Curve Skeleton and Moments of Area Supported Beam Parametrization in Multi-Objective Compliance Structural Optimization

This work addresses the end-to-end virtual automation of structural optimization up to the derivation of a parametric geometry model that can be used for application areas such as additive manufacturing or the verification of the structural optimization result with the finite element method. A holistic design in structural optimization can be achieved with the weighted sum method, which can be automatically parameterized with curve skeletonization and cross-section regression to virtually verify the result and control the local size for additive manufacturing. is investigated in general. In this paper, a holistic design is understood as a design that considers various compliances as an objective function. This parameterization uses the automated determination of beam parameters by so-called curve skeletonization with subsequent cross-section shape parameter estimation based on moments of area, especially for multi-objective optimized shapes. An essential contribution is the linking of the parameterization with the results of the structural optimization, e.g., to include properties such as boundary conditions, load conditions, sensitivities or even density variables in the curve skeleton parameterization. The parameterization focuses on guiding the skeletonization based on the information provided by the optimization and the finite element model. In addition, the cross-section detection considers circular, elliptical, and tensor product spline cross-sections that can be applied to various shape descriptors such as convolutional surfaces, subdivision surfaces, or constructive solid geometry. The shape parameters of these cross-sections are estimated using stiffness distributions, moments of area of 2D images, and convolutional neural networks with a tailored loss function to moments of area. Each final geometry is designed by extruding the cross-section along the appropriate curve segment of the beam and joining it to other beams by using only unification operations. The focus of multi-objective structural optimization considering 1D, 2D and 3D elements is on cases that can be modeled using equations by the Poisson equation and linear elasticity. This enables the development of designs in application areas such as thermal conduction, electrostatics, magnetostatics, potential flow, linear elasticity and diffusion, which can be optimized in combination or individually. Due to the simplicity of the cases defined by the Poisson equation, no experts are required, so that many conceptual designs can be generated and reconstructed by ordinary users with little effort. Specifically for 1D elements, a element stiffness matrices for tensor product spline cross-sections are derived, which can be used to optimize a variety of lattice structures and automatically convert them into free-form surfaces. For 2D elements, non-local trigonometric interpolation functions are used, which should significantly increase interpretability of the density distribution. To further improve the optimization, a parameter-free mesh deformation is embedded so that the compliances can be further reduced by locally shifting the node positions. Finally, the proposed end-to-end optimization and parameterization is applied to verify a linear elasto-static optimization result for and to satisfy local size constraint for the manufacturing with selective laser melting of a heat transfer optimization result for a heat sink of a CPU. For the elasto-static case, the parameterization is adjusted until a certain criterion (displacement) is satisfied, while for the heat transfer case, the manufacturing constraints are satisfied by automatically changing the local size with the proposed parameterization. This heat sink is then manufactured without manual adjustment and experimentally validated to limit the temperature of a CPU to a certain level.:TABLE OF CONTENT III I LIST OF ABBREVIATIONS V II LIST OF SYMBOLS V III LIST OF FIGURES XIII IV LIST OF TABLES XVIII 1. INTRODUCTION 1 1.1 RESEARCH DESIGN AND MOTIVATION 6 1.2 RESEARCH THESES AND CHAPTER OVERVIEW 9 2. PRELIMINARIES OF TOPOLOGY OPTIMIZATION 12 2.1 MATERIAL INTERPOLATION 16 2.2 TOPOLOGY OPTIMIZATION WITH PARAMETER-FREE SHAPE OPTIMIZATION 17 2.3 MULTI-OBJECTIVE TOPOLOGY OPTIMIZATION WITH THE WEIGHTED SUM METHOD 18 3. SIMULTANEOUS SIZE, TOPOLOGY AND PARAMETER-FREE SHAPE OPTIMIZATION OF WIREFRAMES WITH B-SPLINE CROSS-SECTIONS 21 3.1 FUNDAMENTALS IN WIREFRAME OPTIMIZATION 22 3.2 SIZE AND TOPOLOGY OPTIMIZATION WITH PERIODIC B-SPLINE CROSS-SECTIONS 27 3.3 PARAMETER-FREE SHAPE OPTIMIZATION EMBEDDED IN SIZE OPTIMIZATION 32 3.4 WEIGHTED SUM SIZE AND TOPOLOGY OPTIMIZATION 36 3.5 CROSS-SECTION COMPARISON 39 4. NON-LOCAL TRIGONOMETRIC INTERPOLATION IN TOPOLOGY OPTIMIZATION 41 4.1 FUNDAMENTALS IN MATERIAL INTERPOLATIONS 43 4.2 NON-LOCAL TRIGONOMETRIC SHAPE FUNCTIONS 45 4.3 NON-LOCAL PARAMETER-FREE SHAPE OPTIMIZATION WITH TRIGONOMETRIC SHAPE FUNCTIONS 49 4.4 NON-LOCAL AND PARAMETER-FREE MULTI-OBJECTIVE TOPOLOGY OPTIMIZATION 54 5. FUNDAMENTALS IN SKELETON GUIDED SHAPE PARAMETRIZATION IN TOPOLOGY OPTIMIZATION 58 5.1 SKELETONIZATION IN TOPOLOGY OPTIMIZATION 61 5.2 CROSS-SECTION RECOGNITION FOR IMAGES 66 5.3 SUBDIVISION SURFACES 67 5.4 CONVOLUTIONAL SURFACES WITH META BALL KERNEL 71 5.5 CONSTRUCTIVE SOLID GEOMETRY 73 6. CURVE SKELETON GUIDED BEAM PARAMETRIZATION OF TOPOLOGY OPTIMIZATION RESULTS 75 6.1 FUNDAMENTALS IN SKELETON SUPPORTED RECONSTRUCTION 76 6.2 SUBDIVISION SURFACE PARAMETRIZATION WITH PERIODIC B-SPLINE CROSS-SECTIONS 78 6.3 CURVE SKELETONIZATION TAILORED TO TOPOLOGY OPTIMIZATION WITH PRE-PROCESSING 82 6.4 SURFACE RECONSTRUCTION USING LOCAL STIFFNESS DISTRIBUTION 86 7. CROSS-SECTION SHAPE PARAMETRIZATION FOR PERIODIC B-SPLINES 96 7.1 PRELIMINARIES IN B-SPLINE CONTROL GRID ESTIMATION 97 7.2 CROSS-SECTION EXTRACTION OF 2D IMAGES 101 7.3 TENSOR SPLINE PARAMETRIZATION WITH MOMENTS OF AREA 105 7.4 B-SPLINE PARAMETRIZATION WITH MOMENTS OF AREA GUIDED CONVOLUTIONAL NEURAL NETWORK 110 8. FULLY AUTOMATED COMPLIANCE OPTIMIZATION AND CURVE-SKELETON PARAMETRIZATION FOR A CPU HEAT SINK WITH SIZE CONTROL FOR SLM 115 8.1 AUTOMATED 1D THERMAL COMPLIANCE MINIMIZATION, CONSTRAINED SURFACE RECONSTRUCTION AND ADDITIVE MANUFACTURING 118 8.2 AUTOMATED 2D THERMAL COMPLIANCE MINIMIZATION, CONSTRAINT SURFACE RECONSTRUCTION AND ADDITIVE MANUFACTURING 120 8.3 USING THE HEAT SINK PROTOTYPES COOLING A CPU 123 9. CONCLUSION 127 10. OUTLOOK 131 LITERATURE 133 APPENDIX 147 A PREVIOUS STUDIES 147 B CROSS-SECTION PROPERTIES 149 C CASE STUDIES FOR THE CROSS-SECTION PARAMETRIZATION 155 D EXPERIMENTAL SETUP 15

Natural Parameterization

The objective of this project has been to develop an approach for imitating physical objects with an underlying stochastic variation. The key assumption is that a set of “natural parameters” can be extracted by a new subdivision algorithm so they reflect what is called the object’s “geometric DNA”. A case study on one hundred wheat grain crosssections (Triticum aestivum) showed that it was possible to extract thirty-six such parameters and to reuse them for Monte Carlo simulation of “new” stochastic phantoms which possessthe same stochastic behavior as the “original” cross-sections

Adaptation et transformation automatiques des résultats d’optimisation topologique en modèles CAO de structures de poutres

Les méthodes d’optimisation topologique sont de nos jours très populaires et intégrées dans plusieurs logiciels de conception. Elles munissent le concepteur d’un outil de choix pour l’obtention des formes optimisées en phase de conception. Cependant, l’une des principales limites de ces méthodes est l’interprétation du résultat de l’optimisation en un modèle facilement exploitable dans la suite du processus de conception. En effet, seulement un nombre limité d’approches ont été développées en vue de transformer un modèle optimisé en un modèle de CAO (Conception Assistée par Ordinateur). Bien qu’elles soient probantes à bien des égards, elles sont pour la plupart encore limitées aux modèles en 2D et sont semi ou non automatiques, ce qui fait que le concepteur est beaucoup mis à contribution durant l’interprétation du modèle optimisé. Dans cette recherche, une méthodologie d’interprétation d’un résultat d’optimisation topologique est proposée. La méthode d’optimisation utilisée est la méthode SIMP (Solid Isotropic Material with Penalization) qui donne comme résultat une répartition optimale de la matière dans le modèle. En considérant un résultat de la méthode SIMP qui s’oriente vers des structures composées de poutres, l’approche proposée compte deux étapes importantes que sont l’amélioration de la qualité du résultat de l’optimisation et la conversion en modèle CAO du modèle optimisé adapté qui en découle. Le produit est donc un modèle CAO plus facile à exploiter et à fabriquer. Ce dernier est finalement validé par une analyse multidimensionnelle par éléments finis. En plus d’être automatique, l’approche développée retourne des modèles CAO qui représentent bien la forme telle qu’optimisée. Nowadays, topology optimization methods are very popular and integrated into several computer-aided design (CAD) software. They provide the designer with a tool allowing to obtain optimized shapes in the design phase. However, one of the main limitations of these methods is the interpretation of optimization results into CAD models that can be easily used in subsequent design phases. Indeed, only a limited number of approaches have been developed in order to interpret raw optimization results into CAD models. Even if good results can be obtained with some of these methods, it is still limited to 2D models and these methods are semi- or non-automatic. Consequently, the interpretation process requires the designer’s intervention. In this work, a methodology for automatically interpreting three-dimensional topology optimization result into CAD models is proposed. The optimization method used is the SIMP (Solid Isotropic Material with Penalization) method, which results in an optimized distribution of material inside a given volume. Considering results of the SIMP method that tend towards beam-like structures, the proposed approach involves two main stages, which are quality improvement of the optimized result and conversion of the improved optimized shape into a CAD model of beam structure. Therefore, the outcome of this approach is a CAD model that is easier to use and to manufacture. The converted model is finally validated through a multidimensional finite element analysis (FEA). More than being fully automatic, the proposed method also produces CAD models that are good approximations of optimized shapes generated by the SIMP method