A parallel two‐level domain decomposition based one‐shot method for shape optimization problems

A two‐level domain decomposition method is introduced for general shape optimization problems constrained by the incompressible Navier–Stokes equations. The optimization problem is first discretized with a finite element method on an unstructured moving mesh that is implicitly defined without assuming that the computational domain is known and then solved by some one‐shot Lagrange–Newton–Krylov–Schwarz algorithms. In this approach, the shape of the domain, its corresponding finite element mesh, the flow fields and their corresponding Lagrange multipliers are all obtained computationally in a single solve of a nonlinear system of equations. Highly scalable parallel algorithms are absolutely necessary to solve such an expensive system. The one‐level domain decomposition method works reasonably well when the number of processors is not large. Aiming for machines with a large number of processors and robust nonlinear convergence, we introduce a two‐level inexact Newton method with a hybrid two‐level overlapping Schwarz preconditioner. As applications, we consider the shape optimization of a cannula problem and an artery bypass problem in 2D. Numerical experiments show that our algorithm performs well on a supercomputer with over 1000 processors for problems with millions of unknowns. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

A domain decomposition technique applied to the solution of the coupled electro‐mechanical problem

A domain decomposition approach for the solution of the coupled electro‐mechanical problem in dynamics is proposed. The finite element analysis of a coupled electro‐mechanical system is frequently found, for example, in the modelling and design of microsystems and may lead to a burdensome nonlinear problem solution, particularly in the dynamic case. Two versions of the algorithm are proposed: the first one, called single‐level decomposition, exploits the natural partition of the analysis domain given by the two physics to be solved; the second one, called two‐level decomposition, adds a further subdivision of each physics into subdomains. The multilevel domain decomposition strategy here proposed is shown to accurately predict the response of microsystems subjected to electro‐mechanical coupling and to allow for a significant reduction in the computational burden. Copyright © 2012 John Wiley & Sons, Ltd.     

A FETI‐based domain decomposition technique for time‐dependent first‐order systems based on a DAE approach

We present a novel partitioned coupling algorithm to solve first‐order time‐dependent non‐linear problems (e.g. transient heat conduction). The spatial domain is partitioned into a set of totally disconnected subdomains. The continuity conditions at the interface are modeled using a dual Schur formulation where the Lagrange multipliers represent the interface fluxes (or the reaction forces) that are required to maintain the continuity conditions. The interface equations along with the subdomain equations lead to a system of differential algebraic equations (DAEs). For the resulting equations a numerical algorithm is developed, which includes choosing appropriate constraint stabilization techniques. The algorithm first solves for the interface Lagrange multipliers, which are subsequently used to advance the solution in the subdomains. The proposed coupling algorithm enables arbitrary numeric schemes to be coupled with different time steps (i.e. it allows subcycling) in each subdomain. This implies that existing software and numerical techniques can be used to solve each subdomain separately. The coupling algorithm can also be applied to multiple subdomains and is suitable for parallel computers. We present examples showing the feasibility of the proposed coupling algorithm. Copyright © 2008 John Wiley & Sons, Ltd.     

基于神经网络的动态测量误差分解研究

在讨论神经网络方法的基础上，尝试将其应用于动态测量误差分解理论。建立了一个简单的动态测试仿真系统，分别用小波变换方法和神经网络方法对系统输出的总误差进行分解，并对两种方法的处理结果作了比较，指出了神经网络方法在动态误差分解中的实用性和优越性。     

A contact detection algorithm for multi‐sphere particles by means of two‐level‐grid‐searching in DEM simulations

In discrete element method simulations, multi‐sphere particle is extensively employed for modeling the geometry shape of non‐spherical particle. A contact detection algorithm for multi‐sphere particles has been developed through two‐level‐grid‐searching. In the first‐level‐grid‐searching, each multi‐sphere particle is represented by a bounding sphere, and global space is partitioned into identical square or cubic cells of size D, the diameter of the greatest bounding sphere. The bounding spheres are mapped into the cells in global space. The candidate particles can be picked out by searching the bounding spheres in the neighbor cells of the bounding sphere for the target particle. In the second‐level‐grid‐searching, a square or cubic local space of size (D + d) is partitioned into identical cells of size d, the diameter of the greatest element sphere. If two bounding spheres of two multi‐sphere particles are overlapped, the contacts occurring between the element spheres in the target multi‐sphere particle and in the candidate multi‐sphere particle are checked. Theoretical analysis and numerical tests on the memory requirement and contact detection time of this algorithm have been performed to verify the efficiency of this algorithm. The results showed that this algorithm can effectively deal with the contact problem for multi‐sphere particles. Copyright © 2015 John Wiley & Sons, Ltd.     

Mismatching refinement with domain decomposition for the analysis of steady‐state metal forming process

The finite element analysis of three‐dimensional metal forming processes is generally subject to large computational burden due to its non‐linearity. For economic computation, the mismatching refinement, an efficient domain decomposition method with different mesh density for each subdomain, is developed in the present study. A modified velocity alternating scheme for the interface treatment is proposed in order to obtain good convergence and accuracy in the mismatching refinement. As a numerical example, the analysis of the axisymmetric extrusion processes is carried out. The results are discussed for the various velocity update schemes and for the variation of the length of overlapped region. The three‐dimensional extrusion processes for a rectangular section and an E‐section are analysed in order to verify the effectiveness of the proposed method. Comparing the results with those of the conventional method of full region analysis, the accuracy and the computational efficiency of the proposed method are then discussed. Copyright © 2000 John Wiley & Sons, Ltd.     

Performance of fully coupled algebraic multilevel domain decomposition preconditioners for incompressible flow and transport

This study investigates algebraic multilevel domain decomposition preconditioners of the Schwarz type for solving linear systems associated with Newton–Krylov methods. The key component of the preconditioner is a coarse approximation based on algebraic multigrid ideas to approximate the global behaviour of the linear system. The algebraic multilevel preconditioner is based on an aggressive coarsening graph partitioning of the non‐zero block structure of the Jacobian matrix. The scalability of the preconditioner is presented as well as comparisons with a two‐level Schwarz preconditioner using a geometric coarse grid operator. These comparisons are obtained on large‐scale distributed‐memory parallel machines for systems arising from incompressible flow and transport using a stabilized finite element formulation. The results demonstrate the influence of the smoothers and coarse level solvers for a set of 3D example problems. For preconditioners with more than one level, careful attention needs to be given to the balance of robustness and convergence rate for the smoothers and the cost of applying these methods. For properly chosen parameters, the two‐ and three‐level preconditioners are demonstrated to be scalable to 1024 processors. Copyright © 2006 John Wiley & Sons, Ltd.     

Asynchronous multi‐domain variational integrators for non‐linear problems

We develop an asynchronous time integration and coupling method with domain decomposition for linear and non‐linear problems in mechanics. To ensure stability in the time integration and in coupling between domains, we use variational integrators with local Lagrange multipliers to enforce continuity at the domain interfaces. The asynchronous integrator lets each domain step with its own time step, using a smaller time step where required by stability and accuracy constraints and a larger time step where allowed. We show that in practice the time step is limited by accuracy requirements rather than by stability requirements. Copyright © 2008 John Wiley & Sons, Ltd.     

A bubble‐inspired algorithm for finite element mesh partitioning

This paper presents a bubble‐inspired algorithm for partitioning finite element mesh into subdomains. Differing from previous diffusion BUBBLE and Center‐oriented Bubble methods, the newly proposed algorithm employs the physics of real bubbles, including nucleation, spherical growth, bubble–bubble collision, reaching critical state, and the final competing growth. The realization of foaming process of real bubbles in the algorithm enables us to create partitions with good shape without having to specify large number of artificial controls. The minimum edge cut is simply achieved by increasing the volume of each bubble in the most energy efficient way. Moreover, the order, in which an element is gathered into a bubble, delivers the minimum number of surface cells at every gathering step; thus, the optimal numbering of elements in each subdomain has naturally achieved. Because finite element solvers, such as multifrontal method, must loop over all elements in the local subdomain condensation phase and the global interface solution phase, these two features have a huge payback in terms of solver efficiency. Experiments have been conducted on various structured and unstructured meshes. The obtained results are consistently better than the classical kMetis library in terms of the edge cut, partition shape, and partition connectivity. Copyright © 2012 John Wiley & Sons, Ltd.     

A co‐rotational grid‐based model for fabric drapes

Fabric drapes are typical large displacement, large rotation and small strain problems. Compared to conventional geometric non‐linear shell analyses, computational fabric drape analysis is particularly challenging due to the extremely weak bending rigidities of fabrics. Compared to continuum shell finite element methods, grid‐ or particle‐based methods appear to be more successful in high drapeability problems. The latter methods often resort to simple particle mechanics and formulate the elastic energy in terms of the inter‐particle distances and trigonometrical functions of the angles between the straight lines joining adjacent particles. In this paper, the co‐rotational approach and commonly employed assumptions for small strain problems in finite element analysis will be adopted to formulate the elastic energy. It will be seen that the internal force vector and the stiffness matrix are considerably simpler than other grid‐based models, yet the sparsity of the tangential stiffness matrix remains unchanged. A number of examples are considered and the predicted results are promising. Copyright © 2003 John Wiley & Sons, Ltd.     

A Nitsche‐type formulation and comparison of the most common domain decomposition methods in isogeometric analysis

This paper provides a detailed elaboration and assessment of the most common domain decomposition methods for their application in isogeometric analysis. The methods comprise a penalty approach, Lagrange multiplier methods, and a Nitsche‐type method. For the Nitsche method, a new stabilized formulation is developed in the context of isogeometric analysis to guarantee coercivity. All these methods are investigated on problems of linear elasticity and eigenfrequency analysis in 2D. In particular, focus is put on non‐uniform rational B‐spline patches which join nonconformingly along their common interface. Thus, the application of isogeometric analysis is extended to multi‐patches, which can have an arbitrary parametrization on the adjacent edges. Moreover, it has been shown that the unique properties provided by isogeometric analysis, that is, high‐order functions and smoothness across the element boundaries, carry over for the analysis of multiple domains. Copyright © 2013 John Wiley & Sons, Ltd.     

Analysis of photonic crystals using the hybrid finite‐element/finite‐difference time domain technique based on the discontinuous Galerkin method

Two‐dimensional photonic crystal structures are analyzed by a recently developed hybrid technique combining the finite‐element time‐domain (FETD) method and the finite‐difference time‐domain (FDTD) method. This hybrid FETD/FDTD method uses the discontinuous Galerkin method as framework for domain decomposition. To the best of our knowledge, this is the first hybrid FETD/FDTD method that allows non‐conformal meshes between different FETD and FDTD subdomains. It is also highly parallelizable. These properties are very suitable for the computation of periodic structures with curved surfaces. Numerical examples for the computation of the scattering parameters of two‐dimensional photonic bandgap structures are presented as applications of the hybrid FETD/FDTD method. Numerical results demonstrate the efficiency and accuracy of the proposed hybrid method. Copyright © 2012 John Wiley & Sons, Ltd.     

A new neural network‐based model for hysteretic behavior of materials

Cyclic behavior of materials is complex and difficult to model. A combination of hardening rules in classical plasticity is one possibility for modeling this complex material behavior. Neural network (NN) constitutive models have been shown in the past to have the capability of modeling complex material behavior directly from the results of material tests. In this paper, we propose a novel approach for NN‐based modeling of the cyclic behavior of materials. The proposed NN material model uses new internal variables that facilitate the learning of the hysteretic behavior of materials. The same approach can also be used in modeling of the hysteretic behavior of structural systems or structural components under cyclic loadings. The proposed model is shown to be superior to the earlier versions of NN material models. Although the earlier versions of the NN material models were effective in capturing the multi‐axial material behavior, they were only tested under cyclic uni‐axial state of stress. The proposed NN material model is capable of learning the hysteretic behavior of materials under even non‐uniform stress state in multi‐dimensional stress space. The performance of the proposed model is demonstrated through a series of examples using actual experimental data and simulated testing data. Copyright © 2007 John Wiley & Sons, Ltd.     

Strict lower bounds with separation of sources of error in non‐overlapping domain decomposition methods

This article deals with the computation of guaranteed lower bounds of the error in the framework of finite element and domain decomposition methods. In addition to a fully parallel computation, the proposed lower bounds separate the algebraic error (due to the use of a domain decomposition iterative solver) from the discretization error (due to the finite element), which enables the steering of the iterative solver by the discretization error. These lower bounds are also used to improve the goal‐oriented error estimation in a substructured context. Assessments on 2D static linear mechanic problems illustrate the relevance of the separation of sources of error and the lower bounds' independence from the substructuring. We also steer the iterative solver by an objective of precision on a quantity of interest. This strategy consists in a sequence of solvings and takes advantage of adaptive remeshing and recycling of search directions. Copyright © 2016 John Wiley & Sons, Ltd.     

A versatile load balancing framework for parallel applications based on domain decomposition

This paper presents a general load balancing framework and two specific load balancing algorithms that can be used for a wide range of parallel programs based on domain decomposition. Both the framework and the algorithms are versatile in the sense that they work on dedicated and non‐dedicated parallel computers, and on homogeneous and heterogeneous parallel computers. The effectiveness of the load balancing framework is demonstrated for a parallel finite element simulation on a Cray T3E and a non‐dedicated heterogeneous workstation cluster. Copyright © 2000 John Wiley & Sons, Ltd.     

A neural network constitutive model for hyperelasticity based on molecular dynamics simulations

Numerical analysis of the hyperelastic behavior of polymer materials has drawn significant interest from within the field of mechanical engineering. Currently, hyperelastic models based on the energy density function, such as the Neo-Hookean, Mooney-Rivlin, and Ogden models, are used to investigate the hyperelastic responses of materials. Conventionally, constants relating to materials were determined from experimental data by using global least-squares fitting. However, formulating a constitutive equation to capture the complex behavior of hyperelastic materials was difficult owing to the limitations of the analytical model and experimental data. This study addresses these limitations by using a system of neural networks (NNs) to design a data-driven surrogate model without a specific function formula, and employs molecular dynamics (MD) simulations to calculate the massive amount of combined loading data of hyperelastic materials. Thus, MD simulations were used to propose an NN constitutive model for hyperelasticity to derive the constitutive equation to model the complex hyperelastic response. In addition, the probability distributions of the numerical solutions of hyperelasticity are used to characterize the uncertainty of the MD models. These statistical finite element results not only present numerical results with reliability ranges but also scattered distributions of the solution obtained from the MD-based probability distributions.     

Time‐domain soil–structure interaction analysis in two‐dimensional medium based on analytical frequency‐dependent infinite elements

A direct method for soil–structure interaction analysis in two‐dimensional medium is presented in time domain, which is based on the transformation of the analytical frequency‐dependent dynamic stiffness matrix. The present dynamic stiffness matrix for the far‐field region is constructed by assembling stiffness matrices of the analytical frequency‐dependent dynamic infinite elements, so that the equation of motion can be analytically transformed into the time‐domain equation. An efficient procedure is devised to evaluate the dynamic responses in time domain. Verification of the present formulation is carried out by comparing the compliances for a strip foundation on a homogeneous and layered half‐spaces with those obtained by other methods. Numerical analyses are also carried out for the transient responses of an elastic block and tunnel in a homogeneous and a layered half‐space. The comparisons with those by other approaches indicate that the proposed time‐domain method for soil–structure interaction analysis gives good solutions. Copyright © 2000 John Wiley & Sons, Ltd.