Coupling of adaptively refined dual mixed finite elements and boundary elements in linear elasticity

We investigate a coupling of mixed finite elements and Galerkin boundary elements which is stable and leads to symmetric matrices. In the FEM domain, a posteriori error estimates are employed to refine the mesh adaptively. Numerical results are given for plane strain problems.     

Coupling finite elements and auxiliary sources

We consider second-order scalar elliptic boundary value problems on unbounded domains, which model, for instance, electrostatic fields. We propose a discretization that relies on a Trefftz approximation by multipole auxiliary sources in some parts of the domain and on standard mesh-based primal Lagrangian finite elements in other parts. Several approaches are developed and, based on variational saddle point theory, rigorously analyzed to couple both discretizations across the common interface:1. Least-squares-based coupling using techniques from PDE-constrained optimization.2. Coupling through Dirichlet-to-Neumann operators.3. Three-field variational formulation in the spirit of mortar finite element methods.We compare these approaches in a series of numerical experiments.     

MFX-ANSYS/CFX流固耦合计算及其在“魔方”上的应用

流体和固体之间的耦合作用在工程领域中有非常广泛的应用,对这类问题的求解一直是工程上的难点。本文介绍了一种流固耦合求解方法,采用CFX和ANSYS软件分别对流体域和结构域进行求解,其耦合面的数据交换以MFX-ANSYS/CFX为平台,并通过实例验证了流体和结构之间的耦合效应。同时,基于"魔方"高性能计算平台,探讨了这类问题的并行求解算法以及在不同处理器数目下的计算效率,为该类方法的大规模计算以及更好地发挥并行计算优势提供参考。     

A mesh-geometry based method for coupling 1D and 3D elements

When conducting a finite element analysis (FEA) one way to reduce the total number of degrees of freedom is to use a mixed-dimensional model. Using beam elements to model long and slender components can significantly reduce the total number of elements. Problems arise when trying to connect elements with different dimensions in part due to incompatible degrees of freedom between different types of finite elements. This paper focuses on problems that occur in coupling beams and solids, which means coupling 1D and 3D finite elements. This paper presents a mesh-based solution to these problems only using specific arrangements of classical 1D and 3D finite elements without requiring the use of additional constraint equations. Two alternative solutions are detailed, evaluated and compared in this paper through series of computational experiments. The implementation of both solutions is also presented and involves mesh and geometry processing operations along with an adaptation of boundary representation (BREP) classical data structures.     

Boundary element analysis of elastic contact problems using gap finite elements

Gap finite elements provide a practical approach for dealing with elastic contact problems, and it is not possible to derive something similar with boundary elements. This work introduces a simple technique for the analysis of elastic contact problems by coupling a gap finite element subregion with boundary element subregions. The developed algorithm proves to be accurate and reliable, combining the advantages within contact problems of both the gap finite element and the boundary element method.     

Asymptotic boundary conditions with immersed finite elements for interface magnetostatic/electrostatic field problems with open boundary

Many of the magnetostatic/electrostatic field problems encountered in aerospace engineering, such as plasma sheath simulation and ion neutralization process in space, are not confined to finite domain and non-interface problems, but characterized as open boundary and interface problems. Asymptotic boundary conditions (ABC) and immersed finite elements (IFE) are relatively new tools to handle open boundaries and interface problems respectively. Compared with the traditional truncation approach, asymptotic boundary conditions need a much smaller domain to achieve the same accuracy. When regular finite element methods are applied to an interface problem, it is necessary to use a body-fitting mesh in order to obtain the optimal convergence rate. However, immersed finite elements possess the same optimal convergence rate on a Cartesian mesh, which is critical to many applications. This paper applies immersed finite element methods and asymptotic boundary conditions to solve an interface problem arising from electric field simulation in composite materials with open boundary. Numerical examples are provided to demonstrate the high global accuracy of the IFE method with ABC based on Cartesian meshes, especially around both interface and boundary. This algorithm uses a much smaller domain than the truncation approach in order to achieve the same accuracy.     

Finite state machine-based optimization of data parallel regular domain problems applied in low-level image processing

A popular approach to providing nonexperts in parallel computing with an easy-to-use programming model is to design a software library consisting of a set of preparallelized routines, and hide the intricacies of parallelization behind the library's API. However, for regular domain problems (such as simple matrix manipulations or low-level image processing applications-in which all elements in a regular subset of a dense data field are accessed in turn) speedup obtained with many such library-based parallelization tools is often suboptimal. This is because interoperation optimization (or: time-optimization of communication steps across library calls) is generally not incorporated in the library implementations. We present a simple, efficient, finite state machine-based approach for communication minimization of library-based data parallel regular domain problems. In the approach, referred to as lazy parallelization, a sequential program is parallelized automatically at runtime by inserting communication primitives and memory management operations whenever necessary. Apart from being simple and cheap, lazy parallelization guarantees to generate legal, correct, and efficient parallel programs at all times. The effectiveness of the approach is demonstrated by analyzing the performance characteristics of two typical regular domain problems obtained from the field of low-level image processing. Experimental results show significant performance improvements over nonoptimized parallel applications. Moreover, obtained communication behavior is found to be optimal with respect to the abstraction level of message passing programs.     

HPC‐GAP: engineering a 21st‐century high‐performance computer algebra system

Symbolic computation has underpinned a number of key advances in Mathematics and Computer Science. Applications are typically large and potentially highly parallel, making them good candidates for parallel execution at a variety of scales from multi‐core to high‐performance computing systems. However, much existing work on parallel computing is based around numeric rather than symbolic computations. In particular, symbolic computing presents particular problems in terms of varying granularity and irregular task sizes that do not match conventional approaches to parallelisation. It also presents problems in terms of the structure of the algorithms and data. This paper describes a new implementation of the free open‐source GAP computational algebra system that places parallelism at the heart of the design, dealing with the key scalability and cross‐platform portability problems. We provide three system layers that deal with the three most important classes of hardware: individual shared memory multi‐core nodes, mid‐scale distributed clusters of (multi‐core) nodes and full‐blown high‐performance computing systems, comprising large‐scale tightly connected networks of multi‐core nodes. This requires us to develop new cross‐layer programming abstractions in the form of new domain‐specific skeletons that allow us to seamlessly target different hardware levels. Our results show that, using our approach, we can achieve good scalability and speedups for two realistic exemplars, on high‐performance systems comprising up to 32000 cores, as well as on ubiquitous multi‐core systems and distributed clusters. The work reported here paves the way towards full‐scale exploitation of symbolic computation by high‐performance computing systems, and we demonstrate the potential with two major case studies. © 2016 The Authors. Concurrency and Computation: Practice and Experience Published by John Wiley & Sons Ltd.     

Adaptive refinement analysis using hybrid-stress transition elements

In this paper, 4-node to 7-node hybrid-stress transition elements are developed for automatic adaptive refinement analysis of plane elasticity problems. The displacement-based transition quadrilateral elements are first adopted and applied to refinement analysis using both full and reduced integration schemes. As the stress field over the displacement-based transition elements is not continuous, a more smooth stress pattern is desirable and could enhance the performance of the element. Indeed, continuous stress field of various orders can be easily introduced into a displacement-based element through a variational procedure based on the Hellinger–Reissner functional. Of the same kinematics and displacement pattern, the resulting hybrid-stress transition elements are more superior to the displacement-based elements in possessing a more continuous high quality stress field within the element. The hybrid-stress transition elements are tested with classical benchmark examples, and the results indicate that hybrid-stress transition elements are consistently more efficient than the displacement-based counterparts in adaptive refinement analysis. A more economical rank-deficient version of hybrid-stress transition elements is also available. While they are less expensive to evaluate, they enjoy a very similar convergence rate as the rank-sufficient hybrid-stress transition elements.     

Stabilized extended finite elements for the approximation of saddle point problems with unfitted interfaces

We address a two-phase Stokes problem, namely the coupling of two fluids with different kinematic viscosities. The domain is crossed by an interface corresponding to the surface separating the two fluids. We observe that the interface conditions allow the pressure and the velocity gradients to be discontinuous across the interface. The eXtended Finite Element Method (XFEM) is applied to accommodate the weak discontinuity of the velocity field across the interface and the jump in pressure on computational meshes that do not fit the interface. Numerical evidence shows that the discrete pressure approximation may be unstable in the neighborhood of the interface, even though the spatial approximation is based on inf-sup stable finite elements. It means that XFEM enrichment locally violates the satisfaction of the stability condition for mixed problems. For this reason, resorting to pressure stabilization techniques in the region of elements cut by the unfitted interface is mandatory. In alternative, we consider the application of stabilized equal order pressure/velocity XFEM discretizations and we analyze their approximation properties. On one side, this strategy increases the flexibility on the choice of velocity and pressure approximation spaces. On the other side, symmetric pressure stabilization operators, such as local pressure projection methods or the Brezzi–Pitkaranta scheme, seem to be effective to cure the additional source of instability arising from the XFEM approximation. We will show that these operators can be applied either locally, namely only in proximity of the interface, or globally, that is on the whole domain when combined with equal order approximations. After analyzing the stability, approximation properties and the conditioning of the scheme, numerical results on benchmark cases will be discussed, in order to thoroughly compare the performance of different variants of the method.     

非结构网格上求解中子输运方程的并行流水线Sn扫描算法

间断有限元离散纵标方法(Sn)是广泛应用于求解高维非定常中子输运方程的数值方法,它涉及几何网格空间、速度相空间和中子能群的离散,计算量很大．该文基于非结构网格,提出了基于区域分解的并行流水线Sn扫描算法,通过设计具有不同内在并行度和通信面体比的区域分解方法和队列插入算法,对两个不同物理模型,分别使用两台并行机的92个和256个CPU,获得72倍和78倍以上的加速．可扩展性能分析表明,算法的性能非常依赖于并行机的点对点通信延迟．     

Discontinuous finite elements and godunov-type methods for the eulerian equations of gas dynamics

We investigate the numerical solution of typical problems in gas dynamics by discontinuous finite element methods, and compare the results with computations performed with variants of Godunov's conservative method.A one-dimensional shock wave problem with reflection, and a three-dimensional shock tube-type problem with convergent-divergent nozzle geometry are analyzed. For the one-dimensional problem we also present results obtained with a variant of Glimm's method. In one dimension, finite elements give valuable results, although they need a substantially larger computing time; in three space dimensions discontinuous elements appear to be too cumbersome, in the present form, to lend themselves to an efficient treatment of time-dependent shock wave problems.     

A new era in scientific computing: Domain decomposition methods in hybrid CPU–GPU architectures

Recent advances in graphics processing units (GPUs) technology open a new era in high performance computing. Applications of GPUs to scientific computations are attracting a lot of attention due to their low cost in conjunction with their inherently remarkable performance features and the recently enhanced computational precision and improved programming tools. Domain decomposition methods (DDM) constitute today an important category of methods for the solution of highly demanding problems in simulation-based applied science and engineering. Among them, dual domain decomposition methods have been successfully applied in a variety of problems in both sequential as well as in parallel/distributed processing systems. In this work, we demonstrate the implementation of the FETI method to a hybrid CPU–GPU computing environment. Parametric tests on implicit finite element structural mechanics benchmark problems revealed the tremendous potential of this type of hybrid computing environment as a result of the full exploitation of multi-core CPU hardware resources and the intrinsic software and hardware features of the GPUs as well as the numerical properties of the solution method.     

Gradient elements of the knapsack polytope

The knapsack problem is a classical NP-hard problem of special interest in combinatorial mathematics and complexity theory. Particularly interesting are the properties of the knapsack problem domain. In this paper we study the set of gradient elements of the knapsack polytope. They are related with the well-known greedy algorithm for finding approximate solutions to the optimization knapsack problem and constitute an important subset of the extremal elements (vertices) of the knapsack polytope. We obtain a tight bound for the number of distinct gradient elements, which is an exponential function of the problem dimension. We also use the greedy-algorithmic approach and the concept of gradient elements to devise a polynomial algorithm for a large subclass of a linear Diophantine problem which is a variant of the knapsack problem. Some optimization issues related to the problems considered are discussed as well. Received: September 1999 / Accepted: December 2000     

Boundary element formulation for two-dimensional creep problems using isoparametric quadratic elements

A boundary element formulation for creep and time-dependent material behaviour problems based on an initial strain approach is presented. The details of numerical algorithm are shown where isoparametric quadratic elements are used both for the boundary elements and the quadrilateral domain cells. The Euler method with automatic time-step control scheme is implemented for time integration. Two creep power laws, time-hardening and strain-hardening, are employed to analyse a number of problems, including a square plate, a plate with a circular hole and a plate with a semi-circular notch subjected to a uniaxial load. The results are compared with analytical solutions where available and the corresponding finite element solutions.     

计算生物学中的高性能计算(Ⅱ)—序列分析

序列分析是高性能计算应用的一个重要方向。随着高通量测序技术的发展,基因数据呈现爆炸性增长,对高性能计算的需求也更加迫切。介绍了高性能计算在序列分析中的应用和序列分析算法的并行实现,包括序列比对、检索、重测序、拼接等。     

Solid-to-shell transition elements for the computation of interlaminar stresses

This paper presents an accurate and practical technique for coupling shell element models to three-dimensional continuum finite element models. The compatibility between these two types of formulations is enforced by degenerating a continuum element through kinematic constraints compatible with shell deformations. Two formulations of two-dimensional/three-dimensional transition elements are presented. The first and simplest formulation is based on the Mindlin-Reissner plate assumptions, and is found to perform well in a variety of problems involving the analysis of geometrically linear/non-linear laminated structures. The second formulation is based on a higher-order shell theory that allows stretching in the through-the-thickness direction. This additional freedom virtually eliminates the interlaminar normal stress boundary layer that can form in lower-order transition elements. Finally, the coupling of two-dimensional to three-dimensional subdomains is enriched with the use of an interface element, which can be used in conjunction with either transition formulation. The interface element improves the efficiency of the solid-to-shell transition modeling scheme by allowing the independent selection of optimal mesh sizes in the shell and the three-dimensional regions of the model.     

Anisotropic interpolation error estimates for isoparametric quadrilateral finite elements

Anisotropic local interpolation error estimates are derived for quadrilateral and hexahedral Lagrangian finite elements with straight edges. These elements are allowed to have diameters with different asymptotic behaviour in different space directions. The case of affine elements (parallel-epipeds) with arbitrarily high degree of the shape functions is considered first. Then, a careful examination of the multi-linear map leads to estimates for certain classes of more general, isoparametric elements. As an application, the Galerkin finite element method for a reaction diffusion problem in a polygonal domain is considered. The boundary layers are resolved using anisotropic trapezoidal elements.     

The Raincore API for clusters of networking elements

Clustering technology offers a way to increase overall reliability and performance of Internet information flow by strengthening one link in the chain without adding others. We have implemented this technology in a distributed computing architecture for network elements. The architecture, called Raincore, originated in the Reliable Array of Independent Nodes, or RAIN, research collaboration between the California Institute of Technology and the US National Aeronautics and Space Agency's Jet Propulsion Laboratory. The RAIN project focused on developing high-performance, fault-tolerant, portable clustering technology for spaceborne computing . The technology that emerged from this project became the basis for a spinoff company, Rainfinity, which has the exclusive intellectual property rights to the RAIN technology. The authors describe the Raincore conceptual architecture and distributed services, which are designed to make it easy for developers to port their applications to run on top of a cluster of networking elements. We include two applications: a Web server prototype that was part of the original RAIN research project and a commercial firewall cluster product from Rainfinity