formerly ANSYS Structural)
Tell me about "Ansys Mechanical"!
Developed in 1970 by Swanson Analysis Systems Inc. (SASI). Based on APDL (Ansys Parametric Design Language).
Current Affiliation: Ansys Inc.
Wow, the story about structural analysis is super interesting! Tell me more.
File Formats and Interoperability
Are there any points to note when transferring data between different software?