Topology Optimization With Load Uncertainty as an Inhomogeneous Eigenvalue Problem

2015 ◽  
Author(s):  
Wei Song ◽  
Euihark Lee ◽  
Hae Chang Gea ◽  
Limei Xu
Keyword(s):  
Topology Optimization ◽  
Eigenvalue Problem ◽  
Optimization Problem ◽  
External Loading ◽  
Worst Case ◽  
Load Uncertainty ◽  
Critical Loading ◽  
Structure Response ◽  
Gradient Based ◽  
Upper Level

In this paper, a non-probabilistic based topology optimization method under an external load uncertainty is presented. In traditional topology optimization problems, external loadings that apply to structures are always assumed as deterministic, but an external loading with uncertainties is very common in many practical engineering applications. In this paper, load uncertainty is described as an unknown-but-bounded model and the maximum possible strain energy based topology optimization formulation under an uncertain load is solved for the worst case condition. This optimization problem can be rewritten as a two-level optimization problem: the upper level optimization problem is a deterministic topology optimization under a critical loading of the worst structure response, and the lower level optimization problem is to determine the critical loading corresponding to the worst structure response. The challenge of the lower level optimization problem is on its non-convexity which makes many gradient based search methods ineffective. To overcome this issue, the lower level optimization problem is reformulated based on the KKT optimality conditions as an inhomogeneous eigenvalue problem and is solved for the critical loading corresponding to the worst structure response. After the worst loading case is identified, the upper level problem can be solved through the existing gradient based optimization algorithms. The effectiveness of the proposed topology optimization under unknown-but-bounded external loading uncertainty is demonstrated through a few numerical examples.

Topology Optimization With Unknown-but-Bounded Load Uncertainty

2014 ◽  
Author(s):  
Xike Zhao ◽  
Wei Song ◽  
Hae Chang Gea ◽  
Limei Xu
Keyword(s):  
Topology Optimization ◽  
Strain Energy ◽  
Optimization Problem ◽  
Solution Procedure ◽  
Convex Model ◽  
Load Uncertainty ◽  
Topology Optimization Problem ◽  
Structure Response ◽  
Gradient Based ◽  
Upper Level

In this paper, convex modeling based topology optimization with load uncertainty is presented. The load uncertainty is described using the non-probabilistic based unknown-but-bounded convex model, and the strain energy based topology optimization problem under uncertain loads is formulated. Unlike the conventional deterministic topology optimization problem, the maximum possible strain energy under uncertain loads is selected as the new objective in order to achieve a safe solution. Instead of obtaining approximated solutions as used before, an exact solution procedure is presented. The problem is first formulated as a single level optimization problem, and then rewritten as a two-level optimization problem. The upper level optimization problem is solved as a deterministic topology optimization with the load which generated from the worst structure response in the lower level problem. The lower level optimization problem is to identify this worst structure response, and it is found equivalent to an inhomogeneous eigenvalue problem. Three different cases are discussed for accurately evaluating the global optima of the lower level optimization problem, while the corresponding sensitivities are derived individually. With the function value and sensitivity information ready, the upper level optimization problem can be solved through existing gradient based optimization algorithms. The effectiveness of the proposed convex modeling based topology optimization is demonstrated through different numerical examples.

Topology Optimization Under Independent Multi-Load With Uncertainty

2016 ◽  
Author(s):  
Hae Chang Gea ◽  
Xing Liu ◽  
Euihark Lee ◽  
Limei Xu
Keyword(s):  
Topology Optimization ◽  
Optimization Problem ◽  
Optimization Method ◽  
Engineering Practice ◽  
Worst Case ◽  
Load Uncertainty ◽  
Topology Optimization Problem ◽  
Level Problem ◽  
Upper Level ◽  
Mean Compliance

In this paper, topology optimization under multiple independent loadings with uncertainty is presented. In engineering practice, load uncertainty can be found in many applications. From the literature, researchers have focused mainly on problems containing only a single uncertain external load. However, such idealistic problems may not be very useful in engineering practice. Problems involving multi-loadings with uncertainty are more commonly found in engineering applications. This paper presents a method to solve a system which contains multiple independent loadings with load uncertainty. First, a two-level optimization problem is formulated. The upper level problem is a typical topology optimization problem to minimize the mean compliance in the design using the worst case conditions. The lower level optimization problem is to solve for the worst loadings corresponding to the critical structure response. At the lower level formulation, an unknown-but-bounded model is used to define uncertain loadings. There are two challenges in finding the worst loading case: non-convexity and multi-loadings. The non-convexity problem is addressed by reformulating the problem as an inhomogeneous eigenvalue problem by applying the KKT optimality conditions and the multi-uncertain loadings problem is solved by an iterative method. After the worst loadings are generated, the upper level problem can be solved by a general topology optimization method. The effectiveness of the proposed method is demonstrated by numerical examples.

Non-Probabilistic Based Topology Optimization Under External Load Uncertainty With Eigenvalue-Superposition of Convex Models

2013 ◽  
Author(s):  
Xike Zhao ◽  
Hae Chang Gea ◽  
Wei Song
Keyword(s):  
Topology Optimization ◽  
External Load ◽  
Optimization Problems ◽  
Optimization Methods ◽  
Optimization Method ◽  
Convex Model ◽  
Structure Response ◽  
Gradient Based ◽  
Bounded Convex ◽  
Topology Optimization Method

In this paper the Eigenvalue-Superposition of Convex Models (ESCM) based topology optimization method for solving topology optimization problems under external load uncertainties is presented. The load uncertainties are formulated using the non-probabilistic based unknown-but-bounded convex model. The sensitivities are derived and the problem is solved using gradient based algorithm. The proposed ESCM based method yields the material distribution which would optimize the worst structure response under the uncertain loads. Comparing to the deterministic based topology optimization formulation the ESCM based method provided more reasonable solutions when load uncertainties were involved. The simplicity, efficiency and versatility of the proposed ESCM based topology optimization method can be considered as a supplement to the sophisticated reliability based topology optimization methods.

scholarly journals Topology optimization for fail-safe designs using moving morphable components as a representation of damage

2021 ◽  
Author(s):  
Hampus Hederberg ◽  
Carl-Johan Thore
Keyword(s):  
Topology Optimization ◽  
Structural Damage ◽  
Optimization Problem ◽  
Computational Cost ◽  
Independent Method ◽  
Element Mesh ◽  
Convex Problem ◽  
Gradient Based ◽  
Fail Safe ◽  
Moving Morphable Components

AbstractDesigns obtained with topology optimization (TO) are usually not safe against damage. In this paper, density-based TO is combined with a moving morphable component (MMC) representation of structural damage in an optimization problem for fail-safe designs. Damage is inflicted on the structure by an MMC which removes material, and the goal of the design problem is to minimize the compliance for the worst possible damage. The worst damage is sought by optimizing the position of the MMC to maximize the compliance for a given design. This non-convex problem is treated using a gradient-based solver by initializing the MMC at multiple locations and taking the maximum of the compliances obtained. The use of MMCs to model damage gives a finite element-mesh-independent method, and by allowing the components to move rather than remain at fixed locations, more robust structures are obtained. Numerical examples show that the proposed method can produce fail-safe designs with reasonable computational cost.

Efficient Sensitivity Analysis for Multibody Dynamics Systems Using an Iterative Steps Method With Application in Topology Optimization

2011 ◽  
Author(s):  
Guang Dong ◽  
Zheng-Dong Ma ◽  
Gregory Hulbert ◽  
Noboru Kikuchi ◽  
Sudhakar Arepally ◽  
...  
Keyword(s):  
Sensitivity Analysis ◽  
Topology Optimization ◽  
Multibody Dynamics ◽  
Optimization Problem ◽  
Finite Difference Methods ◽  
Analysis Method ◽  
Topology Optimization Problem ◽  
Design Variables ◽  
Sensitivity Analysis Method ◽  
Rotational Motions

Efficient and reliable sensitivity analyses are critical for topology optimization, especially for multibody dynamics systems, because of the large number of design variables and the complexities and expense in solving the state equations. This research addresses a general and efficient sensitivity analysis method for topology optimization with design objectives associated with time dependent dynamics responses of multibody dynamics systems that include nonlinear geometric effects associated with large translational and rotational motions. An iterative sensitivity analysis relation is proposed, based on typical finite difference methods for the differential algebraic equations (DAEs). These iterative equations can be simplified for specific cases to obtain more efficient sensitivity analysis methods. Since finite difference methods are general and widely used, the iterative sensitivity analysis is also applicable to various numerical solution approaches. The proposed sensitivity analysis method is demonstrated using a truss structure topology optimization problem with consideration of the dynamic response including large translational and rotational motions. The topology optimization problem of the general truss structure is formulated using the SIMP (Simply Isotropic Material with Penalization) assumption for the design variables associated with each truss member. It is shown that the proposed iterative steps sensitivity analysis method is both reliable and efficient.

Topology Optimization for the Single Phase Flow Using Two Point Flux Approximation Model

2013 ◽  
Author(s):  
Guang Dong ◽  
Yulan Song
Keyword(s):  
Porous Media ◽  
Topology Optimization ◽  
Optimization Problem ◽  
Single Phase ◽  
Flow In Porous Media ◽  
Phase Flow ◽  
Approximation Model ◽  
Single Phase Flow ◽  
Topology Optimization Problem ◽  
Flux Approximation

The topology optimization method is extended to solve a single phase flow in porous media optimization problem based on the Two Point Flux Approximation model. In particular, this paper discusses both strong form and matrix form equations for the flow in porous media. The design variables and design objective are well defined for this topology optimization problem, which is based on the Solid Isotropic Material with Penalization approach. The optimization problem is solved by the Generalized Sequential Approximate Optimization algorithm iteratively. To show the effectiveness of the topology optimization in solving the single phase flow in porous media, the examples of two-dimensional grid cell TPFA model with impermeable regions as constrains are presented in the numerical example section.

Iterative Parametric Approach for Quadratically Constrained Bi-Level Multiobjective Quadratic Fractional Programming

2020 ◽  
Vol 17 (11) ◽  
pp. 5046-5051
Author(s):  
Vandana Goyal ◽  
Namrata Rani ◽  
Deepak Gupta
Keyword(s):  
Fractional Programming ◽  
Optimization Problem ◽  
Programming Model ◽  
Pareto Optimal Solutions ◽  
Parametric Approach ◽  
Second Stage ◽  
Quadratic Fractional Programming ◽  
Upper Level ◽  
Quadratically Constrained ◽  
Constraint Method

The paper proposed an iterative parametric approach procedure for solving Bi-level Multiobjective Quadratic Fractional Programming model. The Model is divided into two levels-upper and lower. In the first stage of the approach, a set of pareto optimal solutions of upper Level is obtained by converting the problem into equivalent single non-fractional parametric objective optimization problem by using parametric vector and ε-constraint method. Then for the second stage, the solution of upper level is followed by the lower level decision maker while finding solution with the proposed algorithm to obtain the best preferred solution. A numerical example is solved in the last to validate the feasibility of the approach.

Finding Better Local Optima in Topology Optimization via Tunneling

2018 ◽  
Author(s):  
Shanglong Zhang ◽  
Julián A. Norato
Keyword(s):  
Topology Optimization ◽  
Optimization Problems ◽  
Structural Response ◽  
Free Form ◽  
Local Optima ◽  
Design Representation ◽  
Tunneling Method ◽  
Optimization Parameters ◽  
Gradient Based ◽  
Geometric Primitives

Topology optimization problems are typically non-convex, and as such, multiple local minima exist. Depending on the initial design, the type of optimization algorithm and the optimization parameters, gradient-based optimizers converge to one of those minima. Unfortunately, these minima can be highly suboptimal, particularly when the structural response is very non-linear or when multiple constraints are present. This issue is more pronounced in the topology optimization of geometric primitives, because the design representation is more compact and restricted than in free-form topology optimization. In this paper, we investigate the use of tunneling in topology optimization to move from a poor local minimum to a better one. The tunneling method used in this work is a gradient-based deterministic method that finds a better minimum than the previous one in a sequential manner. We demonstrate this approach via numerical examples and show that the coupling of the tunneling method with topology optimization leads to better designs.

scholarly journals Data-Driven Additive Manufacturing Constraints for Topology Optimization

2018 ◽  
Author(s):  
Benjamin M. Weiss ◽  
Joshua M. Hamel ◽  
Mark A. Ganter ◽  
Duane W. Storti
Keyword(s):  
Topology Optimization ◽  
Additive Manufacturing ◽  
Optimal Designs ◽  
Data Driven ◽  
Constraint Function ◽  
Manufacturing Constraint ◽  
Gradient Based ◽  
Fixed Radius ◽  
Different Shapes ◽  
Moving Morphable Components

The topology optimization (TO) of structures to be produced using additive manufacturing (AM) is explored using a data-driven constraint function that predicts the minimum producible size of small features in different shapes and orientations. This shape- and orientation-dependent manufacturing constraint, derived from experimental data, is implemented within a TO framework using a modified version of the Moving Morphable Components (MMC) approach. Because the analytic constraint function is fully differentiable, gradient-based optimization can be used. The MMC approach is extended in this work to include a “bootstrapping” step, which provides initial component layouts to the MMC algorithm based on intermediate Solid Isotropic Material with Penalization (SIMP) topology optimization results. This “bootstrapping” approach improves convergence compared to reference MMC implementations. Results from two compliance design optimization example problems demonstrate the successful integration of the manufacturability constraint in the MMC approach, and the optimal designs produced show minor changes in topology and shape compared to designs produced using fixed-radius filters in the traditional SIMP approach. The use of this data-driven manufacturability constraint makes it possible to take better advantage of the achievable complexity in additive manufacturing processes, while resulting in typical penalties to the design objective function of around only 2% when compared to the unconstrained case.

Multi-Component Topology Optimization for Powder Bed Additive Manufacturing (MTO-A)

2018 ◽  
Author(s):  
Yuqing Zhou ◽  
Tsuyoshi Nomura ◽  
Kazuhiro Saitou
Keyword(s):  
Topology Optimization ◽  
Additive Manufacturing ◽  
Simultaneous Optimization ◽  
Previous Attempt ◽  
Numerical Instability ◽  
Feature Size ◽  
Powder Bed ◽  
Nonlinear Projection ◽  
Gradient Based ◽  
Length Width

This paper presents a gradient-based multi-component topology optimization (MTO) method for structures assembled from components made by powder bed additive manufacturing. It is built upon our previous work on the continuously-relaxed MTO framework utilizing the concept of fractional component membership. The previous attempt on the integration of the relaxed MTO framework with additive manufacturing constraints, however, suffered from numerical instability for larger size problems, limiting its application to 2D low-resolution examples. To overcome this difficulty, this paper proposes an improved MTO formulation based on a design field regularization and a nonlinear projection of component membership variables, with a focus on powder bed additive manufacturing. For each component, constraints on the maximum allowable build volume (i.e., length, width, and height), the elimination of enclosed voids, and the minimum printable feature size are imposed during the simultaneous optimization of the overall base topology and component partitioning. The scalability of the new MTO formulation is demonstrated by a few 2D examples with much higher resolution than previously reported, and the first reported 3D example of MTO.

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