Discrete Linear System Of Equations Research Articles

A Generic Finite Element Framework on Parallel Tree-Based Adaptive Meshes

In this work we formally derive and prove the correctness of the algorithms and data structures in a parallel, distributed-memory, generic finite element framework that supports h-adaptivity on computational domains represented as forest-of-trees. The framework is grounded on a rich representation of the adaptive mesh suitable for generic finite elements that is built on top of a low-level, light-weight forest-of-trees data structure handled by a specialized, highly parallel adaptive meshing engine, for which we have identified the requirements it must fulfill to be coupled into our framework. Atop this two-layered mesh representation, we build the rest of data structures required for the numerical integration and assembly of the discrete system of linear equations. We consider algorithms that are suitable for both subassembled and fully-assembled distributed data layouts of linear system matrices. The proposed framework has been implemented within the FEMPAR scientific software library, using p4est as a practical forest-of-octrees demonstrator. A strong scaling study of this implementation when applied to Poisson and Maxwell problems reveals remarkable scalability up to 32.2K CPU cores and 482.2M degrees of freedom. Besides, a comparative performance study of FEMPAR and the state-of-the-art deal.ii finite element software shows at least comparative performance, and at most factor 2-3 improvements in the h-adaptive approximation of a Poisson problem with first- and second-order Lagrangian finite elements, respectively.

Linear least-squares method for global luminescent oil film skin friction field analysis.

A data analysis method based on the linear least-squares (LLS) method was developed for the extraction of high-resolution skin friction fields from global luminescent oil film (GLOF) visualization images of a surface in an aerodynamic flow. In this method, the oil film thickness distribution and its spatiotemporal development are measured by detecting the luminescence intensity of the thin oil film. From the resulting set of GLOF images, the thin oil film equation is solved to obtain an ensemble-averaged (steady) skin friction field as an inverse problem. In this paper, the formulation of a discrete linear system of equations for the LLS method is described, and an error analysis is given to identify the main error sources and the relevant parameters. Simulations were conducted to evaluate the accuracy of the LLS method and the effects of the image patterns, image noise, and sample numbers on the results in comparison with the previous snapshot-solution-averaging (SSA) method. An experimental case is shown to enable the comparison of the results obtained using conventional oil flow visualization and those obtained using both the LLS and SSA methods. The overall results show that the LLS method is more reliable than the SSA method and the LLS method can yield a more detailed skin friction topology in an objective way.

An Adaptive Mesh Refinement Strategy for Immersed Boundary/Interface Methods

An adaptive mesh refinement strategy is proposed in this paper for the Immersed Boundary and Immersed Interface methods for two-dimensional elliptic interface problems involving singular sources. The interface is represented by the zero level set of a Lipschitz function φ(x,y). Our adaptive mesh refinement is done within a small tube of |φ(x,y)|≤ δ with finer Cartesian meshes. The discrete linear system of equations is solved by a multigrid solver. The AMR methods could obtain solutions with accuracy that is similar to those on a uniform fine grid by distributing the mesh more economically, therefore, reduce the size of the linear system of the equations. Numerical examples presented show the efficiency of the grid refinement strategy.

Inverse Estimation of the Thermal Conductivity in a One-Dimensional Domain by Taylor Series Approach

The Taylor series approximation is developed for the inverse estimation of thermal conductivity in a one-dimensional domain. The differential governing equation of heat conduction is converted to a discrete system of linear equations in matrix form using the temperature measurement and heat generation at the grid points as well as surface heat flux. The unknown thermal conductivity is estimated by solving the linear algebraic equations directly without iterations. The features of the present method are that no prior information about the functional form of the thermal conductivity is required, nor are any initial guesses or iterations in the calculation process needed. The accuracy and robustness of the present method are verified by comparing the results with the analytical solutions for constant, spatial- and temperature-dependent thermal conductivities. The results show that the inverse solutions are in good agreement with the exact solutions.

Divergence-free wavelet solution to the Stokes problem

In this paper, we use divergence-free wavelets to give an adaptive solution to the velocity field of the Stokes problem. We first use divergence-free wavelets to discretize the divergence-free weak formulation of the Stokes problem and obtain a discrete positive definite linear system of equations whose coefficient matrix is quasi-sparse; Secondly, an adaptive scheme is used to solve the discrete linear system of equations and the error estimation and complexity analysis are given.

A Multigrid Preconditioner for the Mixed Formulation of Linear Plane Elasticity

In this paper, we develop a multigrid preconditioner for the discrete system of linear equations that results from the mixed formulation of the linear plane elasticity problem using the Arnold-Winther elements. This, in turn, can be reduced to the problem of finding a multigrid preconditioner for the form $(\cdot,\cdot)+(\vdiv\cdot,\vdiv\cdot)$ in the symmetric matrix space resulting from Arnold-Winther elements. Since the form is not uniformly elliptic, a Helmholtz-type decomposition is essential. The Arnold-Winther finite element space gives rise to nonnested multilevel spaces adding difficulty to the analysis. We prove that for the variable V-cycle multigrid preconditioner, the condition number of the preconditioned system is independent of the number of levels. The results of numerical experiments are also presented.

First-order system least squares (FOSLS) for coupled fluid-elastic problems

Mathematical models for the mechanical coupling between a moving fluid and an elastic solid are inherently nonlinear because the shape of the Eulerian fluid domain is not known a priori – it is at least partially determined by the displacement of the elastic solid. In this paper, a first-order system least squares finite element formulation is used to solve the nonlinear system of model equations using different iteration techniques, including an approach where the equations are fully coupled and two other approaches in which the equations are decoupled. The discrete linear system of equations is solved using an algebraic multigrid solver as a preconditioner for a conjugate gradient iteration. The numerical results show that the approach is optimal in the sense that computational cost is proportional to the degrees of freedom. The results also show that the choice of iteration method, from fully coupled to fully decoupled, does not significantly effect computational cost, but it does influence the error in the solution.

The natural element method in solid mechanics

The application of the Natural Element Method (NEM) to boundary value problems in two-dimensional small displacement elastostatics is presented. The discrete model of the domain Ω consists of a set of distinct nodes N, and a polygonal description of the boundary ∂Ω. In the Natural Element Method, the trial and test functions are constructed using natural neighbour interpolants. These interpolants are based on the Voronoi tessellation of the set of nodes N. The interpolants are smooth (C∞) everywhere, except at the nodes where they are C0. In one-dimension, NEM is identical to linear finite elements. The NEM interpolant is strictly linear between adjacent nodes on the boundary of the convex hull, which facilitates imposition of essential boundary conditions. A methodology to model material discontinuities and non-convex bodies (cracks) using NEM is also described. A standard displacement-based Galerkin procedure is used to obtain the discrete system of linear equations. Application of NEM to various problems in solid mechanics, which include, the patch test, gradient problems, bimaterial interface, and a static crack problem are presented. Excellent agreement with exact (analytical) solutions is obtained, which exemplifies the accuracy and robustness of NEM and suggests its potential application in the context of other classes of problems—crack growth, plates, and large deformations to name a few. © 1998 John Wiley & Sons, Ltd.

An exact analytic solution to segmented-mirror adaptive-optics control

An innovative method of phase compensation employs a segmented primary mirror in place of a conventional deformable mirror. Segmentation of the primary mirror offers an inexpensive method to produce large, active telescopes or beam directors. The success of this approach for adaptive optics hinges on the ability to control the segments. The segmented mirror will adopt the requisite conjugate beacon phase front to the level of precision of the wave-front-sensor measurements if it is made to behave like a continuous deformable membrane. We show that this is equivalent to applying the measured wave-front slopes directly to the segments and then matching adjacent edge midpoints. This discrete linear system of equations for the segment pistons is singular, and no general solution exists. We have succeeded in analytically solving for the segmented-mirror configuration that minimizes the sum of the squares of the differences in height of adjacent segment midpoints. This solution results from the identification of constraints on the surface. The constraints have a natural geometric interpretation as continuity loops. The knowledge of this constructive solution eliminates the need to do iterations or the need to develop iterative control algorithms. This solution functional can be found in advance for any particular mirror design and relates segment-midpoint height differences to the measured input tilt field in a fully deterministic and unique way.

A numerical method of solution of the Boltzmann equation in a weakly ionized heterogeneous gas

We give a numerical method of solution of the Boltzmann equation for a weakly ionized gas based on the study of the characteristic curves defined by the free motion of particles. First, we perform a systematic study of the topological properties of these characteristic lines when collisions are present. We show that their distribution not only gives us a method of attack for the numerical resolution but also allows us to predict, even before solving the Boltzmann equation, the complete behavior of the distribution function. Second, we define several discrete schemes and we prove that they are convergent and stable. We also give a brief proof of the convergence of the iterative process associated with the resolution of the discrete linear system of equations. Finally, we give the results obtained for helium gas, both the transmission factor of electrons and the isotropic part of the distribution function. The main feature to be emphasized is the existence of a distribution presenting a shape with a succession of “knobs” and “hollows” characteristic of a nonequilibrium state.