Schur Complement Method Research Articles

Flexible aircraft gust encounter simulation using subspace projection model reduction

Reduced-order modelling of multidisciplinary computational fluid dynamics can enable routine industrial gust load analysis. Key to the herein presented model reduction approach is that it modularly builds upon the widely accepted modal structural reduced-order model, where high-fidelity aerodynamics are simply projected onto the structural degrees-of-freedom, while opening a novel route to account for dominant modal aerodynamics in multidisciplinary edge-of-the-envelope flight physics. The influence of the flexible structure is captured by aeroelastic eigenmodes which originate in the structural equations. Such global aeroelastic modes for a large aircraft case with nearly 50 million degrees-of-freedom are computed for the first time using exact aerodynamics from computational fluid dynamics via the Schur complement method. The influence of atmospheric gusts is captured by modes from proper orthogonal decomposition applied to sinusoidal gust responses at discrete frequencies. Afterwards, the two sets of modes are combined and the linearised operator of the Reynolds-averaged Navier–Stokes equations coupled with a modal structural model is projected onto the resulting subspace. The constructed low-dimensional model can be solved rapidly for practical gust analysis giving accurate agreement with the full-order reference results throughout. Feasibility of aeroelastic model reduction, using the industry-grade computational fluid dynamics package DLR–TAU, for a relevant use case in transonic flight gust encounter is demonstrated.

Scalability of a parallel Schur complement method given the limitations for memory

In decomposition methods, the memory costs for solving the interface problem are increasing significantly with the number of subdomains increasing. Using of the Schur complement method allows you to reduce the number of iterations when the system is being solved. At the same time, the Schur complement matrix takes up more memory in comparison with global stiffness matrix. This imposes restrictions on the maximum size of the problem for which you can apply this method. Different approaches to reduce the costs and limitations of memory on stage of the construction and solving of the interface system exist. Parallel algorithm of the construction S with distributed storage of the matrix are considered when implementing using OpenMP and MPI technologies. This approach allows not only to reduce the limits on the maximum size of the solved problem, but also to resolve conflicts of shared memory access by increasing the number of independent parallel tasks.

A Finite Element Approach for the Dual Rudin--Osher--Fatemi Model and Its Nonoverlapping Domain Decomposition Methods

We consider a finite element discretization for the dual Rudin--Osher--Fatemi model using a Raviart--Thomas basis for $H_0 (\mathrm{div};\Omega)$. Since the proposed discretization has splitting property for the energy functional, which is not satisfied for existing finite difference based discretizations, it is more adequate for designing domain decomposition methods. In this paper, a primal domain decomposition method is proposed, which resembles the classical Schur complement method for the second order elliptic problems, and it achieves $O(1/n^2)$ convergence. A primal-dual domain decomposition method based on the method of Lagrange multipliers on the subdomain interfaces is also considered. Local problems of the proposed primal-dual domain decomposition method can be solved in linear convergence rate. Numerical results for the proposed methods are provided.

Schur complement solver for Quantum Monte-Carlo simulations of strongly interacting fermions

We present a non-iterative solver based on the Schur complement method for sparse linear systems of special form which appear in Quantum Monte-Carlo (QMC) simulations of strongly interacting fermions on the lattice. While the number of floating-point operations for this solver scales as the cube of the number of lattice sites, for practically relevant lattice sizes it is still significantly faster than iterative solvers such as the Conjugate Gradient method in the regime of strong inter-fermion interactions, for example, in the vicinity of quantum phase transitions. The speed-up is even more dramatic for the solution of multiple linear systems with different right-hand sides. We present benchmark results for QMC simulations of the tight-binding models on the hexagonal graphene lattice with on-site (Hubbard) and non-local (Coulomb) interactions, and demonstrate the potential for further speed-up using GPU.

A Schur-complement model-order-reduction technique for the finite element solution of transient elastohydrodynamic lubrication problems

The current work presents a reduced order finite element modeling framework for the solution of transient elastohydrodynamic lubrication (EHL) problems. The model order reduction technique is based on a Schur-complement method, applied to the elastic part within EHL problems. As such, it is exact and introduces no additional errors to the solution, with respect to the standard non-reduced finite element model. The technique reduces by one, the dimension of the linear elasticity part within the EHL problem. The use of the Schur-complement method however leads to a semi-dense Jacobian matrix. This is why the technique is complemented with a splitting procedure, allowing it to retrieve a standard finite-element-like sparsity pattern. In terms of computational performance, it is shown through a set of numerical experiments that the proposed reduced model offers a speed-up in computational times of the order of 15:1, compared to the equivalent full model, without any compromise on the accuracy of the solution.

Techniques for Efficient Computation of Electric Fields Generated by Transcranial Direct-Current Stimulation

Enhancement of computational efficiency is highly critical for finite-element analysis of electric fields generated by transcranial direct-current stimulation (tDCS) in order to foster field-analysis-based customized brain stimulation in practical scenarios. In this communication, we applied domain decomposition method (DDM) and adaptive mesh refinement method to the analysis of tDCS. DDM is likely to be particularly useful for tDCS field analysis problems with extracephalic reference electrodes. Our simulation results demonstrated that the DDM adopting the Schur complement method could reduce the overall computational time by 15% compared to the conventional single-domain analysis. On the other hand, to verify the enhancement of computational efficiency by adaptive mesh refinement, we used a realistic human head model with two sponge electrodes attached on the scalp surface. The distribution of numerical error estimated using an a posteriori error estimation method demonstrated that high errors were mostly concentrated on the edges and corners of the sponge electrodes. The overall solution accuracy could be remarkably enhanced by adding about 250 nodes around the high-error regions.

Stress-aware large-scale mesh editing using a domain-decomposed multigrid solver

In this paper, we develop a domain-decomposed subspace and multigrid solver to analyze the stress distribution for large-scale finite element meshes with millions of degrees of freedom. Through the domain decomposition technique, the shape editing directly updates the data structure of local finite element matrices. Doing so avoids the expensive factorization step in a direct solver and provides users with a progressive feedback of the stress distribution corresponding to the mesh operations: a fast preview is achieved through the subspace solver, and the multigrid solver refines the preview result if the user needs to examine the stress distribution carefully at certain design stages. Our system constructs the subspace for stress analysis using reduced constrained modes and builds a three-level multigrid solver through the algebraic multigrid method. We remove mid-edge nodes and lump unknowns with the Schur complement method. The updating and solving of the large global stiffness matrix are implemented in parallel after the domain decomposition. Experimental results show that our solver outperforms the parallel Intel MKL solver. Speedups of ▪ can be achieved for large-scale meshes with reasonable pre-computation costs when setting the stopping criterion of the multigrid solver to be 1e−3 relative error.

Optimal number of Schur subdomains: Application to semi-implicit finite volume discretization of semilinear reaction diffusion problem

The purpose of this paper is to establish a new numerical approach to solve, in two dimensions, a semilinear reaction diffusion equation combining finite volume method and Schur complement method. We applied our method for q = 2 non-overlapping subdomains and then we generalized in the case of several subdomains (q≥2). A large number of numerical test cases shows the efficiency and the good accuracy of the proposed approach in terms of the CPU time and the order of the error, when increasing the number of subdomains, without using the parallel computing. After several variations of the number of subdomains and the mesh grid, we remark two significant results. On the one hand, the increase related to the number of subdomains does not affect the order of the error, on the other hand, for each mesh grid when we augment the number of subdomains, the CPU time reaches the minimum for a specific number of subdomains. In order to have the minimum CPU time, we resorted to a statistical study between the optimal number of subdomains and the mesh grid.

Parallel Schur Complement Techniques Based on Multiprojection Methods

The simulation of several physical phenomena, arising from a wide class of scientific fields, requires solving large sparse linear systems effectively. In order to exploit appropriately the available supercomputing infrastructures, designing efficient and scalable solvers for large sparse linear systems is necessary. A hybrid parallel algebraic iterative method based on the Schur complement method and multiprojection techniques that utilizes semiaggregated subdomains is proposed. The graph corresponding to the coefficient matrix of the sparse linear system is partitioned by a graph partitioning algorithm and the Schur complement problem is solved by a preconditioned Krylov subspace method. The preconditioning scheme is based on domain decomposition and the resulting subdomains consist of fine and aggregated components from the inner boundaries. Moreover, two variants for approximating the local Schur complements, derived from the semiaggregation procedure, are proposed in order to improve memory requirements and preprocessing time. The proposed method is suitable for distributed memory systems with multicore nodes and has improved convergence behavior for a large number of workstations. Numerical and comparative results concerning the applicability, the convergence behavior, and the parallel performance of the proposed scheme are given.

Numerical Modelling of Foundation Slabs with use of Schur Complement Method

The paper discusses numerical modelling of foundation slabs with use of advanced numerical approaches, which are suitable for parallel processing. The solution is based on the Finite Element Method with the slab-type elements. The subsoil is modelled with use of Winklertype contact model (as an alternative a multi-parameter model can be used). The proposed modelling approach uses the Schur Complement method to speed-up the computations of the problem. The method is based on a special division of the analyzed model to several substructures. It adds some complexity to the numerical procedures, especially when subsoil models are used inside the finite element method solution. In other hand, this method makes possible a fast solution of large models but it introduces further problems to the process. Thus, the main aim of this paper is to verify that such method can be successfully used for this type of problem. The most suitable finite elements will be discussed, there will be also discussion related to finite element mesh and limitations of its construction for such problem. The core approaches of the implementation of the Schur Complement Method for this type of the problem will be also presented. The proposed approach was implemented in the form of a computer program, which will be also briefly introduced. There will be also presented results of example computations, which prove the speed-up of the solution - there will be shown important speed-up of solution even in the case of on-parallel processing and the ability of bypass size limitations of numerical models with use of the discussed approach.

Deployment dynamics simulation and ground test of a large space hoop truss antenna reflector

In recent years, the space community has paid much attention to the development of the hoop truss antenna of satellite so as to improve the quality of telecommunication of high frequency. The antenna reflector is mainly composed of a mesh net and a deployable ring truss, which can be considered as a rigid-flexible multibody system. During the research process of the hoop truss antenna of satellite, an accurate and effective dynamics model has to be built. In order to describe the flexible components in this system accurately, the Absolute-Coordinate-Based (ACB) method combining the Absolute Nodal Coordinate Formulation (ANCF) and Natural Coordinate Formulation (NCF) is used to predict the deployment dynamics of the reflector antenna. Furthermore, the parallel computation methodology based on the multilevel decomposition method and the Schur Complement method is used to improve the computational efficiency. Using the parallel computation methodology, the dynamic simulation of a rigid-flexible multibody system is efficiently completed based on the Message Passing Interface. To analyze the asynchronous deployment phenomenon in the ground test, the decrease of the driving force is applied to the antenna reflector model. Finally, based on the above method, a computational program is developed via the Fortran language to analyze the deployment dynamics of the hoop truss mesh antennas. Using the program, the second phase of the deployment process of the hoop truss mesh antennas is simulated, and the computational results are verified by a deployment experiment on the 4 m aperture antenna reflector.

Legendre–Galerkin spectral-element method for the biharmonic equations and its applications

A spectral-element method based on the Legendre–Galerkin approximation is presented to solve the two-dimensional biharmonic equations. Rigorous error analysis is carried out to establish the convergence of the method. By constructing appropriate basis functions which satisfy the boundary conditions of the differential equations, the discrete variational formulation is reduced to linear system with sparse and symmetric matrices, which can be efficiently solved by a fast Schur-complement method. Accuracy test is provided to confirm the convergence rate of the theoretical results. Finally, the proposed method is applied to calculate the displacement of an elastic plate under a uniform applied load and stream function of zero Reynolds number flow in a driven cavity, the results compare well with established results.

A Schur Complement Preconditioner for Scalable Parallel Fluid Simulation

We present an algorithmically efficient and parallelized domain decomposition based approach to solving Poisson’s equation on irregular domains. Our technique employs the Schur complement method, which permits a high degree of parallel efficiency on multicore systems. We create a novel Schur complement preconditioner which achieves faster convergence, and requires less computation time and memory. This domain decomposition method allows us to apply different linear solvers for different regions of the flow. Subdomains with regular boundaries can be solved with an FFT-based Fast Poisson Solver. We can solve systems with 1,024 3 degrees of freedom, and demonstrate its use for the pressure projection step of incompressible liquid and gas simulations. The results demonstrate considerable speedup over preconditioned conjugate gradient methods commonly employed to solve such problems, including a multigrid preconditioned conjugate gradient method.

A Schur Complement Preconditioner for Scalable Parallel Fluid Simulation

We present an algorithmically efficient and parallelized domain decomposition based approach to solving Poisson’s equation on irregular domains. Our technique employs the Schur complement method, which permits a high degree of parallel efficiency on multicore systems. We create a novel Schur complement preconditioner which achieves faster convergence, and requires less computation time and memory. This domain decomposition method allows us to apply different linear solvers for different regions of the flow. Subdomains with regular boundaries can be solved with an FFT-based Fast Poisson Solver. We can solve systems with 1,024 3 degrees of freedom, and demonstrate its use for the pressure projection step of incompressible liquid and gas simulations. The results demonstrate considerable speedup over preconditioned conjugate gradient methods commonly employed to solve such problems, including a multigrid preconditioned conjugate gradient method.

A schur complement preconditioner for scalable parallel fluid simulation

We present an algorithmically efficient and parallelized domain decomposition based approach to solving Poisson’s equation on irregular domains. Our technique employs the Schur complement method, which permits a high degree of parallel efficiency on multicore systems. We create a novel Schur complement preconditioner which achieves faster convergence, and requires less computation time and memory. This domain decomposition method allows us to apply different linear solvers for different regions of the flow. Subdomains with regular boundaries can be solved with an FFT-based Fast Poisson Solver. We can solve systems with 1,0243 degrees of freedom, and demonstrate its use for the pressure projection step of incompressible liquid and gas simulations. The results demonstrate considerable speedup over preconditioned conjugate gradient methods commonly employed to solve such problems, including a multigrid preconditioned conjugate gradient method.

A Schur Complement Method for Optimum Experimental Design in the Presence of Process Noise

Optimization problems arising in Optimum Experimental Design (OED) applications require repeated covariance matrix evaluations of the underlying Parameter Estimation (PE) problems. The complexity of this task grows quickly with the problem dimensions, especially when process noise is considered. In this paper, a Schur complement method is proposed to alleviate this problem by translating the covariance matrix evaluation into the solution of a sparse linear system and the inversion of a small-scale matrix. The method is used as a building block for the open-source software package casiopeia, a powerful and easy-to-use environment for OED and PE. The performance of the software and the proposed Schur complement method is assessed on a numerical example.

Intel Xeon Phi acceleration of Hybrid Total FETI solver

This paper describes an approach for acceleration of the Hybrid Total FETI (HTFETI) domain decomposition method using the Intel Xeon Phi coprocessors. The HTFETI method is a memory bound algorithm which uses sparse linear BLAS operations with irregular memory access pattern. The presented local Schur complement (LSC) method has regular memory access pattern, that allows the solver to fully utilize the Intel Xeon Phi fast memory bandwidth.This translates to speedup over 10.9 of the HTFETI iterative solver when solving 3 billion unknown heat transfer problem (3D Laplace equation) on almost 400 compute nodes. The comparison is between the CPU computation using sparse data structures (PARDISO sparse direct solver) and the LSC computation on Xeon Phi. In the case of the structural mechanics problem (3D linear elasticity) of size 1 billion DOFs the respective speedup is 3.4.The presented speedups are asymptotic and they are reached for problems requiring high number of iterations (e.g., ill-conditioned problems, transient problems, contact problems). For problems which can be solved with under hundred iterations the local Schur complement method is not optimal. For these cases we have implemented sparse matrix processing using PARDISO also for the Xeon Phi accelerators.

Numerical analysis of coupled heat and moisture transport in masonry

Coupled analysis of heat and moisture transport in real world masonry structures deserves a special attention because the spatial discretization by the finite element method leads usually to large number of degrees of freedom. Thin mortar layers and large bricks or stones have very different material properties and the finite element mesh has to be able to describe correct temperature and moisture fields in mortar and in its vicinity in the blocks. This paper describes two possible solutions of such problems. The first solution is based on the domain decomposition method executed on parallel computers, where the Schur complement method is used with respect to non-symmetric systems of algebraic equations. The second alternative method is the application of a multi-scale approach in connection with a processor farming method, where the whole structure is described by a reasonably coarse finite element mesh, called the macro-scale problem, and the material parameters are obtained from the lower-level problems, called the meso-scale problem, by a homogenization procedure. In this procedure, the macro-problem is assigned to the master processor and the meso-scale problems belong to the slave processors in the processor farm.

Finite element analysis of switched reluctance motor with rotor position based control

This research presents a field-circuit coupled parallel finite element model of a switched reluctance motor embedded in a simple closed loop control system. The parallel numerical model is based on the Schur-complement method coupled with an iterative solver. The used control system is the rotor position based control, which is applied to the FEM model. The results and parallel performance of the voltage driven finite element model are compared with the results from the current driven model. Moreover, the results of the start-up of the loaded motor show why the model accuracy is important in the control loop.

A scalable schur-complement fluids solver for heterogeneous compute platforms

We present a scalable parallel solver for the pressure Poisson equation in fluids simulation which can accommodate complex irregular domains in the order of a billion degrees of freedom, using a single server or workstation fitted with GPU or Many-Core accelerators. The design of our numerical technique is attuned to the subtleties of heterogeneous computing, and allows us to benefit from the high memory and compute bandwidth of GPU accelerators even for problems that are too large to fit entirely on GPU memory. This is achieved via algebraic formulations that adequately increase the density of the GPU-hosted computation as to hide the overhead of offloading from the CPU, in exchange for accelerated convergence. Our solver follows the principles of Domain Decomposition techniques, and is based on the Schur complement method for elliptic partial differential equations. A large uniform grid is partitioned in non-overlapping subdomains, and bandwidth-optimized (GPU or Many-Core) accelerator cards are used to efficiently and concurrently solve independent Poisson problems on each resulting subdomain. Our novel contributions are centered on the careful steps necessary to assemble an accurate global solver from these constituent blocks, while avoiding excessive communication or dense linear algebra. We ultimately produce a highly effective Conjugate Gradients preconditioner, and demonstrate scalable and accurate performance on high-resolution simulations of water and smoke flow.