Non-overlapping Subdomains Research Articles

ES-PIM with Cell Death and Birth Technique for Simulating Heat Transfer in Concrete Dam Construction Process

The meshfree edge-based smoothed point interpolation method (ES-PIM) is extended to the simulation of transient heat transfer problems for concrete dams in the construction process. The present ES-PIM consists of the following ingredients: (1) a novel cell death and birth technique is proposed to deal with the existence or nonexistence of the concrete layers, and hence the construction process of dams can be easily simulated; (2) a lumped diagonalization procedure for specific heat matrix is implemented to improve the stability and efficiency of transient computations; and (3) the triangular background cells are used to construct the nonoverlapping smoothing integral subdomains, and the cell-based T3- and T6/3-schemes are adopted for the node selection in the formation of PIM shape functions. On the basis of the weakened weak formulation, the ES-PIM is spatially stable and convergent to the exact solution. The ES-PIM can achieve better results in accuracy and convergence rate, in comparison with the finite element method using the same meshes.

A p-type finite element solution for the simulation of O2 transport in articular cartilage tissue: heterogeneous and porous media

The partial pressure of oxygen (pO2) is suggested to have a regulatory effect on chondrocyte biosynthetic activities, and its effect during expansion is unknown. The authors hypothesize that oxygen tension due to mechanical deformation or swelling could be as important as direct mechanical effects on cell biosynthetic activities. While there are plenty of studies on measuring and/or modelling pO2 in articular cartilage (AC) for static (rest) conditions, to the best of the authors' knowledge there are very few such studies on pO2 in AC for dynamic conditions such as swelling or tissue deformation. In this study, it is attempted to develop a model to study the dynamics of oxygen transport in AC. A high-precision hybrid element is designed using the p-type finite element method, by which diffusion and convection are incorporated as a single element. A domain decomposition method is used that allows the use of a different type of discretization with independent discretization variables in non-overlapping sub-domains, for a generic three-dimensional approach to elliptic boundary value problems of order 2 or higher. The formulation developed in this study might be used in determining the necessary flow conditions to cultivate tissue constructs in tissue repair and tissue engineering.

An efficient modified hierarchical domain decomposition for two-dimensional magnetotelluric forward modelling

We use 2-D magnetotelluric (MT) problems as a feasibility study to demonstrate that 3-D MT problems can be solved with a direct solver, even on a standard single processor PC. The scheme used is a hierarchical domain decomposition (HDD) method in which a global computational domain is uniformly split into many smaller non-overlapping subdomains. To make it more efficient, two modifications are made to the standard HDD method. Instead of three levels as in the standard HDD method, we classify the unknowns into four classes: the interiors, the horizontal interfaces, the vertical interfaces and the intersections. Four sets of smaller systems of equations are successively solved with a direct method (an LU factorization). The separation significantly reduces the large memory requirements of a direct solver. It also reduces the CPU time to almost half that of the standard HDD method although it is still slower than the conventional finite difference (FD) method. To further enhance the speed of the code, a red-black ordering is applied to solve the horizontal and vertical interface reduced systems. Numerical experiments on a 2-D MT problem of a given size running on a single processor machine shows that CPU time and memory used are almost constant for any resistivity models, frequencies and modes. This is a clear advantage of our algorithm and is of particular importance if the method is applied to 3-D problems. We show that our new method results in reductions in both memory usage and CPU time for large enough domains when compared to the standard FD and HDD schemes. In addition, we also introduce a ‘memory minimization map’, a graphical tool we can use instead of trial-and-error to pre-select the optimal size of subdomains, which yield the best performance in both CPU time and memory even running on a serial machine.

An efficient time‐domain method analysis of quasiperiodic structures by a finite‐element tearing and interconnecting algorithm

AbstractBased on the finite‐element approximation and nonoverlapping domain decomposition, an efficient tearing and interconnecting algorithm of the finite‐element time‐domain method is presented. Using edge‐based finite element, the method divides the computation domain into several nonoverlapping subdomains and computes electric fields in each subdomain by solving the vector curl–curl wave equation. Adjacent subdomains are related to each other by Dirichlet and Neumann boundary condition. By use of the repetition of the quasiperiod structures, the method further reduces the memory requirement and CPU time. The time‐dependent formulation employs an unconditional stable time‐integration method based on the Newmark‐beta method. An anisotropic perfectly matched layers absorbing material terminated by the first‐order boundary condition is applied for the truncation of lattices. Numerical results demonstrate that our proposed method is extremely efficient for the time‐domain analysis of the quasiperiodic structures, such as electromagnetic band gap structures.© 2010 Wiley Periodicals, Inc. Microwave Opt Technol Lett 52: 1072–1078, 2010; Published online in Wiley InterScience (www.interscience. wiley.com). DOI 10.1002/mop.25100

Bilevel Approach of a Decomposed Topology Optimization Problem

In topology optimization problems, we are often forced to deal with large-scale numerical problems, so that the domain decomposition method occurs naturally. Consider a typical topology optimization problem, the minimum compliance problem of a linear isotropic elastic continuum structure, in which the constraints are the partial differential equations of linear elasticity. We subdivide the partial differential equations into two subproblems posed on non-overlapping sub-domains. In this paper, we consider the resulting problem as multilevel one and show that it can be written as one level problem

Analysis of Highly Conducting Lamellar Gratings With Multidomain Pseudospectral Method

In this paper, multidomain pseudospectral (MDPS) method is adopted to analyze the diffraction of electromagnetic waves by the metallic lamellar grating. This method is based on applying a spectral accuracy at the Chebyshev collocation points to the spatial derivatives in Helmholtz equation, and then dividing the computational domain into nonoverlapping subdomains. Finally, the physical boundary conditions at the subdomain interfaces enforce the subdomains to a global system. The numerical examples for 1-D binary gratings composed of different highly conducting materials are presented. The validity of results by MDPS is compared with commonly used rigorous coupled wave analysis for TM polarization. The numerical evidence shows that the developed method has better stability and higher efficiency.

Parallel computation approaches for flexible multibody dynamics simulations

Finite element based formulations for flexible multibody systems are becoming increasingly popular and as the complexity of the configurations to be treated increases, so does the computational cost. It seems natural to investigate the applicability of parallel processing to this type of problems; domain decomposition techniques have been used extensively for this purpose. In this approach, the computational domain is divided into non-overlapping sub-domains, and the continuity of the displacement field across sub-domain boundaries is enforced via the Lagrange multiplier technique. In the finite element literature, this approach is presented as a mathematical algorithm that enables parallel processing. In this paper, the divided system is viewed as a flexible multibody system, and the sub-domains are connected by kinematic constraints. Consequently, all the techniques applicable to the enforcement of constraints in multibody systems become applicable to the present problem. In particular, it is shown that a combination of the localized Lagrange multiplier technique with the augmented Lagrange formulation leads to interesting solution strategies.

Mesh domain decomposition in the finite-difference solution of Maxwell’s equations

We discuss the mesh domain decomposition when studying optical diffraction from sub-wavelength structures using the finite-difference solution of Maxwell’s equations. Special consideration is given to the case of decomposition into non-overlapping sub-domains. Theoretical estimates of the algorithms’ and computational procedures’ speedup at various decomposition parameters are compared.

CBFEM-MPI: A Parallelized Version of Characteristic Basis Finite Element Method for Extraction of 3-D Interconnect Capacitances

In this paper, we present a novel, non-iterative domain decomposition method, which has been parallelized by using the message passing interface (MPI) library, and used to efficiently extract the capacitance matrixes of 3-D interconnect structures, by employing characteristic basis functions (CBFs) in the context of the finite element method (FEM). In this method, which is called CBFEM-MPI, the computational domain is partitioned into a number of nonoverlapping subdomains in which the CBFs are constructed by employing a procedure that is different from what has been used in the past for generating them. Specifically, the CBFs for the problem at hand are determined by calculating the potentials due to a finite number of point charges located hypothetically along the boundaries of the conductors. Two major advantages of the method are substantial reduction of the matrix size - enabling the use of direct solvers - and the utilization of parallel processing techniques that provide a substantial decrease in the overall computation time. Numerical examples are included to illustrate the application of the proposed approach to a number of representative problems.

Parallelized Characteristic Basis Finite Element Method (CBFEM-MPI)—A non-iterative domain decomposition algorithm for electromagnetic scattering problems

In this paper, we introduce a parallelized version of a novel, non-iterative domain decomposition algorithm, called Characteristic Basis Finite Element Method (CBFEM-MPI), for efficient solution of large-scale electromagnetic scattering problems, by utilizing a set of specially defined characteristic basis functions (CBFs). This approach is based on the decomposition of the computational domain into a number of non-overlapping subdomains wherein the CBFs are generated by employing a novel procedure, which differs from all those that have been used in the past. Clearly, the CBFs are obtained by calculating the fields radiated by a finite number of dipole-type sources, which are placed hypothetically along the boundary of the conducting object. The major advantages of the proposed technique are twofold: (i) it provides a substantial reduction in the matrix size, and thus, makes use of direct solvers efficiently and (ii) it enables the utilization of parallel processing techniques that considerably decrease the overall computation time. We illustrate the application of the proposed approach via several 3D electromagnetic scattering problems.

A modified upwind difference domain decomposition method for convection–diffusion equations

A new, efficient and noniterative type of domain decomposition method is presented for three-dimensional convection–diffusion equations, whose convection term is handled by modified upwind differences. The whole domain is decomposed into several nonoverlapping subdomains. The values on inter-domain boundaries are calculated by reduced-dimensional solver and the solutions in subdomains are computed by implicit upwind procedures completely in parallel. With certain weakened stability constraint, error estimates in discrete l 2 norm is discussed by introducing new inner products and techniques of prior estimates. Finally, the theoretical results are illustrated by linear and nonlinear numerical experiments.

A Fast Domain Decomposition Method for Solving Three-Dimensional Large-Scale Electromagnetic Problems

An efficient algorithm based on domain decomposition method (DDM) and partial basic solution vectors (PBSV) technique is proposed for solving three-dimensional (3-D), large-scale, finite periodic electromagnetic problems, such as photonic or electromagnetic bandgap structures, frequency selective surfaces. The entire computational domain is divided into many smaller nonoverlapping subdomains. A Robin-type condition is introduced at the interfaces between subdomains to enforce the field continuity. With the help of a set of dual unknowns, each subdomain can be tackled independently. Because of geometric repetitions, all the sudomains can be classified into a few building blocks, which can be dealt with by an improved PBSV algorithm. Thus, the original problem becomes a much smaller one which involves the unknowns only at the interfaces. The resulting linear system of equations is solved by a block symmetric successive over relaxation (SSOR) preconditioned Krylov subspace method. Once the unknowns at the interfaces have been obtained, the final solution on each subdomain can easily be calculated independently. Some numerical examples are provided and show the method is scalable with the number of subdomains.

Molecular epitopes of the ankyrin-spectrin interaction.

Isoforms of ankyrin and its binding partner spectrin are responsible for a number of interactions in a variety of human cells. Conflicting evidence, however, had identified two different, non-overlapping human erythroid ankyrin subdomains, Zu5 and 272, as the minimum binding region for beta-spectrin. Complementary studies on the ankyrin-binding domain of spectrin have been somewhat more conclusive yet have not presented binding in terms of well-phased, integral numbers of spectrin repeats. Thus, the objective of this study was to clearly define and characterize the minimal ankyrin-spectrin binding epitopes. Circular dichroism (CD) wavelength spectra of the aforementioned ankyrin subdomains show that these fragments are 30-60% unstructured. In contrast, human erythroid beta-spectrin repeats 13, 14, 15, and 16 (prepared in all combinations of two adjacent repeats) demonstrated proper folding and stability as determined by CD and tryptophan wavelength and heat denaturation scans. Native polyacrylamide gel electrophoresis (PAGE) gel shifts as well as affinity pull-down assays implicated Zu5 and beta-spectrin repeats 14-15 as the minimum binding epitopes. These results were confirmed by analytical ultracentrifugation to sedimentation equilibrium by which a 1:1 complex was obtained if and only if Zu5 was mixed with beta-spectrin constructs containing repeats 14 and 15 in tandem. Surface plasmon resonance yielded a K D of 15.2 nM for binding of beta-spectrin fragments to the ankyrin subdomain Zu5, accounting for all of the binding observed between the intact molecules. Collectively, these results show the 14th and 15th beta-spectrin repeats comprise the minimal, phased region of beta-spectrin, which binds ankyrin at the Zu5 subdomain with high affinity.

Monotone iterates for solving systems of semilinear elliptic equations and applications

Consider monotone finite difference iterative algorithms for solving coupled systems of semilinear elliptic equations. A monotone domain decomposition algorithm based on a modification of Schwarz alternating method and on decomposition of a computational domain into nonoverlapping subdomains is constructed. Advantages of the algorithm are that the algorithm solves only linear discrete systems at each iterative step, converges monotonically to the exact solution of the nonlinear discrete problem, and is potentially parallelisable. The monotone domain decomposition algorithm is applied to a gas-liquid interaction model. Numerical experiments confirm theoretical results.

A Domain Decomposition Method With Nonconformal Meshes for Finite Periodic and Semi-Periodic Structures

We present a domain decomposition method as a preconditioner for Krylov-type solvers to model complex electromagnetic problems containing periodicities. The method reduces memory requirements by decomposing the original problem into several nonoverlapping sub-domains. The 1st order transmission condition is employed on interfaces between adjacent sub-domains to enforce continuity of electromagnetic fields and to ensure the sub-domain problems are well-posed. By following the spirit of duality paring a symmetric system is obtained. To reduce the computational burdens of the present method, the finite element tearing and interconnecting like algorithm is adopted. This algorithm results in the computation of the so-called Green's function, which can be compressed efficiently via a rank-revealing matrix factorization algorithm. The final system matrix is solved by Krylov solvers instead of classical stationary solvers. To improve the convergence of iterative solvers, several robust implementation details are discussed and the choice of some popular Krylov-subspace solvers is studied through numerical examples.

Efficient parallel PML algorithms for truncating finite difference time domain simulations

In this paper, the parallel implementation of the stretched coordinate perfectly matched layer (SC-PML) and the wave equation PML (WE-PML) formulations is presented for truncating three-dimensional (3-D) finite difference time domain (FDTD) grids. In the proposed parallel algorithms, the FDTD computational domain is divided into contiguous non-overlapping subdomains using two-dimensional topology and the interprocessor communications between the neighboring subdomains are carried out by using the message passing interface (MPI) system. The performance of the proposed parallel algorithms has been studied by using a point source radiating in 3-D domains and performed on a network of PCs interconnected with Ethernet. It has been observed that the WE-PML parallel algorithm is approximately 2.3 faster than the SC-PML parallel algorithm.

A parallel approach for self-adjoint singular perturbation problems using Numerov's scheme

In this paper, we considered singularly perturbed self-adjoint boundary-value problems and proposed a computational technique based on Numerov's scheme, which is also suitable for parallel computing. The whole domain is divided into three non-overlapping subdomains, and corresponding subproblems are obtained by using zeroth-order approximations of the solution at the boundaries of these subproblems. The subproblems corresponding to boundary-layer regions are solved using Numerov's method after the introduction of suitable stretching variables and the solution of the reduced problem is taken as an approximate solution in the outer region. A numerical example is provided to show the efficiency and accuracy of the technique.

Numerical analysis of a two-level preconditioner for the diffusion equation with an anisotropic diffusion tensor

We develop and analyze a new two-level preconditioner for algebraic systems arising from finite volume discretization of 3D anisotropic diffusion problems on prismatic meshes. The preconditioner is based on partitioning of the mesh in the (x,y)-plane into non-overlapping subdomains and on a special coarsening algorithm. The condition number of the preconditioned matrix does not depend on the coefficients in the diffusion operator. Numerical experiments confirm the theoretical results and reveal the competitiveness of the new preconditioner with respect to the algebraic multigrid preconditioner.

Optimal B-spline collocation method for self-adjoint singularly perturbed boundary value problems

We present a B-spline collocation method of higher order for a class of self-adjoint singularly perturbed boundary value problems. The essential idea in this method is to divide the domain of the differential equation into three non-overlapping subdomains and solve the regular problems obtained by transforming the differential equation with respective boundary conditions on these subdomains using the present higher order B-spline collocation method. The boundary conditions at the transition points are obtained by the asymptotic approximation of order zero to the solution of the problem. The convergence analysis is given and the method is shown to have optimal order convergence; by collocating the perturbed differential equation, which is satisfied by a special cubic spline interpolate of the true solution. Numerical experiments are conducted to demonstrate the efficiency of the method.

Preconditioning Schur complement matrices based on an aggregation multigrid method for shell structures

An aggregation multigrid method is utilized in constructing a preconditioner for a Schur complement system of automatically partitioned, non-overlapping subdomains. Preserving the relationship of the partitioned subdomains, we apply a rigid body based aggregation method, which employ geometric data, as a coarsening procedure. And then, we derive a new Schur complement coarse grid matrix by an approach of a condensation after the coarsening procedure. Therefore, we generate a multi-level preconditioner of a Krylov subspace method for Schur complement matrices using the coarse grid matrix. Through numerical experiments, the proposed preconditioner shows efficient performance and robust convergences irrespect of the size of elements and subdomains. It also shows better performance than the preconditioned conjugate gradient method (PCG) for the partitioned system and the aggregation multigrid method for the original domain in shell problems of structural mechanics.