Exploiting semantics of temporal multi‐scale methods to optimize multi‐level mesh partitioning

Multi‐scale problems are often solved by decomposing the problem domain into multiple subdomains, solving them independently using different levels of spatial and temporal refinement, and coupling the subdomain solutions back to obtain the global solution. Most commonly, finite elements are used for spatial discretization, and finite difference time stepping is used for time integration. Given a finite element mesh for the global problem domain, the number of possible decompositions into subdomains and the possible choices for associated time steps is exponentially large, and the computational costs associated with different decompositions can vary by orders of magnitude. The problem of finding an optimal decomposition and the associated time discretization that minimizes computational costs while maintaining accuracy is nontrivial. Existing mesh partitioning tools, such as METIS, overlook the constraints posed by multi‐scale methods and lead to suboptimal partitions with a high performance penalty. We present a multi‐level mesh partitioning approach that exploits domain‐specific knowledge of multi‐scale methods to produce nearly optimal mesh partitions and associated time steps automatically. Results show that for multi‐scale problems, our approach produces decompositions that outperform those produced by state‐of‐the‐art partitioners like METIS and even those that are manually constructed by domain experts. Copyright © 2017 John Wiley & Sons, Ltd.     

An enhanced parallel sub‐domain generation method for mesh partitioning in parallel finite element analysis

This paper describes an optimization and artificial intelligence‐based approach for solving the mesh partitioning problem for explicit parallel dynamic finite element analysis. The Sub‐Domain Generation Method (SGM) (Topping, Khan, Parallel Finite Element Computations. Saxe‐Coburg Publications: Edinburgh, U.K., 1996) is briefly introduced with its virtues and drawbacks. This paper describes the enhancement of the SGM algorithm (ESGM) by the introduction of a new, non‐convex bisection procedure and a new Genetic Algorithm (GA) module, which is better tuned for this particular optimization problem. Example decompositions are given and comparisons made between parallel versions of the ESGM, the SGM and other decomposition methods. The scalability of the ESGM is examined by using a range of examples. Copyright © 2000 John Wiley & Sons, Ltd.     

Robust compliance topology optimization based on the topological derivative concept

In this paper, we present an approach for robust compliance topology optimization under volume constraint. The compliance is evaluated considering a point‐wise worst‐case scenario. Analogously to sequential optimization and reliability assessment, the resulting robust optimization problem can be decoupled into a deterministic topology optimization step and a reliability analysis step. This procedure allows us to use topology optimization algorithms already developed with only small modifications. Here, the deterministic topology optimization problem is addressed with an efficient algorithm based on the topological derivative concept and a level‐set domain representation method. The reliability analysis step is handled as in the performance measure approach. Several numerical examples are presented showing the effectiveness of the proposed approach. Copyright © 2015 John Wiley & Sons, Ltd.     

Mesh partitioning for implicit computations via iterative domain decomposition: Impact and optimization of the subdomain aspect ratio

Optimal domain decomposition methods have emerged as powerful iterative algorithms for parallel implicit computations. Their key preprocessing step is mesh partitioning, where research has focused so far on the automatic generation of load-balanced subdomains with minimum interface nodes. In this paper, we emphasize the importance of the subdomain aspect ratio as a mesh partitioning factor, and highlight its impact on the convergence rate of an optimal domain decomposition based iterative method. We also present a fast optimization algorithm for improving the aspect ratio of existing mesh partitions, and illustrate it with several examples from fluid dynamics and structural mechanics applications. For a stiffened shell problem decomposed by the optimal Recursive Spectral Bisection scheme and solved by the FETI method, this optimization algorithm is shown to improve the solution time by a factor equal to 1·54 and to restore numerical scalability.     

A bilevel game theoretic approach to optimum design of flywheels

Multiobjective optimization problems arise frequently in mechanical design. One approach to solving these types of problems is to use a game theoretic formulation. This article illustrates the application of a bilevel, leader–follower model for solving an optimum design problem. In particular, the optimization problem is modelled as a Stackelberg game. The partitioning of variables between the leader and follower problem is discussed and a variable partitioning metric is introduced to compare various variable partitions. A computational procedure based on variable updating using sensitivity information is developed for exchanging information between the follower and leader problems. The proposed approach is illustrated through the design of a flywheel. The two objective functions used for the design problem include maximizing the kinetic energy stored in the flywheel while simultaneously minimizing the manufacturing cost.     

Multiresolution subspace-based optimization method for inverse scattering problems

This paper investigates an approach to inverse scattering problems based on the integration of the subspace-based optimization method (SOM) within a multifocusing scheme in the framework of the contrast source formulation. The scattering equations are solved by a nested three-step procedure composed of (a) an outer multiresolution loop dealing with the identification of the regions of interest within the investigation domain through an iterative information-acquisition process, (b) a spectrum analysis step devoted to the reconstruction of the deterministic components of the contrast sources, and (c) an inner optimization loop aimed at retrieving the ambiguous components of the contrast sources through a conjugate gradient minimization of a suitable objective function. A set of representative reconstruction results is discussed to provide numerical evidence of the effectiveness of the proposed algorithmic approach as well as to assess the features and potentialities of the multifocusing integration in comparison with the state-of-the-art SOM implementation.     

On structural shape optimization using an embedding domain discretization technique

This contribution presents a novel approach to structural shape optimization that relies on an embedding domain discretization technique. The evolving shape design is embedded within a uniform finite element background mesh which is then used for the solution of the physical state problem throughout the course of the optimization. We consider a boundary tracking procedure based on adaptive mesh refinement to separate between interior elements, exterior elements, and elements intersected by the physical domain boundary. A selective domain integration procedure is employed to account for the geometric mismatch between the uniform embedding domain discretization and the evolving structural component. Thereby, we avoid the need to provide a finite element mesh that conforms to the structural component for every design iteration, as it is the case for a standard Lagrangian approach to structural shape optimization. Still, we adopt an explicit shape parametrization that allows for a direct manipulation of boundary vertices for the design evolution process. In order to avoid irregular and impracticable design updates, we consider a geometric regularization technique to render feasible descent directions for the course of the optimization. Copyright © 2016 John Wiley & Sons, Ltd.     

Domain decomposition of stochastic PDEs: Theoretical formulations

We present a novel theoretical framework for the domain decomposition of uncertain systems defined by stochastic partial differential equations. The methodology involves a domain decomposition method in the geometric space and a functional decomposition in the probabilistic space. The probabilistic decomposition is based on a version of stochastic finite elements based on orthogonal decompositions and projections of stochastic processes. The spatial decomposition is achieved through a Schur‐complement‐based domain decomposition. The methodology aims to exploit the full potential of high‐performance computing platforms by reducing discretization errors with high‐resolution numerical model in conjunction to giving due regards to uncertainty in the system. The mathematical formulation is numerically validated with an example of waves in random media. Copyright © 2008 John Wiley & Sons, Ltd.     

A domain decomposition tool for boundary element methods

Domain decomposition boundary element methods have become increasingly popular over the last several years for a variety of reasons. In particular, these methods reduce the storage and CPU requirements, can result in sparse linear systems, are easy to parallelize, and, when used in conjunction with a dual reciprocity method, can significantly improve the conditioning of the associated linear system. Nevertheless, for complex geometries, determining an appropriate decomposition of the domain can be extremely difficult. A domain decomposition tool based on a graph partitioning algorithm is presented to automate the process and provide quality decompositions.     

Iterative mesh partitioning optimization for parallel nonlinear dynamic finite element analysis with direct substructuring

 This work presents a novel iterative approach for mesh partitioning optimization to promote the efficiency of parallel nonlinear dynamic finite element analysis with the direct substructure method, which involves static condensation of substructures' internal degrees of freedom. The proposed approach includes four major phases – initial partitioning, substructure workload prediction, element weights tuning, and partitioning results adjustment. The final three phases are performed iteratively until the workloads among the substructures are balanced reasonably. A substructure workload predictor that considers the sparsity and ordering of the substructure matrix is used in the proposed approach. Several numerical experiments conducted herein reveal that the proposed iterative mesh partitioning optimization often results in a superior workload balance among substructures and reduces the total elapsed time of the corresponding parallel nonlinear dynamic finite element analysis. Received 22 August 2001 / Accepted 20 January 2002     

Efficient representation of turbulent flows using data-enriched finite elements

Efficient modal decomposition of high-dimensional turbulent flow data is an important first step for data reduction, analysis, and low-dimensional predictive modeling. The conventional modal decomposition techniques, such as proper orthogonal and dynamic mode decompositions, aim to represent the system response using spatially global basis vectors that span a broad spatial domain. A significant challenge facing approaches based on global domain decomposition is the rapid increase in both the amount of training data and the number of modes that must be retained for an accurate representation of convection dominated turbulent flows. An alternative generalized finite element (GFEM) based approach is explored for efficient representation of high-dimensional fluid flow data. Here, the standard finite element interpolation method is enriched with numerical functions that are learned from a small amount of high-fidelity training data over spatially localized subdomains. The GFEM approach is demonstrated on a 3D flow past a cylinder at Reynolds number of 100 000 and flows inside a 2D lid-driven cavity over a range of Reynolds numbers. Compared with a global proper orthogonal decomposition, the GFEM-based approach increases efficiency in reconstructing the datasets while also substantially reducing the amounts of training data.     

Autonomous decentralized finite element method and its applications

In this paper, a new solution procedure using the finite element technique in order to solve problems of structure analysis is proposed. This procedure is called the autonomous decentralized finite element method because it is based on the characteristic autonomy and decentralization in life or biological systems (life‐like approach). The fundamental approach is developed according to an idea of cellular automata manipulation by the new neighbourhood model. The finite element method with an algorithm of the relaxation method is adopted as the solution procedure in this approach. The proposed procedure demonstrates that it is a powerful means of numerical analysis for many kinds of structural problems that are structural morphogenesis, structural optimization and structural inverse problems. Our procedure is applied to numerical analysis of three simple plane models: (1) The structural shape analysis problem for the prescribed displacement mode of a truss structure, (2) An adaptive structure remodelling problem on an elastic continuum, (3) An identification problem of thermal conductivity on a continuum. The effectiveness and validity of our idea are shown from their numerical results. Copyright © 2003 John Wiley & Sons, Ltd.     

Penalized-likelihood image reconstruction for digital holography

Conventional numerical reconstruction for digital holography using a filter applied in the spatial-frequency domain to extract the primary image may yield suboptimal image quality because of the loss in high-frequency components and interference from other undesirable terms of a hologram. We propose a new numerical reconstruction approach using a statistical technique. This approach reconstructs the complex field of the object from the real-valued hologram intensity data. Because holographic image reconstruction is an ill-posed problem, our statistical technique is based on penalized-likelihood estimation. We develop a Poisson statistical model for this problem and derive an optimization transfer algorithm that monotonically decreases the cost function at each iteration. Simulation results show that our statistical technique has the potential to improve image quality in digital holography relative to conventional reconstruction techniques.     

A polynomial chaos efficient global optimization approach for Bayesian optimal experimental design

This paper proposes a global optimization framework to address the high computational cost and non convexity of Optimal Experimental Design (OED) problems. To reduce the computational burden and the presence of noise in the evaluation of the Shannon expected information gain (SEIG), this framework proposes the coupling of Laplace approximation and polynomial chaos expansions (PCE). The advantage of this procedure is that PCE allows large samples to be employed for the SEIG estimation, practically vanishing the noisy introduced by the sampling procedure. Consequently, the resulting optimization problem may be treated as deterministic. Then, an optimization approach based on Kriging surrogates is employed as the optimization engine to search for the global solution with limited computational budget. Four numerical examples are investigated and their results are compared to state-of-the-art stochastic gradient descent algorithms. The proposed approach obtained better results than the stochastic gradient algorithms in all situations, indicating its efficiency and robustness in the solution of OED problems.     

Numerical shape optimization of cold forging tools by means of FEM/BEM simulation

The procedure of a numerical shape optimization of the cold forging tool geometry allows for a reduction and a homogenization of tool load stresses. This procedure considers the workpiece-tool-machine interaction by means of the FEM/BEM coupling. The coupling requires modifications to insure accurate integration in TOSCA software for further shape optimization. The resulting distribution of the nodal displacements and changes of tool geometry are discussed and analyzed. Additionally, the numerical validation of the results by means of mechanical simulation of the optimized geometry with the extended FEM/BEM model is given. The equivalent stresses distribution at the end of lateral extrusion in the tool and in workpiece is presented. The numerical shape optimization leads to the maximum equivalent stress value reduction on the press shoulder on 856 MPa, corresponding to a percentage decrease of 24.3 % in comparison with the initial geometry. The approach for the compensation of load induced workpiece deviations, involving the described optimization procedure, is presented as well.     

A reconstruction procedure for microwave nondestructive evaluation based on a numerically computed Green's function

This paper describes a new microwave diagnostic tool for nondestructive evaluation. The approach, developed in the spatial domain, is based on the numerical computation of the inhomogeneous Green's function in order to fully exploit all the available a priori information of the domain under test. The heavy reduction of the computational complexity of the proposed procedure (with respect to standard procedures based on the free-space Green's function) is also achieved by means of a customized hybrid-coded genetic algorithm. In order to assess the effectiveness of the method, the results of several simulations are presented and discussed.     

Global optimum of microstructure parameters in the CMWP line-profile-analysis method by combining Marquardt-Levenberg and Monte-Carlo procedures

Line profile analysis of X-ray and neutron diffraction patterns is a powerful tool for determining the microstructure of crystalline materials. The Convolutional-Multiple-Whole-Profile (CMWP) procedure is based on physical profile functions for dislocations, domain size, stacking faults and twin boundaries. Order dependence, strain anisotropy, hkl dependent broadening of planar defects and peak shape are used to separate the effect of different lattice defect types. The Marquardt-Levenberg (ML) numerical optimization procedure has been used successfully to determine crystal defect types and densities. However, in more complex cases like hexagonal materials or multiple phases the ML procedure alone reveals uncertainties. In a new approach the ML and a Monte-Carlo statistical method are combined in an alternative manner. The new CMWP procedure eliminates uncertainties and provides globally optimized parameters of the microstructure.     

Topology optimization of microwave waveguide filters

We present a density‐based topology optimization approach for the design of metallic microwave insert filters. A two‐phase optimization procedure is proposed in which we, starting from a uniform design, first optimize to obtain a set of spectral varying resonators followed by a band gap optimization for the desired filter characteristics. This is illustrated through numerical experiments and comparison to a standard band pass filter design. It is seen that the carefully optimized topologies can sharpen the filter characteristics and improve performance. Furthermore, the obtained designs share little resemblance to standard filter layouts, and hence, the proposed design method offers a new design tool in microwave engineering. Copyright © 2017 John Wiley & Sons, Ltd.     

Improving multiresolution topology optimization via multiple discretizations

Unlike the traditional topology optimization approach that uses the same discretization for finite element analysis and design optimization, this paper proposes a framework for improving multiresolution topology optimization (iMTOP) via multiple distinct discretizations for: (1) finite elements; (2) design variables; and (3) density. This approach leads to high fidelity resolution with a relatively low computational cost. In addition, an adaptive multiresolution topology optimization (AMTOP) procedure is introduced, which consists of selective adjustment and refinement of design variable and density fields. Various two‐dimensional and three‐dimensional numerical examples demonstrate that the proposed schemes can significantly reduce computational cost in comparison to the existing element‐based approach. Copyright © 2012 John Wiley & Sons, Ltd.