[Paper Review] On smooth or 0/1 designs of the fixed-mesh element-based topology optimization

Xiaodong Huang|arXiv (Cornell University)|Jun 8, 2020

Topology Optimization in Engineering39 references4 citations

TL;DR

This paper proposes a floating projection topology optimization (FPTO) method for fixed-mesh finite element analysis that enables both smooth and 0/1 designs in element-based topology optimization. By combining upper and lower bounds with a floating projection constraint, the method effectively simulates 0/1 design behavior without relying solely on material penalization, reducing compliance overestimation in boundary intermediate elements and enabling explicit structural topologies via the ersatz material model.

ABSTRACT

The traditional element-based topology optimization based on material penalization typically aims at a 0/1 design. Our numerical experiments reveal that the compliance of a smooth design is overestimated when material properties of boundary intermediate elements under the fixed-mesh finite element analysis are interpolated with a material penalization model. This paper proposes a floating projection topology optimization (FPTO) method for seeking a smooth design using the ersatz material model or a 0/1 design using a material penalization model. The proposed floating projection constraint combining with the upper and lower bounds heuristically simulates 0/1 constraints of design variables in the original discrete optimization problem. Numerical examples demonstrate the capability of the proposed element-based topology optimization approach in obtaining 0/1 or smooth designs for 2D and 3D compliance minimization problems. The proposed topology optimization approach can be easily implemented under the framework of the fixed-mesh finite element analysis and provides an alternative way to form explicit topologies of structures, especially when the ersatz material model is adopted.

Motivation & Objective

• To address the overestimation of compliance in smooth designs caused by material penalization in fixed-mesh finite element analysis.
• To develop a method that achieves either 0/1 or smooth designs without relying solely on material penalization.
• To provide a robust, easily implementable topology optimization framework under fixed-mesh FEA.
• To simulate 0/1 constraints in discrete optimization problems using heuristic bounds and projection.
• To enable explicit structural topologies, especially when using the ersatz material model.

Proposed method

• Proposes a floating projection constraint (FPTO) that combines upper and lower bounds to simulate 0/1 design constraints in the original discrete optimization problem.
• Uses the ersatz material model to represent intermediate elements with effective material properties, enabling smooth designs.
• Applies material penalization models to achieve 0/1 designs, with the FPTO constraint minimizing intermediate density artifacts.
• Implements the method within a fixed-mesh finite element analysis framework, avoiding mesh dependency.
• Employs a heuristic projection mechanism that dynamically adjusts design variable bounds during optimization.
• Integrates the FPTO constraint into standard element-based topology optimization algorithms for seamless implementation.

Experimental results

Research questions

• RQ1 Why does material penalization lead to overestimated compliance in smooth designs when using fixed-mesh FEA?
• RQ2 Can a single optimization framework produce both 0/1 and smooth designs without changing the core algorithm?
• RQ3 How effective is the floating projection constraint in simulating 0/1 behavior without relying on high penalization parameters?
• RQ4 What is the impact of the ersatz material model on the quality and manufacturability of smooth designs?
• RQ5 Can the proposed method produce explicit, interpretable topologies suitable for downstream manufacturing?

Key findings

• The FPTO method successfully reduces compliance overestimation in smooth designs by mitigating the influence of intermediate boundary elements under material penalization.
• The method achieves both 0/1 and smooth designs using the same framework, with the ersatz material model enabling clear, explicit topologies.
• Numerical examples in 2D and 3D demonstrate the method's robustness and effectiveness in compliance minimization problems.
• The floating projection constraint effectively simulates 0/1 constraints, reducing reliance on high penalization exponents that may cause convergence issues.
• The approach is easily implementable within standard fixed-mesh finite element analysis, enhancing compatibility with existing FEA software.
• The results show that the ersatz material model produces manufacturable, interpretable designs when used in conjunction with FPTO.