Parallel Multifrontal Solver Research Articles

Comparison of the Structure of Equation Systems and the GPU Multifrontal Solver for Finite Difference, Collocation and Finite Element Method

Abstract The article is an in-depth comparison of numerical solvers and corresponding solution pro- cesses of the systems of algebraic equations resulting from finite difference, collocation, and finite element approximations. The paper considers recently developed isogeometric versions of the collocation and finite element methods, employing B-splines for the computations and ensuring Cp − 1 continuity on the borders of elements for the B-splines of the order p . For solving the systems, we use our GPU implementation of the state-of-the-art parallel multifrontal solver, which leverages modern GPU architectures and allows to reduce the complexity. We analyze the structures of linear equation systems resulting from each of the methods and how different matrix structures lead to different multifrontal solver elimination trees. The paper also considers the flows of multifrontal solver depending on the originally employed method.

Massively parallel structured direct solver for equations describing time-harmonic qP-polarized waves in TTI media

We considered the discretization and approximate solutions of equations describing time-harmonic qP-polarized waves in 3D inhomogeneous anisotropic media. The anisotropy comprises general (tilted) transversely isotropic symmetries. We are concerned with solving these equations for a large number of different sources. We considered higher-order partial differential equations and variable-order finite-difference schemes to accommodate anisotropy on the one hand and allow higher-order accuracy — to control sampling rates for relatively high frequencies — on the other hand. We made use of a nested dissection based domain decomposition in a massively parallel multifrontal solver combined with hierarchically semiseparable matrix compression techniques. The higher-order partial differential operators and the variable-order finite-difference schemes require the introduction of separators with variable thickness in the nested dissection; the development of these and their integration with the multifrontal solver is the main focus of our study. The algorithm that we developed is a powerful tool for anisotropic full-waveform inversion.

A finite element multifrontal method for 3D CSEM modeling in the frequency domain

There has been a recent increase in the use of controlled-source electromagnetic (CSEM) surveys in the exploration for oil and gas. We developed a modeling scheme for 3D CSEM modeling in the frequency domain. The electric field was decomposed in primary and secondary components to eliminate the singularity originated by the source term. The primary field was calculated using a closed form solution, and the secondary field was computed discretizing a second-order partial differential equation for the electric field with the edge finite element. The solution to the linear system of equations was obtained using a massive parallel multifrontal solver, because such solvers are robust for indefinite and ill-conditioned linear systems. Recent trends in parallel computing were investigated for their use in mitigating the computational overburden associated with the use of a direct solver, and of its feasibility for 3D CSEM forward modeling with the edge finite element. The computation of the primary field was parallelized, over the computational domain and the number of sources, using a hybrid model of parallelism. When using a direct solver, the attainment of multisource solutions was only competitive if the same factors are used to achieve a solution for multi right-hand sides. This aspect was also investigated using the presented methodology. We tested our proposed approach using 1D and 3D synthetic models, and they demonstrated that it is robust and suitable for 3D CSEM modeling using a distributed memory system. The codes could thus be used to help design new surveys, as well to estimate subsurface conductivities through the implementation of an appropriate inversion scheme.

Parallel multi-frontal solver for multi-physics p adaptive problems

The paper presents a parallel direct solver for multi-physics problems. The solver is dedicated for solving problems resulting from adaptive Finite Element Method computations. The concept of finite element is actually replaced by the concept of the node. The computational mesh consists of several nodes, related to element vertices, edges, faces and interiors. The ordering of unknowns in the solver is performed on the level of nodes. The concept of the node can be efficiently utilized in order to recognize unknowns that can be eliminated at a given node of the elimination tree. The solver is tested on the exemplary three dimensional multi-physics problem involving the computations of the linear acoustics coupled with linear elasticity. The three dimensional tetrahedral mesh generation and the solver algorithm are modeled by using graph grammar formalism. The execution time and the memory usage of the solver are compared with the MUMPS solver.

Parallel multi-frontal solver for p adaptive finite element modeling of multi-physics computational problems

The paper presents a parallel direct solver for multi-physics problems. The solver is dedicated for solving problems resulting from adaptive finite element method computations. The concept of finite element is actually replaced by the concept of the node. The computational mesh consists of several nodes, related to element vertices, edges, faces and interiors. The ordering of unknowns in the solver is performed on the level of nodes. The concept of the node can be efficiently utilized in order to recognize unknowns that can be eliminated at a given node of the elimination tree. The solver is tested on the exemplary three-dimensional multi-physics problem involving the computations of the linear acoustics coupled with linear elasticity. The three-dimensional tetrahedral mesh generation and the solver algorithm are modeled by using graph grammar formalism. The execution time and the memory usage of the solver are compared with the MUMPS solver.

Finite element modelling of the squeeze casting process

PurposeThis paper sets out to present developments of a numerical model of squeeze casting process.Design/methodology/approachThe entire process is modelled using the finite element method. The mould filling, associated thermal and thermomechanical equations are discretized using the Galerkin method. The front in the filling analysis is followed using volume of fluid method and the advection equation is discretized using the Taylor Galerkin method. The coupling between mould filling and the thermal problem is achieved by solving the thermal equation explicitly at the end of each time step of the Navier Stokes and advection equations, which allows one to consider the actual position of the front of the filling material. The thermomechanical problem is defined as elasto‐visco‐plastic described in a Lagrangian frame and is solved in the staggered mode. A parallel version of the thermomechanical program is presented. A microstructural solidification model is applied.FindingsDuring mould filling a quasi‐static Arbitrary Lagrangian Eulerian (ALE) is applied and the resulting temperatures distribution is used as the initial condition for the cooling phase. During mould filling the applied pressure can be used as a control for steering the distribution of the solidified fractions.Practical implicationsThe presented model can be used in engineering practice. The industrial examples are shown.Originality/valueThe quasi‐static ALE approach was found to be applicable to model the industrial SQC processes. It was found that the staggered scheme of the solution of the thermomechanical problem could parallelize using a multifrontal parallel solver.

Parallel computation of a damage localization problem using parallel multifrontal solver

In this paper, nonlinear parallel structural analyses are performed using a distributed memory sparse direct multifrontal linear solver. The linear solver is fully parallel, working with substructures determined by a parallel graph partitioner. The parallel performance of the nonlinear parallel algorithm is demonstrated using damage localization problems for two and three dimensional crack models. Convergence studies of the predicted local variation of damage at the crack tip are performed using discretizations with up to one million degrees of freedom. Implementation issues for damage localization problems are discussed, including the selection of independent variables and the use of Riks’ continuation method.

다중 프런트 해법을 이용한 비선형 구조문제의 병렬계산

본 논문에서는, 병렬 다중 프런트 해법을 이용한 비선형 병렬 구조해석 과정을 소개하고 이의 응용예제로써 2차원 및 3차원 균열 모델에서의 손상 국부화를 모사하였다. 비선형 유한요소 해석과 연속체 손상역학과 관련된 병렬 알고리듬을 구현하였고, 비선형 손상해석에서 사용되는 많은 변수들을 저장하기 위해서는 메모리가 많이 필요하므로 이를 줄이는 방안을 고려하였다. 그리고 손상 진전으로 인한 변형도 연화가 발생할 때 해를 구하기 위해 사용한 Riks의 연속법도 병렬 계산에 맞도록 수정하였다. 수치예제로써, 최대 약 1백만 자유도를 가진 모델들을 사용하여 손상 국부화 문제를 해석하였는데, 이 예제에서 비선형 병렬 알고리듬의 병렬효율과 자유도 변화에 따른 균열 끝단에서의 손상 변화를 살펴보았다. In this paper, nonlinear parallel structural analyses are introduced by using the parallel multifrontal solver and damage localization for 2D and 3D crack models is presented as the application of nonlinear parallel computation. The parallel algorithms related with nonliear reduce the amount of memory used is carried out because many variables should be utilized for this highly nonlinear damage analysis. Also, Riks' continuation method is parallelized to search the solution when strain softening occurs due to damage evolution. For damage localization problem, several computational models having up to around 1-million degree of freedoms are used. The parallel performance in this nonlinear parallel algorithm is shown through these examples and the local variation of damage at crack tip is compared among the models with different degree of freedoms.

Direct Numerical Simulation of Composite Structures

Fiber-reinforced composites are not chemical compounds but physical mixtures of fiber and matrix, and the constituents are bonded together. Therefore, it is natural and essential to adopt a full microscopic model and directly analyze the model with no assumptions for local deformation or local loading conditions, in order to understand and predict not only the averaged or homogenized behaviors but also the detailed microscopic behaviors of composite structures. However, in spite of the necessity, full microscopic models of composite structures have rarely been dealt with, mainly because of their difficulties arising in the actual computation of the finite element model with immense degree of freedoms. In this work, to overcome the difficulties and analyze full microscopic models of composite structures, an efficient parallel multifrontal solver, which can utilize distributed computing resources unlimitedly, is developed and applied to the Direct Numerical Simulation (DNS) of composite structures. Using the developed code, feasibility studies are carried out to observe whether or not the proposed solver is adequate for the DNS of composite structures, such as in parallel performance and others. As an example of DNS, virtual experiments are carried out, where material constants are directly obtained through DNS. Comparisons with previous experimental and theoretical results show the promising possibility that the present virtual experiments by the DNS of composite structures can be an alternative way to obtain material constants without expensive real experiments. Additionally, a microscopic model with defects is also analyzed and compared with a perfect model to validate the potential of the DNS in dealing with irregularity of microscopic models of composite structures coming from imperfection. The examples show the future direction of analysis and design of composite structures as well as the usefulness of the proposed DNS methodology.

Large-Scale Structural Analysis by Parallel Multifrontal Solver Through Internet-Based Personal Computers

Internet supercomputing methodology is introduced, and the concept is realized for large-scale finite element analysis. This is enabled by an efficient out-of-core parallel solver, which is based on domainwise or elementwise multifrontal approach. The primary resources of Internet supercomputing are numerous idling personal computers connected by the Internet irrespective of their locations. The computing ability of hundreds or thousands personal computers connected by the Internet can be as powerful as that of supercomputer such as the IBM SP system if these personal computers are utilized for solving a problem simultaneously through an efficient parallel computing algorithm. Under the described concept, a virtual supercomputing system InterSup I is constructed and tested. To establish the InterSup I system, 64 personal computer nodes, which are located in several places and connected by the Internet, are conscripted. When the established InterSup I system is used, linear static analyses of a finite element model having around 2 × 10 6 degrees of freedom are done through the developed parallel multifrontal solver. Some numerical tests have been carried out to investigate the affordability and effectiveness of Internet supercomputing. The possibility of Internet supercomputing through Internet personal computers as an alternative supercomputing power for high-performance computing is provided in this research.

Large-scale structural analysis by parallel multifrontal solver through Internet-based personal computers

Internet supercomputing methodology is introduced, and the concept is realized for large-scale finite element analysis. This is enabled by an efficient out-of-core parallel solver, which is based on domainwise or elementwise multifrontal approach. The primary resources of Internet supercomputing are numerous idling personal computers connected by the Internet irrespective of their locations. The computing ability of hundreds or thousands personal computers connected by the Internet can be as powerful as that of supercomputer such as the IBM SP system if these personal computers are utilized for solving a problem simultaneously through an efficient parallel computing algorithm. Under the described concept, a virtual supercomputing system InterSup I is constructed and tested. To establish the InterSup I system, 64 personal computer nodes, which are located in several places and connected by the Internet, are conscripted. When the established InterSup I system is used, linear static analyses of a finite element model having around 2 × 10 6 degrees of freedom are done through the developed parallel multifrontal solver. Some numerical tests have been carried out to investigate the affordability and effectiveness of Internet supercomputing. The possibility of Internet supercomputing through Internet personal computers as an alternative supercomputing power for high-performance computing is provided in this research.