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Isogeometric Topology Optimization Based on Topological Derivatives

Guilherme Henrique Teixeira, Nepomuk Krenn, Peter Gangl, Benjamin Marussig

TL;DR

This work tackles topology optimization for 2D linear elasticity where remeshing is challenging. It combines isogeometric analysis with a level-set representation guided by topological derivatives, solving on a fixed domain D using an immersed formulation to update geometry without remeshing, and employing a Lagrangian-based derivative to derive an analytical expression for the topological sensitivity. A key contribution is the explicit topological-derivative expression and a robust update scheme via spherical linear interpolation, along with cut-element handling through accurate quadrature and smoothing. Numerical experiments on Cantilever and Quarter Ring geometries show that higher-degree basis functions for the state solution improve accuracy and convergence, while a linear level-set discretization often suffices to capture the topology, highlighting the method’s efficiency and robustness for isogeometric topology optimization without remeshing.

Abstract

Topology optimization is a valuable tool in engineering, facilitating the design of optimized structures. However, topological changes often require a remeshing step, which can become challenging. In this work, we propose an isogeometric approach to topology optimization driven by topological derivatives. The combination of a level-set method together with an immersed isogeometric framework allows seamless geometry updates without the necessity of remeshing. At the same time, topological derivatives provide topological modifications without the need to define initial holes [7]. We investigate the influence of higher-degree basis functions in both the level-set representation and the approximation of the solution. Two numerical examples demonstrate the proposed approach, showing that employing higher-degree basis functions for approximating the solution improves accuracy, while linear basis functions remain sufficient for the level-set function representation.

Isogeometric Topology Optimization Based on Topological Derivatives

TL;DR

This work tackles topology optimization for 2D linear elasticity where remeshing is challenging. It combines isogeometric analysis with a level-set representation guided by topological derivatives, solving on a fixed domain D using an immersed formulation to update geometry without remeshing, and employing a Lagrangian-based derivative to derive an analytical expression for the topological sensitivity. A key contribution is the explicit topological-derivative expression and a robust update scheme via spherical linear interpolation, along with cut-element handling through accurate quadrature and smoothing. Numerical experiments on Cantilever and Quarter Ring geometries show that higher-degree basis functions for the state solution improve accuracy and convergence, while a linear level-set discretization often suffices to capture the topology, highlighting the method’s efficiency and robustness for isogeometric topology optimization without remeshing.

Abstract

Topology optimization is a valuable tool in engineering, facilitating the design of optimized structures. However, topological changes often require a remeshing step, which can become challenging. In this work, we propose an isogeometric approach to topology optimization driven by topological derivatives. The combination of a level-set method together with an immersed isogeometric framework allows seamless geometry updates without the necessity of remeshing. At the same time, topological derivatives provide topological modifications without the need to define initial holes [7]. We investigate the influence of higher-degree basis functions in both the level-set representation and the approximation of the solution. Two numerical examples demonstrate the proposed approach, showing that employing higher-degree basis functions for approximating the solution improves accuracy, while linear basis functions remain sufficient for the level-set function representation.

Paper Structure

This paper contains 13 sections, 23 equations, 11 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Description
  3. Linear Elasticity Problem
  4. Immersed Isogeometric Approach
  5. Minimization Problem
  6. Discretized Level-set Representation
  7. Topological Derivatives
  8. Cut Elements
  9. Numerical Results
  10. Cantilever
  11. Quarter Ring
  12. Conclusion
  13. Acknowledgement

Figures (11)

  • Figure 1: Different types of optimization: a) Parameter Optimization; b) Shape Optimization; c) Topology Optimization
  • Figure 2: Representation of the domain problem: a) Domain $\Omega$ defined by the level-set; b) Domain $\Omega$ as a subset of domain $D$; c) Domain $\Omega$ inside of a B-spline background mesh defined from the knot vector of the geometry $D$
  • Figure 3: Distribution of the Greville abscissae on the elements for different polynomial degrees and basis functions defined by a) $\Xi=\{0 \:0 \: 0 \: 0.5 \: 1 \: 1 \: 1\}$. b) $\Xi=\{0 \: 0 \: 0 \: 0 \: 0.5 \: 1 \: 1 \: 1 \: 1\}$. c) $\Xi=\{0 \: 0 \: 0 \: 0 \: 0 \: 0.5 \: 1 \: 1 \: 1 \: 1 \: 1\}$
  • Figure 4: Type identification of the elements for assembling of the material property $\alpha$: a) Domain $D$ divided into two regions by a level-set function. b) Domains $D$ discretized as the background mesh. c) Identification of the elements. In yellow, the elements are located outside of $\Omega$. In blue, inside of $\Omega$ and in pink, the cut elements
  • Figure 5: Approaches to treat the cut elements
  • ...and 6 more figures