Nonconforming Subdomain Research Articles

About Revisiting Domain Decomposition Methods for Poroelasticity

In this paper, we revisit well-established domain decomposition (DD) schemes to perform realistic simulations of coupled flow and poroelasticity problems on parallel computers. We define distinct solution schemes to take into account different transmission conditions among subdomain boundaries. Indeed, we examine two different approaches, i.e., Dirichlet-Neumann (DN) and the mortar finite element method (MFEM), and we recognize their advantages and disadvantages. The MFEM significantly lessens the computational cost of reservoir compaction and subsidence calculations by dodging the conforming Cartesian grids that arise from the pay-zone onto its vicinity. There is a manifest necessity of producing non-matching interfaces between the reservoir and its neighborhood. We thus employ MFEM over nonuniform rational B-splines (NURBS) surfaces to stick these non-conforming subdomain parts. We then decouple the mortar saddle-point problem (SPP) using the Dirichlet-Neumann domain decomposition (DNDD) scheme. We confirm that this procedure is proper for calculations at the field level. We also carry comprehensive comparisons between the conventional and non-matching solutions to prove the method’s accuracy. Examples encompass linking finite element codes for slightly compressible single-phase and poroelasticity. We have used this program to a category of problems ranking from near-borehole applications to whole field subsidence estimations.

2-D Reduced Model for Eddy Currents Calculation in Litz Wire and Its Application for Switched Reluctance Machine

The strong correlation between the level of eddy current losses and the winding geometry shows the necessity to pay attention to the manner of disposition of coils in machine slots. The conductor type, whether it is solid or stranded, also has a crucial influence. This paper suggests an electromagnetic analysis of complex stranded conductors, such as litz wire (LW) and twisted wire. Each conductor is composed of individual strands that are woven or twisted in the slot throughout the machine length. Combined with several electric circuit relationships that interpret the strands transposition, a 2-D finite-element (FE) model is developed in this paper so that the copper losses in each strand can be calculated individually. A model reduction is also proposed in the case of LWs. This allows the benefit of only one complete FE solution to find fast solutions in the slots’ domains when any variation of geometrical data occurrs. It allows adapting non-conforming subdomains’ meshes and seeing that it prevents the Newton–Raphson iterations and the moving band technique, the model reduction presents clear interests in repetitive analysis, such as winding optimization processes.

A parallel coupling strategy for the Chimera and domain decomposition methods in computational mechanics

Domain Decomposition Methods (DDMs) are techniques that divide the solution of a PDE on a domain into smaller solutions on smaller subdomains coupling them using a certain strategy. They are used for essentially two purposes: designing parallel solvers and/or coupling subdomains with different meshes, different numerical approximations, etc. In this paper we are interested in this last category. One example of application is the Chimera method. In that sense, the Chimera method can be viewed as a preprocess technique plus a DDM on overlapping and non-conforming subdomains. The coupling technique of DDM is usually achieved via transmission conditions to impose the continuities of the unknown and its flux across the subdomain boundaries. We propose in this work an alternative coupling strategy, intervening as a preprocess method. It consists in connecting the nodes of one subdomain with the nodes of the adjacent subdomains via newly created elements. In this way, the multi-domain character of a DDM disappears, making it a parallel, implicit and versatile method. We discuss in this paper the relation between the proposed method and the existing coupling strategies. We also present some convergence results as well as some applications to the Navier–Stokes equations and other PDE’s.

A Three-Level BDDC Algorithm for Mortar Discretizations

In this paper, a three-level balancing domain decomposition by constraints (BDDC) algorithm is developed for the solutions of large sparse algebraic linear systems arising from the mortar discretization of elliptic boundary value problems. The mortar discretization is considered on geometrically nonconforming subdomain partitions. In two-level BDDC algorithms, the coarse problem needs to be solved exactly. However, its size will increase with the increase of the number of the subdomains. To overcome this limitation, the three-level algorithm solves the coarse problem inexactly while a good rate of convergence is maintained. This is an extension of previous work: the three-level BDDC algorithms for standard finite element discretization. Estimates of the condition numbers are provided for the three-level BDDC method, and numerical experiments are also discussed.

A BDDC Method for Mortar Discretizations Using a Transformation of Basis

A BDDC (balancing domain decomposition by constraints) method is developed for elliptic equations, with discontinuous coefficients, discretized by mortar finite element methods for geometrically nonconforming partitions in both two and three space dimensions. The coarse component of the preconditioner is defined in terms of one mortar constraint for each edge/face, which is the intersection of the boundaries of a pair of subdomains. A condition number bound of the form $C\max_i\{(1+\log(H_i/h_i))^2\}$ is established under certain assumptions on the geometrically nonconforming subdomain partition in the three-dimensional case. Here $H_i$ and $h_i$ are the subdomain diameters and the mesh sizes, respectively. In the geometrically conforming case and the geometrically nonconforming cases in two dimensions, no assumptions on the subdomain partition are required. This BDDC preconditioner is also shown to be closely related to the Neumann-Dirichlet version of the FETI-DP algorithm. The results are illustrated by numerical experiments which confirm the theoretical results.

A FETI-DP Formulation of Three Dimensional Elasticity Problems with Mortar Discretization

In this paper, a FETI-DP formulation for three dimensional elasticity on nonmatching grids over geometrically nonconforming subdomain partitions is considered. To resolve the nonconformity of the finite elements, a mortar matching condition is imposed on the subdomain interfaces (faces). A FETI-DP algorithm is then built by enforcing the mortar matching condition in dual and primal ways. In order to make the FETI-DP algorithm scalable, a set of primal constraints, which include average and momentum constraints over interfaces, are selected from the mortar matching condition. A condition number bound, $C(1+\log(H/h))^2$, is then proved for the FETI-DP formulation for the elasticity problems with discontinuous material parameters. Only some faces need to be chosen as primal faces on which the average and momentum constraints are imposed. Numerical results are included.

A BDDC Algorithm for Mortar Discretization of Elasticity Problems

A balancing domain decomposition by constraints (BDDC) algorithm is developed for compressible elasticity problems in three dimensions with mortar discretization on geometrically nonconforming subdomain partitions. Material parameters of the elasticity problems may have jump across the subdomain interface. Coarse basis functions in the BDDC algorithm are constructed from primal constraints on faces, which are similar to the average matching condition and the moment matching condition considered in [A. Klawonn and O. B. Widlund, Comm. Pure Appl. Math., 59 (2006), pp. 1523–1572] and [H. H. Kim, A FETI-DP Formulation of Three Dimensional Elasticity Problems with Mortar Discretization, Technical report, New York University, 2005]. A condition number bound is proved to be C(1+log(H/h))$^3$ for geometrically nonconforming partitions and to be C(1+log(H/h))$^2$ for geometrically conforming partitions. The bound is not affected by the jump of the material parameters across the subdomain interface. Numerical results are included.

Domain decomposition technique for aeroacoustic simulations

A computational tool for the simulation of time harmonic sound propagation in open domains is proposed. It consists of a two step technique. In the first step the viscous flow field is calculated, solving the incompressible Navier–Stokes equations. In the second step the acoustic field is obtained solving the Lighthill's wave equation. The two calculations are performed on two different grids, both formed by non-conforming subdomains. The Lighthill's wave equation is solved in the frequency domain, with a high order accurate compact finite difference scheme. To test the acoustic solver, the phenomenon of wave diffraction past an obstacle is calculated. The results confirm that the interface transmission conditions as well the non-reflecting boundary conditions do not affect the accuracy of the scheme. The complete aeroacoustic technique is applied to the simulation of the noise radiation from a cavity with a grazing subsonic laminar boundary layer. The flow field presents a periodic vortex shedding past the cavity, the vortical flow being a noise source. The numerical results show the existence of the directivity character of the acoustic radiation.

A Staggered-Grid Multidomain Spectral Method for the Compressible Navier–Stokes Equations

We present a flexible, non-conforming staggered-grid Chebyshev spectral multidomain method for the solution of the compressible Navier–Stokes equations. In this method, subdomain corners are not included in the approximation, thereby simplifying the subdomain connectivity. To allow for local refinement of the polynomial order or subdomain subdivision, non-conforming subdomains are treated by a mortar method. Spectral accuracy is shown in one- and two-dimensional linear and non-linear problems. Application is made to four compressible flow problems: the Couette flow, a steady boundary layer over a flat plate, steady transonic flow in a nozzle, and unsteady flow over a cylinder at a Reynolds number of 75.