Image Matching Assessment of Attainable Topology via Kriging-Interpolated Level-Sets

2014 ◽  
Author(s):  
David Guirguis ◽  
Mohamed Aly ◽  
Karim Hamza ◽  
Hesham Hegazi
Keyword(s):  
Topology Optimization ◽  
Level Set ◽  
Image Matching ◽  
Level Sets ◽  
Optimization Techniques ◽  
Design Domain ◽  
Structural Topology ◽  
Level Set Function ◽  
Design Variables ◽  
Gradient Optimization

Level-set methods are domain classification techniques that are gaining popularity in the recent years for structural topology optimization. Level sets classify a domain into two or more categories (such as material and void) by examining the value of a scalar level-set function (LSF) defined in the entire design domain. In most level-set formulations, a large number of design variables, or degrees of freedom is used to define the LSF, which implicitly defines the structure. The large number of design variables makes non-gradient optimization techniques all but ineffective. Kriging-interpolated level sets (KLS) on the other hand are formulated with an objective to enable non-gradient optimization by defining the design variables as the LSF values at few select locations (knot points) and using a Kriging model to interpolate the LSF in the rest of the design domain. A downside of concern when adopting KLS, is that using too few knot points may limit the capability to represent complex shapes, while using too many knot points may cause difficulty for non-gradient optimization. This paper presents a study of the effect of number and layout of the knot points in KLS on the capability to represent complex topologies in single and multi-component structures. Image matching error metrics are employed to assess the degree of mismatch between target topologies and those best-attainable via KLS. Results are presented in a catalogue-style in order to facilitate appropriate selection of knot-points by designers wishing to apply KLS for topology optimization.

A Kriging-Interpolated Level-Set Approach for Structural Topology Optimization

2013 ◽  
Vol 136 (1) ◽  
Author(s):  
Karim Hamza ◽  
Mohamed Aly ◽  
Hesham Hegazi
Keyword(s):  
Topology Optimization ◽  
Level Set ◽  
Best Practice ◽  
Hybrid Genetic Algorithm ◽  
Mesh Size ◽  
Test Problems ◽  
Structural Topology Optimization ◽  
Structural Topology ◽  
Level Set Function ◽  
Design Variables

Level-set approaches are a family of domain classification techniques that rely on defining a scalar level-set function (LSF), then carrying out the classification based on the value of the function relative to one or more thresholds. Most continuum topology optimization formulations are at heart, a classification problem of the design domain into structural materials and void. As such, level-set approaches are gaining acceptance and popularity in structural topology optimization. In conventional level set approaches, finding an optimum LSF involves solution of a Hamilton-Jacobi system of partial differential equations with a large number of degrees of freedom, which in turn, cannot be accomplished without gradients information of the objective being optimized. A new approach is proposed in this paper where design variables are defined as the values of the LSF at knot points, then a Kriging model is used sto interpolate the LSF values within the rest of the domain so that classification into material or void can be performed. Perceived advantages of the Kriging-interpolated level-set (KLS) approach include alleviating the need for gradients of objectives and constraints, while maintaining a reasonable number of design variables that is independent from the mesh size. A hybrid genetic algorithm (GA) is then used for solving the optimization problem(s). An example problem of a short cantilever is studied under various settings of the KLS parameters in order to infer the best practice recommendations for tuning the approach. Capabilities of the approach are then further demonstrated by exploring its performance on several test problems.

An Explicit Level-Set Approach for Structural Topology Optimization

2013 ◽  
Author(s):  
Karim Hamza ◽  
Mohamed Aly ◽  
Hesham Hegazi
Keyword(s):  
Topology Optimization ◽  
Level Set ◽  
Best Practice ◽  
Hybrid Genetic Algorithm ◽  
Mesh Size ◽  
Test Problems ◽  
Structural Topology Optimization ◽  
Structural Topology ◽  
Level Set Function ◽  
Design Variables

Level-set approaches are a family of domain classification techniques that rely on defining a scalar level-set function (LSF), then carrying out the classification based on the value of the function relative to one or more thresholds. Most continuum topology optimization formulations are at heart, a classification problem of the design domain into structural materials and void. As such, level-set approaches are gaining acceptance and popularity in structural topology optimization. In conventional level set approaches, finding an optimum LSF involves solution of a Hamilton-Jacobi system of partial differential equations with a large number of degrees of freedom, which in turn, cannot be accomplished without gradients information of the objective being optimized. A new approach is proposed in this paper where design variables are defined as the explicit values of the LSF at knot points, then a Kriging model is used to interpolate the LSF values within the rest of the domain so that classification into material or void can be performed. Perceived advantages of the explicit level-set (ELS) approach include alleviating the need for gradients of objectives and constraints, while maintaining a reasonable number of design variables that is independent from the mesh size. A hybrid genetic algorithm (GA) is then used for solving the optimization problem(s). An example problem of a short cantilever is studied under various settings of the ELS parameters in order to infer the best practice recommendations for tuning the approach. Capabilities of the approach are then further demonstrated by exploring its performance on several test problems.

scholarly journals A local solution approach for level-set based structural topology optimization in isogeometric analysis

2020 ◽  
Vol 7 (4) ◽  
pp. 514-526
Author(s):  
Zijun Wu ◽  
Shuting Wang ◽  
Renbin Xiao ◽  
Lianqing Yu
Keyword(s):  
Topology Optimization ◽  
Level Set ◽  
Isogeometric Analysis ◽  
Iteration Process ◽  
Solid Material ◽  
Optimization Approach ◽  
Structural Topology ◽  
Zero Level ◽  
Solution Approach ◽  
Level Set Function

Abstract This paper develops a new topology optimization approach for minimal compliance problems based on the parameterized level set method in isogeometric analysis. Here, we choose the basis functions as level set functions. The design variables are obtained with Greville abscissae based on the corresponding collocation points. The zero-level set boundaries are derived from the level set function values of the interpolation points in all knot spans. In the optimization iteration process, the whole design domain is discretized into two types of subdomains around the zero-level set boundaries, undesign area with void materials and redesign domain with solid materials. To decrease the size of equations and the computational consumptions, only the solid material area is recalculated and the void material area is discarded according to the high accuracy of isogeometric analysis. Numerical examples demonstrate the validity of the proposed optimization method.

A Level Set Method With a Bounded Diffusion for Structural Topology Optimization

2018 ◽  
Vol 140 (7) ◽  
Author(s):  
Benliang Zhu ◽  
Rixin Wang ◽  
Hai Li ◽  
Xianmin Zhang
Keyword(s):  
Topology Optimization ◽  
Level Set ◽  
Jacobi Equation ◽  
Structural Topology ◽  
Signed Distance Function ◽  
Diffusion Term ◽  
Signed Distance ◽  
Level Set Function ◽  
Set Function ◽  
Hamilton Jacobi Equation

In level-set-based topology optimization methods, the spatial gradients of the level set field need to be controlled to avoid excessive flatness or steepness at the structural interfaces. One of the most commonly utilized methods is to generalize the traditional Hamilton−Jacobi equation by adding a diffusion term to control the level set function to remain close to a signed distance function near the structural boundaries. This study proposed a new diffusion term and built it into the Hamilton-Jacobi equation. This diffusion term serves two main purposes: (I) maintaining the level set function close to a signed distance function near the structural boundaries, thus avoiding periodic re-initialization, and (II) making the diffusive rate function to be a bounded function so that a relatively large time-step can be used to speed up the evolution of the level set function. A two-phase optimization algorithm is proposed to ensure the stability of the optimization process. The validity of the proposed method is numerically examined on several benchmark design problems in structural topology optimization.

Generative Design of Multi-Material Hierarchical Structures via Concurrent Topology Optimization and Conformal Geometry Method

2019 ◽  
Author(s):  
Long Jiang ◽  
Shikui Chen ◽  
Xianfeng David Gu
Keyword(s):  
Topology Optimization ◽  
Conformal Mapping ◽  
Level Set ◽  
Basis Function ◽  
Material Properties ◽  
Structural Topology Optimization ◽  
Structural Topology ◽  
Signed Distance ◽  
Level Set Function ◽  
Set Function

Abstract Topology optimization has been proved to be an automatic, efficient and powerful tool for structural designs. In recent years, the focus of structural topology optimization has evolved from mono-scale, single material structural designs to hierarchical multimaterial structural designs. In this research, the multi-material structural design is carried out in a concurrent parametric level set framework so that the structural topologies in the macroscale and the corresponding material properties in mesoscale can be optimized simultaneously. The constructed cardinal basis function (CBF) is utilized to parameterize the level set function. With CBF, the upper and lower bounds of the design variables can be identified explicitly, compared with the trial and error approach when the radial basis function (RBF) is used. In the macroscale, the ‘color’ level set is employed to model the multiple material phases, where different materials are represented using combined level set functions like mixing colors from primary colors. At the end of this optimization, the optimal material properties for different constructing materials will be identified. By using those optimal values as targets, a second structural topology optimization is carried out to determine the exact mesoscale metamaterial structural layout. In both the macroscale and the mesoscale structural topology optimization, an energy functional is utilized to regularize the level set function to be a distance-regularized level set function, where the level set function is maintained as a signed distance function along the design boundary and kept flat elsewhere. The signed distance slopes can ensure a steady and accurate material property interpolation from the level set model to the physical model. The flat surfaces can make it easier for the level set function to penetrate its zero level to create new holes. After obtaining both the macroscale structural layouts and the mesoscale metamaterial layouts, the hierarchical multimaterial structure is finalized via a local-shape-preserving conformal mapping to preserve the designed material properties. Unlike the conventional conformal mapping using the Ricci flow method where only four control points are utilized, in this research, a multi-control-point conformal mapping is utilized to be more flexible and adaptive in handling complex geometries. The conformally mapped multi-material hierarchical structure models can be directly used for additive manufacturing, concluding the entire process of designing, mapping, and manufacturing.

scholarly journals A Sequential Approach for Aerodynamic Shape Optimization with Topology Optimization of Airfoils

2021 ◽  
Vol 26 (2) ◽  
pp. 34
Author(s):  
Isaac Gibert Martínez ◽  
Frederico Afonso ◽  
Simão Rodrigues ◽  
Fernando Lau
Keyword(s):  
Topology Optimization ◽  
Shape Optimization ◽  
Volume Fraction ◽  
Optimization Techniques ◽  
Free Form ◽  
Aerodynamic Shape Optimization ◽  
Sequential Approach ◽  
Airfoil Surface ◽  
Aerodynamic Shape ◽  
Design Variables

The objective of this work is to study the coupling of two efficient optimization techniques, Aerodynamic Shape Optimization (ASO) and Topology Optimization (TO), in 2D airfoils. To achieve such goal two open-source codes, SU2 and Calculix, are employed for ASO and TO, respectively, using the Sequential Least SQuares Programming (SLSQP) and the Bi-directional Evolutionary Structural Optimization (BESO) algorithms; the latter is well-known for allowing the addition of material in the TO which constitutes, as far as our knowledge, a novelty for this kind of application. These codes are linked by means of a script capable of reading the geometry and pressure distribution obtained from the ASO and defining the boundary conditions to be applied in the TO. The Free-Form Deformation technique is chosen for the definition of the design variables to be used in the ASO, while the densities of the inner elements are defined as design variables of the TO. As a test case, a widely used benchmark transonic airfoil, the RAE2822, is chosen here with an internal geometric constraint to simulate the wing-box of a transonic wing. First, the two optimization procedures are tested separately to gain insight and then are run in a sequential way for two test cases with available experimental data: (i) Mach 0.729 at α=2.31°; and (ii) Mach 0.730 at α=2.79°. In the ASO problem, the lift is fixed and the drag is minimized; while in the TO problem, compliance minimization is set as the objective for a prescribed volume fraction. Improvements in both aerodynamic and structural performance are found, as expected: the ASO reduced the total pressure on the airfoil surface in order to minimize drag, which resulted in lower stress values experienced by the structure.

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