MMS Convergence Rate Calculation
MMS Convergence Rate Calculation: Theoretical Foundations
Overview
Professor! Today's topic is about calculating the MMS convergence order, right? What does that involve?
Evaluation of discretization error using the L2 norm and the method for calculating convergence order via log-log plot. Implementation bugs are detected by comparing with theoretical order.
So, if you cut corners on the discretization error part using the norm, you'll pay for it later. I'll keep that in mind!
Governing Equations
Discretization Method
How do you actually solve this equation on a computer?
We use spatial discretization by the Finite Element Method (FEM). Assemble the element stiffness matrix and construct the global stiffness equation.
Transform into the weak form (variational form) and use formulation by the Galerkin method using 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 on 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 perform the MMS convergence order calculation method?
| Tool Name | Developer/Current | 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
Do the origins of each software have some dramatic stories?
MSC Nastran / NX Nastran
Next is about MSC Nastran. What's the content?
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?
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 structural analysis talk is super interesting! Tell me more.
File Formats and Interoperability
Are there any points to note when transferring data between different software?
| Format | Extension | Type | Overview |
|---|---|---|---|
| STEP | .stp/.step | CAD geometry | International standard format for 3D model exchange between different CAD systems |
| IGES | .igs/.iges | CAD geometry | Initial Graphics Exchange Specification. An older format still widely used, but STEP is preferred for newer systems |
| STL | .stl | Triangular mesh | Stereolithography format. Used for surface representation as triangles. Common in 3D printing |
| Gmsh | .msh | Mesh file | Open-source mesh format. Compatible with many CAE solvers and post-processors |
| Nastran format | .bdf/.dat | FEM analysis file | Proprietary format for Nastran. Defines elements, materials, loads, and constraints |
| Abaqus Input | .inp | FEM analysis file | Text-based input format for Abaqus. Defines parts, material properties, assembly, steps, and loads |
| ANSYS CDB | .cdb | FEM analysis file | ANSYS binary database format. Stores complete model information including geometry and mesh |