Topology Optimization (SIMP Method)
Topology Optimization (SIMP Method): Theoretical Foundations
What is Topology Optimization?
Professor, what is topology optimization?
Topology optimization optimizes the presence or absence of material (0/1) within a design domain. It automatically determines where to place holes and where to leave material. Proposed by Bendsøe & Kikuchi in 1988.
SIMP Method
SIMP (Solid Isotropic Material with Penalization) is the most widely used topology optimization method. It assigns a design variable $\rho_e$ (density from 0 to 1) to each element:
The penalty exponent $p$ (typically $p = 3$) suppresses intermediate densities, pushing them towards 0/1.
Optimization Problem
Typical formulation:
"Find the stiffest structure while keeping the material below $V^*$," right?
Exactly. FEM calculates displacement for each iteration → calculates sensitivity (change in objective function when each element's density is changed) → updates density → iterates until convergence.
Summary
- Optimizes material placement within the design domain — Automatically determines hole position/shape
- SIMP method: $E_e = \rho_e^p E_0$ — Density method. $p = 3$ is standard
- Compliance minimization + Volume constraint — The most basic formulation
- OptiStruct, Abaqus TOSCA, Ansys — Commercial implementations
The "SIMP" in SIMP method was named by Bendsoe (1989)
The representative topology optimization method SIMP (Solid Isotropic Material with Penalization) is a method formulated by Bendsoe (1989) by simplifying the homogenization method of Bendsoe & Kikuchi (1988). By expressing the density ρ of each element as a continuous variable from 0 to 1 and stiffness as Eρ^p, the penalty parameter p suppresses intermediate densities, yielding a clear material distribution that is almost 0 or 1. The origin of the name comes from Bendsoe writing "Solid Isotropic..." in the title of his 1989 paper, which later became an acronym.
Computational Methods for Topology Optimization (SIMP Method)
SIMP Method Algorithm
1. Set initial density for all elements to $\rho = V^*/V_{total}$
2. Calculate displacement and stress using FEM
3. Calculate sensitivity $\partial C / \partial \rho_e$ (Adjoint method)
4. Update density (OC method or MMA method)
5. Iterate until convergence (typically 50–200 iterations)
Solvers
- OptiStruct (Altair) — Industry standard for topology optimization
- Abaqus TOSCA — Integrated with Abaqus
- Ansys Topology Optimization — Built into Workbench
- TopOpt (Technical University of Denmark) — Open-source MATLAB code
Summary
- Iteration of FEM + sensitivity calculation + density update — 50–200 iterations
- OptiStruct is the industry standard — Automotive/Aerospace
- TopOpt — Free MATLAB code. Ideal for education/research
SIMP without density filtering produces a checkerboard pattern
It has long been known that running SIMP topology optimization without density filtering causes a numerical pathology called the "checkerboard pattern", where adjacent elements alternately become ρ=0/1. The Helmholtz PDE (partial differential equation) filter proposed by Bourdin (2001) naturally controls the minimum member size (rmin) and is now incorporated into the standard implementations of OptiStruct, Tosim, and ABAQUS. Setting rmin in correspondence with manufacturing constraints (minimum wall thickness, draft angle) allows simultaneous management of design aesthetics and manufacturability.
Topology Optimization (SIMP Method) in Practice
Topology Optimization in Practice
Automotive lightweighting (brackets, suspension arms), aerospace (structural parts), 3D printing (freeform shapes).
Practical Checklist
- [ ] Are design and non-design domains correctly defined?
- [ ] Do load and boundary conditions include all cases (multiple load cases)?
- [ ] Is the volume constraint appropriate? (typically 30–50%)
- [ ] Is the optimization result clear 0/1? (few gray elements?)
- [ ] Were manufacturing constraints included? (minimum thickness, symmetry, draft direction)
- [ ] Was the optimization result converted to CAD and verification FEM performed?
The Airbus A380 wing attachment bracket is a masterpiece of SIMP optimization
The Airbus A380 cabin ceiling panel attachment bracket (first flight 2006) is famous in the industry as a part designed using SIMP topology optimization with OptiStruct. It achieved a 30% weight reduction compared to the conventional manually designed part while meeting fatigue life constraints, winning Altair's Engineering Impact Award. OptiStruct is now used as the standard topology optimization tool across all Airbus aircraft models, with over 1000 part optimizations conducted annually using this tool.
Topology Optimization (SIMP Method): Software & Solver Comparison
Topology Optimization Tools
- OptiStruct (Altair) — Industry standard. Supports topology, shape, and size optimization. Widespread in automotive and aerospace
- Abaqus TOSCA (Dassault Systemes) — Integrated with Abaqus. Good for nonlinear optimization
- Ansys Topology Optimization (Ansys Workbench) — Built-in topology optimization. Good integration with structural analysis
- Hyperstudy (Altair) — Multi-physics optimization platform. Supports parametric and topology optimization
- TopOpt (Technical University of Denmark) — Open-source MATLAB code. Ideal for education and research. Simple 88-line code available
- Inspire (Altair) — Lightweight design software. Focuses on concept-level topology optimization with quick visualization