Sparse Matrix and Solver Objects for Parallel Finite Element Simulation of Multi-field Problems

As computers can be connected to each other through the internet, we are able to use a remote computer even if it is located overseas in the same way as using a computer which is located nearby. This paper describes a parallel finite element system using the domain decomposition method, which is implemented on a computer network made from remote computers at the Internet. The above system was successfully applied to the 3 D structural analyses with a high parallel efficiency.

Sparse matrix is used in many modern industrial applications, to provide low cost and high performance solution. Unlike a number of previous works in the field of distribution networks, which analyzed the sparse matrix calculations as integral part of its calculation engine, this paper presents the considerations and analysis for the utilization of commercial of-the-shelf sparse matrix solvers (COTS SMS). Analysis is given for the parallel direct sparse solver utilized in unbalanced and unsymmetrical Distribution System Power Flow solution, which maximizes parallelization on multi-core processors. Optimal matrix population algorithm and container class from the power network object model is proposed for the fast population, search and access to the matrix elements. Optimal use cases of symmetrical and unsymmetrical matrices for the given method are analyzed. Test results for the Current Injection—Power Flow algorithm on a Three-Phase large scale system are presented which show the performance scales for symmetric and unsymmetrical matrices over a scope of system sizes and for different network topologies.

HHL algorithm is a basic quantum algorithm, which is mainly used to solve linear equations. It has a wide range of potential applications in science, engineering, finance, economics and other fields. Although the current scale of quantum computers is small, we can still demonstrate basic quantum algorithms. In this paper, taking the HHL quantum algorithm as an example, the quantum circuit of the HHL quantum algorithm corresponding to 4 qubits and 7 qubits is given, and the simulation verification is carried out on the IBM qiskit development platform. The results show that when the input matrix is a second-order matrix, the fidelity is very high, but when the input is a fourth-order matrix, the probability that the result is close to the real value is very high only when the sparsity of the matrix is relatively strong. However, when the sparsity of the input matrix is very weak, the result will be very different from the real value.

With the rapid growth of WAN infrastructures and development of Grid middleware, the cluster-of-clusters has become a realistic methodology for executing computationdemanding applications. While distributed computing or loosely connected applications have been successfully ported to the Grid environment, few tightly connected applications such as parallel finite element analysis (FEA) have been attempted. In this paper we focus on an iterative solver which is often used in FEA: the conjugate gradient (CG) method. By using both predictions and numerical experiments we evaluate the performance of the CG method parallelized via domain decomposition. From numerical experiments we measure the inter-cluster execution time and find it to be close to that obtained by the slowest cluster. At the level of performance currently found in most WAN systems, as long as the number of processes is kept adequately small, the increase in communication cost due to WAN is small relative to the computation cost. For a very large test model which realistically requires the utilization of remote resources, the work ratio stays above 86% even for 64 processes. Especially for very large models, using the distributed environment is expected to be a practical methodology even for parallel computations with communication as frequent as the one found in FEA. To evaluate the feasibility of parallel FEA on the C-of-C, we have proposed methods for a priori finding the optimal number of processes.

In parallel computing based on finite element analysis, domain decomposition is a key technique for its preprocessing. Generally, a domain decomposition of a mesh can be realized through partitioning of a graph which is converted from a finite element mesh. This paper discusses the method for graph partitioning and the way to actualize mesh partitioning. Relevant softwares are introduced, and the data structure and key functions of Metis and ParMetis are introduced. The writing, compiling, and testing of the mesh partitioning interface program based on these key functions are performed. The results indicate some objective law and characteristics to guide the users who use the graph partitioning algorithm and software to write PFEM program, and ideal partitioning effects can be achieved by actualizing mesh partitioning through the program. The interface program can also be used directly by the engineering researchers as a module of the PFEM software. So that it can reduce the application of the threshold of graph partitioning algorithm, improve the calculation efficiency, and promote the application of graph theory and parallel computing.

For large-scale finite element analysis, using a numerical method to reduce the problem size is a standard strategy to analyze a problem efficiently. For the analysis with reduction, the most time-consuming process is reducing the degrees of freedom, not solving the global sparse system in the standard analysis. Therefore, improving the efficiency of the reduction process is the key to further speedup the analyses of large-scale problems. Nonlinear analysis is much more computationally intensive than linear elastic analysis. There is a need to investigate a more efficient reduction method for nonlinear problems. This paper presents a new domain decomposition method for nonlinear substructures, which can greatly reduce the analyses time for large-scale nonlinear problems. In the proposed method, the nonlinear behavior of a substructure is updated by adding correcting modes, instead of totally repeating the whole reduction process. The efficiency can be further improved by cooperating with a parallel processing technique. The proposed method was implemented in a parallel object-oriented finite element analysis program, and its performance and accuracy were verified on various examples.

This paper addresses the secure outsourcing problem for large-scale matrix computation to a public cloud. We propose a novel public-key weave ElGamal encryption (WEE) scheme for encrypting a matrix over the field Zp. The scheme has the echelon transformation property. We can apply a series of elementary row/column operations to transform an encrypted matrix under our WEE scheme into the row/column echelon form. The decrypted result matches the result of the corresponding operations performed on the original matrix. For security, our WEE scheme is shown to be entry irrecoverable for non-zero entries under the computational Diffie-Hellman assumption. By using our WEE scheme, we propose five secure outsourcing protocols of Gaussian elimination, Gaussian-Jordan elimination, matrix determinant, linear system solver, and matrix inversion. Each of these protocols preserves data privacy for clients (data owners). Furthermore, the linear system solver and matrix inversion protocols provide a cheating-resistant mechanism to verify correctness of computation results. Our experimental result shows that our protocols gain efficiency significantly for an outsourcer. Our outsourcing protocol solves a linear system of n = 1, 000 equations and m = 1, 000 unknown variables about 472 times faster than a non-outsourced version. The efficiency gain is more substantial when (n,m) gets larger. For example, when n = 10, 000 and m = 10, 000, the protocol can solve it about 56, 274 times faster. Our protocols can also be easily implemented in parallel computation architecture to get more efficiency improvement.

Research is on-going that examines parallel direct block-diagonal-bordered sparse linear solvers for irregular sparse matrix problems derived from electrical power system applications. Parallel block-diagonal-bordered sparse linear solvers exhibit distinct advantages when compared to current general parallel direct sparse matrix solvers. Our research shows that actual power system matrices can be readily ordered into block-diagonal-bordered form, although load imbalance becomes excessive beyond 16 processors, limiting scalability for a single parallel linear solver within an application. Nevertheless, other dimensions exist in electrical power system applications that can be exploited to efficiently make use of large-scale multi-processors. >

Parallel processing, neural networks and genetic algorithms

Matrix computation is a key technology in various data processing tasks including data mining, machine learning, and information retrieval. Size of matrices has been increasing with the development of computational resources and dissemination of big data. Huge matrices are memory- and computational-time-consuming. Therefore, reducing the size and computational time of huge matrices is a key challenge in the data processing area. We develop MOARLE, a novel matrix computation framework that saves memory space and computational time. In contrast to conventional matrix computational methods that target to sparse matrices, MOARLE can efficiently handle both sparse matrices and dense matrices. Our experimental results show that MOARLE can reduce the memory usage to 2% of the original usage and improve the computational performance by a factor of 124x.

Computer Solution of Large Linear Systems

Many graph-based algorithms in high performance computing (HPC) use approximate solutions due to having algorithms that are computationally expensive or serial in nature. Neural acceleration, i.e., the process of speeding up approximation computation elements via artificial neural networks, is relatively new and has not focused on HPC graph-based algorithms. In this paper, we propose a starting point for applying models for neural acceleration to graph-based HPC algorithms utilizing an understanding of the connectivity computational pattern, recursive neural networks, and graph neural networks. We demonstrate these techniques on the problem related to the utility functions of sparse matrix ordering and fill-in (i.e., zero elements becoming nonzero during factorization) calculations. The problem of sparse matrix ordering is commonly used for issues related to load balancing, improving memory reuse, or reducing computational and memory costs in direct sparse linear solver methods. These utility functions are ideal for demonstration as they comprise a number of different graph-based subproblems, and thus demonstrate the usefulness of our method over a wide range. We show that we can accurately approximate the best ordering and the nonzero count of the sparse factorization matrix while speeding up the calculation by as much as 30.3× over the traditional serial method.

In recent years, a variety of randomized constructions of sketching matrices have been devised, that have been used in fast algorithms for numerical linear algebra problems, such as least squares regression, low-rank approximation, and the approximation of leverage scores. A key property of sketching matrices is that of subspace embedding. In this paper, we study sketching matrices that are obtained from bipartite graphs that are sparse, i.e., have left degree s that is small. In particular, we explore two popular classes of sparse graphs, namely, expander graphs and magical graphs. For a given subspace ${\mathcal{U}} \subseteq {{\mathbb{R}}^n}$ of dimension k, we show that the magical graph with left degree s = 2 yields a (1 ± ϵ) ℓ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> -subspace embedding for ${\mathcal{U}}$, if the number of right vertices (the sketch size) $m = {\mathcal{O}}\left( {{k^2}/{\varepsilon ^2}} \right)$. The expander graph with $s = {\mathcal{O}}(\log k/\varepsilon )$ yields a subspace embedding for $m = {\mathcal{O}}\left( {k\log k/{\varepsilon ^2}} \right)$. We also discuss the construction of sparse sketching matrices with reduced randomness using expanders based on error-correcting codes. Empirical results on various synthetic and real datasets show that these sparse graph sketching matrices work very well in practice.

A Parallel three-dimensional elasto-plastic finite element analysis in a workstation cluster environment

This paper focuses on Deterministic and Reliability Based Design Optimization (DO and RBDO) of composite stiffened panels considering post-buckling regime and progressive failure analysis. The ultimate load that a post-buckled panel can hold is to be maximised by changing the stacking sequence of both skin and stringers composite layups. The RBDO problem looks for a design that collapses beyond the shortening of failure obtained in the DO phase with a target reliability while considering uncertainty in the elastic properties of the composite material. The RBDO algorithm proposed is decoupled and hence separates the Reliability Analysis (RA) from the deterministic optimization. The main code to drive both the DO and RBDO approaches is written in MATLAB and employs Genetic Algorithms (GA) to solve the DO loops because discrete design variables and highly nonlinear response functions are expected. The code is linked with Abaqus to perform parallel explicit nonlinear finite element analyses in order to obtain the structural responses at each generation. The RA is solved through an inverse Most Probable failure Point (MPP) search algorithm that benefits from a Polynomial Chaos Expansion with Latin Hypercube Sampling (PCE-LHS) metamodel when the structural responses are required. The results led to small reductions in the maximum load that the panels can bear but otherwise assure that they will collapse beyond the shortening of failure imposed with a high reliability.