Topology Optimization With Many Right-Hand Sides Using Mirror Descent Stochastic Approximation—Reduction From Many to a Single Sample

2020 ◽  
Vol 87 (5) ◽  
Author(s):  
Xiaojia Shelly Zhang ◽  
Eric de Sturler ◽  
Alexander Shapiro
Keyword(s):  
Topology Optimization ◽  
Linear Systems ◽  
Stochastic Approximation ◽  
Optimization Problems ◽  
Computational Cost ◽  
Single Sample ◽  
Step Size ◽  
Design Quality ◽  
Mirror Descent ◽  
Engineering Designs

Abstract Practical engineering designs typically involve many load cases. For topology optimization with many deterministic load cases, a large number of linear systems of equations must be solved at each optimization step, leading to an enormous computational cost. To address this challenge, we propose a mirror descent stochastic approximation (MD-SA) framework with various step size strategies to solve topology optimization problems with many load cases. We reformulate the deterministic objective function and gradient into stochastic ones through randomization, derive the MD-SA update, and develop algorithmic strategies. The proposed MD-SA algorithm requires only low accuracy in the stochastic gradient and thus uses only a single sample per optimization step (i.e., the sample size is always one). As a result, we reduce the number of linear systems to solve per step from hundreds to one, which drastically reduces the total computational cost, while maintaining a similar design quality. For example, for one of the design problems, the total number of linear systems to solve and wall clock time are reduced by factors of 223 and 22, respectively.

Convergence analysis of online learning algorithm with two-stage step size

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Weilin Nie ◽  
Cheng Wang
Keyword(s):  
Online Learning ◽  
Optimization Problems ◽  
Learning Algorithm ◽  
Computational Cost ◽  
Reproducing Kernels ◽  
Step Size ◽  
Two Stage ◽  
Logarithmic Term ◽  
Convergence Performance ◽  
Online Learning Algorithm

Abstract Online learning is a classical algorithm for optimization problems. Due to its low computational cost, it has been widely used in many aspects of machine learning and statistical learning. Its convergence performance depends heavily on the step size. In this paper, a two-stage step size is proposed for the unregularized online learning algorithm, based on reproducing Kernels. Theoretically, we prove that, such an algorithm can achieve a nearly min–max convergence rate, up to some logarithmic term, without any capacity condition.

scholarly journals An Efficient and High-Resolution Topology Optimization Method Based on Convolutional Neural Networks

2019 ◽  
Author(s):  
Liang Xue ◽  
Jie Liu ◽  
Guilin Wen ◽  
Hongxin Wang
Keyword(s):  
Topology Optimization ◽  
High Resolution ◽  
High Efficiency ◽  
Optimization Problems ◽  
Combined Treatment ◽  
Computational Cost ◽  
Optimization Method ◽  
Element Analysis ◽  
Arbitrary Boundary ◽  
Topology Optimization Method

Topology optimization is a pioneering design method that can provide various candidates with high mechanical properties. However, the high-resolution for the optimum structures is highly desired, normally in turn leading to computationally intractable puzzle, especially for the famous Solid Isotropic Material with Penalization (SIMP) method. In this paper, an efficient and high-resolution topology optimization method is proposed based on the Super-Resolution Convolutional Neural Network (SRCNN) technique in the framework of SIMP. The SRCNN includes four processes, i.e. refining, path extraction & representation, non-linear mapping, and reconstruction. The high computational efficiency is achieved by a pooling strategy, which can balance the number of finite element analysis (FEA) and the output mesh in optimization process. To further reduce the high computational cost of 3D topology optimization problems, a combined treatment method using 2D SRCNN is built as another speeding-up strategy. A number of typical examples justify that the high-resolution topology optimization method adopting SRCNN has excellent applicability and high efficiency for 2D and 3D problems with arbitrary boundary conditions, any design domain shape, and varied load.

scholarly journals Self-Directed Online Machine Learning for Topology Optimization

2021 ◽  
Author(s):  
Changyu Deng ◽  
Yizhou Wang ◽  
Can Qin ◽  
Wei Lu
Keyword(s):  
Topology Optimization ◽  
Optimization Problems ◽  
State Of The Art ◽  
Computational Cost ◽  
Region Of Interest ◽  
Global Optimum ◽  
Training Data ◽  
Computational Time ◽  
Minimization Problems ◽  
Design Variables

Abstract Topology optimization by optimally distributing materials in a given domain requires gradient-free optimizers to solve highly complicated problems. However, with hundreds of design variables or more involved, solving such problems would require millions of Finite Element Method (FEM) calculations whose computational cost is huge and impractical. Here we report a Self-directed Online Learning Optimization (SOLO) which integrates Deep Neural Network (DNN) with FEM calculations. A DNN learns and substitutes the objective as a function of design variables. A small number of training data is generated dynamically based on the DNN's prediction of the global optimum. The DNN adapts to the new training data and gives better prediction in the region of interest until convergence. Our algorithm was tested by compliance minimization problems and fluid-structure optimization problems. It reduced the computational time by 2 ~ 5 orders of magnitude compared with directly using heuristic methods, and outperformed all state-of-the-art algorithms tested in our experiments. This approach enables solving large multi-dimensional optimization problems.

SIMULATION-BASED OPTIMIZATION BY NEW STOCHASTIC APPROXIMATION ALGORITHM

2014 ◽  
Vol 31 (04) ◽  
pp. 1450026 ◽  
Author(s):  
ZI XU ◽  
YINGYING LI ◽  
XINGFANG ZHAO
Keyword(s):  
Approximation Algorithm ◽  
Stochastic Approximation ◽  
Optimization Problems ◽  
Nonlinear Function ◽  
Almost Sure Convergence ◽  
Inventory Problem ◽  
Step Size ◽  
Stochastic Approximation Algorithm ◽  
Simulation Based ◽  
Simulation Based Optimization

This paper proposes one new stochastic approximation algorithm for solving simulation-based optimization problems. It employs a weighted combination of two independent current noisy gradient measurements as the iterative direction. It can be regarded as a stochastic approximation algorithm with a special matrix step size. The almost sure convergence and the asymptotic rate of convergence of the new algorithm are established. Our numerical experiments show that it outperforms the classical Robbins–Monro (RM) algorithm and several other existing algorithms for one noisy nonlinear function minimization problem, several unconstrained optimization problems and one typical simulation-based optimization problem, i.e., (s, S)-inventory problem.

scholarly journals Efficient, high-resolution topology optimization method based on convolutional neural networks

2021 ◽  
Author(s):  
Liang Xue ◽  
Jie Liu ◽  
Guilin Wen ◽  
Hongxin Wang
Keyword(s):  
Topology Optimization ◽  
High Resolution ◽  
High Efficiency ◽  
Optimization Problems ◽  
Combined Treatment ◽  
Computational Cost ◽  
Treatment Method ◽  
Optimization Method ◽  
Arbitrary Boundary ◽  
Topology Optimization Method

AbstractTopology optimization is a pioneer design method that can provide various candidates with high mechanical properties. However, high resolution is desired for optimum structures, but it normally leads to a computationally intractable puzzle, especially for the solid isotropic material with penalization (SIMP) method. In this study, an efficient, high-resolution topology optimization method is developed based on the superresolution convolutional neural network (SRCNN) technique in the framework of SIMP. SRCNN involves four processes, namely, refinement, path extraction and representation, nonlinear mapping, and image reconstruction. High computational efficiency is achieved with a pooling strategy that can balance the number of finite element analyses and the output mesh in the optimization process. A combined treatment method that uses 2D SRCNN is built as another speed-up strategy to reduce the high computational cost and memory requirements for 3D topology optimization problems. Typical examples show that the high-resolution topology optimization method using SRCNN demonstrates excellent applicability and high efficiency when used for 2D and 3D problems with arbitrary boundary conditions, any design domain shape, and varied load.

h-adaptive topology optimization considering variations of material properties and energy error density recovery

2020 ◽  
Vol 37 (9) ◽  
pp. 3209-3241
Author(s):  
Jéderson da Silva ◽  
Jucélio Tomás Pereira ◽  
Diego Amadeu F. Torres
Keyword(s):  
Topology Optimization ◽  
Material Properties ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Computational Cost ◽  
Error Estimator ◽  
Content Type ◽  
Energy Error ◽  
Error Density ◽  
Discretization Errors

Purpose The purpose of this paper is to propose a new scheme for obtaining acceptable solutions for problems of continuum topology optimization of structures, regarding the distribution and limitation of discretization errors by considering h-adaptivity. Design/methodology/approach The new scheme encompasses, simultaneously, the solution of the optimization problem considering a solid isotropic microstructure with penalization (SIMP) and the application of the h-adaptive finite element method. An analysis of discretization errors is carried out using an a posteriori error estimator based on both the recovery and the abrupt variation of material properties. The estimate of new element sizes is computed by a new h-adaptive technique named “Isotropic Error Density Recovery”, which is based on the construction of the strain energy error density function together with the analytical solution of an optimization problem at the element level. Findings Two-dimensional numerical examples, regarding minimization of the structure compliance and constraint over the material volume, demonstrate the capacity of the methodology in controlling and equidistributing discretization errors, as well as obtaining a great definition of the void–material interface, thanks to the h-adaptivity, when compared with results obtained by other methods based on microstructure. Originality/value This paper presents a new technique to design a mesh made with isotropic triangular finite elements. Furthermore, this technique is applied to continuum topology optimization problems using a new iterative scheme to obtain solutions with controlled discretization errors, measured in terms of the energy norm, and a great resolution of the material boundary. Regarding the computational cost in terms of degrees of freedom, the present scheme provides approximations with considerable less error if compared to the optimization process on fixed meshes.

scholarly journals CARMA—Cellular Automata with Refined Mesh Adaptation—The Easy Way of Generation of Structural Topologies

2020 ◽  
Vol 10 (11) ◽  
pp. 3691
Author(s):  
Katarzyna Tajs-Zielińska ◽  
Bogdan Bochenek
Keyword(s):  
Topology Optimization ◽  
Cellular Automata ◽  
Optimization Problems ◽  
Computational Cost ◽  
Mesh Adaptation ◽  
Structural Topology ◽  
Fine Grid ◽  
Fine Mesh ◽  
Design Variables ◽  
Resolution Level

This paper is focused on the development of a Cellular Automata algorithm with the refined mesh adaptation technique and the implementation of this algorithm in topology optimization problems. Traditionally, a Cellular Automaton is created based on regular discretization of the design domain into a lattice of cells, the states of which are updated by applying simple local rules. It is expected that during the topology optimization process the local rules responsible for the evaluation of cell states can drive the solution to solid/void resulting structures. In the proposed approach, the finite elements are equivalent to cells of an automaton and the states of cells are represented by design variables. While optimizing engineering structural elements, the important issue is to obtain well-defined solutions: in particular, topologies with smooth boundaries. The quality of the structural topology boundaries depends on the resolution level of mesh discretization: the greater the number of elements in the mesh, the better the representation of the optimized structure. However, the use of fine meshes implies a high computational cost. We propose, therefore, an adaptive way to refine the mesh. This allowed us to reduce the number of design variables without losing the accuracy of results and without an excessive increase in the number of elements caused by use of a fine mesh for a whole structure. In particular, it is not necessary to cover void regions with a very fine mesh. The implementation of a fine grid is expected mainly in the so-called grey regions where it has to be decided whether a cell becomes solid or void. The benefit of the proposed approach, besides the possibility of obtaining high-resolution, sharply resolved fine optimal topologies with a relatively low computational cost, is also that the checkerboard effect, mesh dependency, and the so-called grey areas can be eliminated without using any additional filtering. Moreover, the algorithm presented is versatile, which allows its easy combination with any structural analysis solver built on the finite element method.

Stress-Constrained Topology Optimization Using Approximate Reanalysis with on-the-Fly Reduced Order Modeling

2021 ◽  
Author(s):  
Manyu Xiao ◽  
Jun Ma ◽  
Dongcheng Lu ◽  
Balaji Raghavan ◽  
Weihong Zhang
Keyword(s):  
Topology Optimization ◽  
Large Scale ◽  
Optimization Problems ◽  
Computational Cost ◽  
Coagulation Factor ◽  
Stress Constraints ◽  
Adjoint Equations ◽  
Stress Constraint ◽  
Reduced Order ◽  
Penalty Factor

Abstract Most of the methods used today for handling local stress constraints in topology optimization, fail to directly address the non-self-adjointness of the stress-constrained topology optimization problem. This in turn could drastically raise the computational cost for an already large-scale problem. These problems involve both the equilibrium equations resulting from finite element analysis (FEA) in each iteration, as well as the adjoint equations from the sensitivity analysis of the stress constraints. In this work, we present a paradigm for large-scale stress-constrained topology optimization problems, where we build a multi-grid approach using an on-the-fly Reduced Order Model (ROM) and the p-norm aggregation function, in which the discrete reduced-order basis functions (modes) are adaptively constructed for both the primal and dual problems. In addition to reducing the computational savings due to the ROM, we also address the computational cost of the ROM learning and updating phases. Both reduced-order bases are enriched according to the residual threshold of the corresponding linear systems, and the grid resolution is adaptively selected based on the relative error in approximating the objective function and constraint values during the iteration. The tests on 2D and 3D benchmark problems demonstrate improved performance with acceptable objective and constraint violation errors. Finally, we thoroughly investigate the influence of relevant stress constraint parameters such as the coagulation factor, stress penalty factor, and the allowable stress value.

Topology and Sizing Optimization of Micromixers Using Graph-Theoretical Representation and Genetic Algorithm

2017 ◽  
Author(s):  
Mitsuo Yoshimura ◽  
Koji Shimoyama ◽  
Takashi Misaka ◽  
Shigeru Obayashi
Keyword(s):  
Genetic Algorithm ◽  
Topology Optimization ◽  
Optimization Problems ◽  
Computational Cost ◽  
Kriging Model ◽  
Local Optimum ◽  
Sizing Optimization ◽  
Local Optima ◽  
Novel Approach ◽  
Design Variables

This paper proposes a novel approach for fluid topology optimization using genetic algorithm. In this study, the enhancement of mixing in the passive micromixers is considered. The efficient mixing is achieved by the grooves attached on the bottom of the microchannel and the optimal configuration of grooves is investigated. The grooves are represented based on the graph theory. The micromixers are analyzed by a CFD solver and the exploration by genetic algorithm is assisted by the Kriging model in order to reduce the computational cost. Three cases with different constraint and treatment for design variables are considered. In each case, GA found several local optima since the objective function is a multi-modal function and each local optimum revealed the specific characteristic for efficient mixing in micromixers. Moreover, we discuss the validity of the constraint for optimization problems. The results show a novel insight for design of micromixer and fluid topology optimization using genetic algorithm.

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