A Hierarchical 3-D Poisson Modified Fourier Solver by Domain Decomposition

We present a Domain Decomposition non-iterative solver for the Poisson equation in a 3-D rectangular box. The solution domain is divided into mostly parallelepiped subdomains. In each subdomain a particular solution of the non-homogeneous equation is first computed by a fast spectral method. This method is based on the application of the discrete Fourier transform accompanied by a subtraction technique. For high accuracy the subdomain boundary conditions must be compatible with the specified inhomogeneous right hand side at the edges of all the interfaces. In the following steps the partial solutions are hierarchically matched. At each step pairs of adjacent subdomains are merged into larger units. In this paper we present the matching algorithm for two boxes which is a basis of the domain decomposition scheme. The hierarchical approach is convenient for parallelization and minimizes the global communication. The algorithm requires O(N ³:log:N) operations, where N is the number of grid points in each direction.     

An effective pseudospectral method for constraint dynamic optimisation problems with characteristic times

Dynamic optimisation problem with characteristic times, widely existing in many areas, is one of the frontiers and hotspots of dynamic optimisation researches. This paper considers a class of dynamic optimisation problems with constraints that depend on the interior points either fixed or variable, where a novel direct pseudospectral method using Legendre–Gauss (LG) collocation points for solving these problems is presented. The formula for the state at the terminal time of each subdomain is derived, which results in a linear combination of the state at the LG points in the subdomains so as to avoid the complex nonlinear integral. The sensitivities of the state at the collocation points with respect to the variable characteristic times are derived to improve the efficiency of the method. Three well-known characteristic time dynamic optimisation problems are solved and compared in detail among the reported literature methods. The research results show the effectiveness of the proposed method.     

Isogeometric shape optimization of photonic crystals via Coons patches

In this paper, we present an approach that extends isogeometric shape optimization from optimization of rectangular-like NURBS patches to the optimization of topologically complex geometries. We have successfully applied this approach in designing photonic crystals where complex geometries have been optimized to maximize the band gaps.Salient features of this approach include the following: (1) multi-patch Coons representation of design geometry. The design geometry is represented as a collection of Coons patches where the four boundaries of each patch are represented as NURBS curves. The use of multiple patches is motivated by the need for representing topologically complex geometries. The Coons patches are used as a design representation so that designers do not need to specify interior control points and they provide a mechanism to compute analytical sensitivities for internal nodes in shape optimization, (2) exact boundary conversion to the analysis geometry with guaranteed mesh injectivity. The analysis geometry is a collection of NURBS patches that are converted from the multi-patch Coons representation with geometric exactness in patch boundaries. The internal NURBS control points are embedded in the parametric domain of the Coons patches with a built-in mesh rectifier to ensure the injectivity of the resulting B-spline geometry, i.e. every point in the physical domain is mapped to one point in the parametric domain, (3) analytical sensitivities. Sensitivities of objective functions and constraints with respect to design variables are derived through nodal sensitivities. The nodal sensitivities for the boundary control points are directly determined by the design parameters and those for internal nodes are obtained via the corresponding Coons patches.     

A Non-Overlapping Domain Decomposition Method for the Advection-Diffusion Problem

The application of a non-overlapping domain decomposition method to the solution of a stabilized finite element method for elliptic boundary value problems is considered. We derive an a-posteriori error estimate which bounds the error on the subdomains by the interface error of the subdomain solutions. As a by-product, some foundation is given to the design of the interface transmission condition. Numerical results support the theoretical results. Furthermore, we adapt a recent result on a-posteriori estimates for singular perturbation problems in order to obtain an a-posteriori estimate for the discrete subdomain solutions.     

A parallel preconditioned conjugate gradient method using domain decomposition and inexact solvers on each subdomain

We describe a preconditioned conjugate gradient solution strategy for a multiprocessor system with message passing architecture. The preconditioner combines two techniques, a Schurcomplement preconditioning over “coupling boundaries” between the subdomains and an arbitrary choice of classic preconditioning for the inner degrees of freedom on each subdomain. All computational work on the single subdomains is carried out in parallel by distributing the subdomain data over the processor network before starting the finite element solution process (including generating the element matrices and assemblying the local subdomain stiffness matrix). The resulting spectral condition number of the entire preconditioner is estimated. For the important example of choosing MIC(0)-*-preconditioning on the subdomains, the condition number obtained is essentially the product of the two condition numbers involved.     

Multi-block Poisson grid generator for cascade simulations

High quality computational grids can greatly enhance the accuracy of turbine and compressor cascade simulations especially when time-dependent results are sought where vortical structures are convected through the computational domain. A technique for generating periodic structured grids for cascade simulations based on the Poisson equations is described. To allow for more complex geometries, the grid can be divided into individual zones or blocks. The grids are generated simultaneously in all blocks, assuring continuity of the grid lines and their slopes across the zonal boundaries. Simple geometric rules can be employed for enforcing orthogonality at block boundaries. The method results in grids with low grid distortion by allowing both, block boundaries and grid points on physical boundaries, to move freely. Results are presented for a linear turbine and a linear compressor cascade.     

On some multiscale algorithms for sector modeling in multiphase flow in porous media

A multiscale algorithm for a multiphase filtration problem is proposed. Filtration fluxes on a fine grid are determined from the solution of the pressure equation on a coarse grid. Further, the domain is decomposed into subdomains with an acceptable number of cells and the full second boundary condition filtration problem is solved using the fluxes. The support operator method has been improved for a complex structure cell for solution of the pressure equation on a coarse grid. This method is a high resolution one: a divergence operator has the approximation of the second order, while fluxes have the approximation of the first order. At the same time, the method allows revealing the solution’s properties related to a fine grid structure.     

Domain decomposition methods for the neutron diffusion problem

The neutronic simulation of a nuclear reactor core is performed using the neutron transport equation, and leads to an eigenvalue problem in the steady-state case. Among the deterministic resolution methods, simplified transport (S_PN$S{P}_N$) or diffusion approximations are often used. The MINOS solver developed at CEA Saclay uses a mixed dual finite element method for the resolution of these problems, and has shown his efficiency. In order to take into account the heterogeneities of the geometry, a very fine mesh is generally required, and leads to expensive calculations for industrial applications. In order to take advantage of parallel computers, and to reduce the computing time and the local memory requirement, we propose here two domain decomposition methods based on the MINOS solver. The first approach is a component mode synthesis method on overlapping subdomains: several eigenmodes solutions of a local problem on each subdomain are taken as basis functions used for the resolution of the global problem on the whole domain. The second approach is an iterative method based on a non-overlapping domain decomposition with Robin interface conditions. At each iteration, we solve the problem on each subdomain with the interface conditions given by the solutions on the adjacent subdomains estimated at the previous iteration. Numerical results on parallel computers are presented for the diffusion model on realistic 2D and 3D cores.     

Zonal finite-volume computations of incompressible flows

The paper addresses convection modelling and flow field zoning procedures destined for finite-volume methods for incompressible steady-state flows with irregular boundaries. A composite oscillation-damping algorithm which is capable of yielding bounded and low diffusive solutions is used to approximate the convection terms of transport equations. To deal with complex geometries, a zonal procedure is introduced into an advanced finite-volume method that uses general non-orthogonal and non-staggered grids. The solution domain is divided into simple subregions each covered by a separate mesh. The flow is solved for concurrently in each of the different zones, and information exchange among the zones is realized through overlapping the grids in the vicinity of the zonal interfaces. The zonal procedure enables the finite-volume method to handle those domains which are extremely difficult to cover with a single grid. Applications to a range of laminar elliptic problems demonstrate the capability of the method for predicting complex flows.     

A parallel viscous flow solver on multi-block overset grids

A multi-block overset grid method is presented to accurately simulate viscous flows around complex configurations. A combination of multi-block and overlapping grids is used to discretize the flow domain. A hierarchical grid system with layers of grids of varying resolution is developed to ensure inter-grid connectivity within a framework suitable for multi-grid and parallel computations. At each stage of the numerical computation, information is exchanged between neighboring blocks across either or both matched block boundaries and overlapping boundaries. Coarse-grain parallel processing is facilitated by the multi-block system. Numerical results of flows over multi-element airfoils and three-dimensional turbulent flows around wing–body aerodynamic configurations demonstrate the utility and efficiency of the method.     

Solid Geometry Processing on Deconstructed Domains

Many tasks in geometry processing are modelled as variational problems solved numerically using the finite element method. For solid shapes, this requires a volumetric discretization, such as a boundary conforming tetrahedral mesh. Unfortunately, tetrahedral meshing remains an open challenge and existing methods either struggle to conform to complex boundary surfaces or require manual intervention to prevent failure. Rather than create a single volumetric mesh for the entire shape, we advocate for solid geometry processing on deconstructed domains, where a large and complex shape is composed of overlapping solid subdomains. As each smaller and simpler part is now easier to tetrahedralize, the question becomes how to account for overlaps during problem modelling and how to couple solutions on each subdomain together algebraically. We explore how and why previous coupling methods fail, and propose a method that couples solid domains only along their boundary surfaces. We demonstrate the superiority of this method through empirical convergence tests and qualitative applications to solid geometry processing on a variety of popular second‐order and fourth‐order partial differential equations.     

An Algorithm for Projecting Points onto a Patched CAD Model

We are interested in building structured overlap-ping grids for geometries defined by Computer-Aided-Design (CAD) packages. Geometric information defining the boundary surfaces of a computation domain is often provided in the form of a collection of possibly hundreds of trimmed patches. The first step in building an overlapping volume grid on such a geometry is to build overlapping surface grids. A surface grid is typically built using hyperbolic grid generation; starting from a curve on the surface, a grid is grown by marching over the surface. A given hyperbolic grid will typically cover many of the underlying CAD surface patches. The fundamental operation needed for building surface grids is that of projecting a point in space onto the closest point on the CAD surface. We describe a fast and robust algorithm for performing this projection which makes use of a fairly coarse global triangulation of the CAD geometry. Before the global triangulation is constructed the connectivity of the model is determined by an edge-matching algorithm which corrects for gaps and overlaps between neighbouring patches. ID="A1" Correspondence and offprint requests to: Dr. W. D. Henshaw, Center for Applied Scientific Computing, L-661, Lawrence Livermore National Laboratory, Livermore, CA 94551, USA. E-mail: henshaw@llnl.gov     

Adaptive and fixed segmented domain decomposition multigrid procedures for internal viscous flows

A segmented domain decomposition multigrid procedure is reviewed. The methodology is outlined for both an adaptive formulation applied primarily to inlet, nozzle and duct flows, and for a fixed, predetermined, subdomain formulation applied for turbomachinery geometries. The procedure has been considered for both low and high speed flows in two- and three-dimensions, and for laminar and turbulent-model applications. With these procedures, the grids remain uniform in each subdomain; although the global grid appears to be highly stretched. This formulation allows for computations with a high degree of accuracy and grid independence at reasonable computer cost. This is generally not possible to replicate with full domain multigrid procedures on stretched grids with large cell aspect ratio.     

3D tetrahedral,unstructured and anisotropic mesh generation with adaptation to natural and multidomain metric

In this paper we present a 3D tetrahedral, unstructured and anisotropic mesh generator that is not based on the Delaunay, frontal or octree method. Instead, it proceeds by local optimizations and uses an anisotropic shape criterion to fit a metric field.Then, we introduce a new 3D metric field that tightens the mesh around interfaces when the calculation domain is divided in several subdomains, and a 3D metric field that places enough elements through each subdomain thickness, without introducing too many nodes in the other directions.Finally, we show some applications for material forming geometries.     

A new parallel domain decomposition method for the adaptive finite element solution of elliptic partial differential equations

We present a new domain decomposition algorithm for the parallel finite element solution of elliptic partial differential equations. As with most parallel domain decomposition methods each processor is assigned one or more subdomains and an iteration is devised which allows the processors to solve their own subproblem(s) concurrently. The novel feature of this algorithm however is that each of these subproblems is defined over the entire domain—although the vast majority of the degrees of freedom for each subproblem are associated with a single subdomain (owned by the corresponding processor). This ensures that a global mechanism is contained within each of the subproblems tackled and so no separate coarse grid solve is required in order to achieve rapid convergence of the overall iteration. Furthermore, by following the paradigm introduced in 15 , it is demonstrated that this domain decomposition solver may be coupled easily with a conventional mesh refinement code, thus allowing the accuracy, reliability and efficiency of mesh adaptivity to be utilized in a well load-balanced manner. Finally, numerical evidence is presented which suggests that this technique has significant potential, both in terms of the rapid convergence properties and the efficiency of the parallel implementation. Copyright © 2001 John Wiley & Sons, Ltd.     

Efficient rendering of multiblock curvilinear grids with complex boundaries

Domain decomposition is a popular technique for solving large computational problems that require data to be divided into smaller sub‐domains. The exact manner of decomposition depends on the computational needs of the algorithm and often introduces irregular boundaries. Each subdomain forms a block of a larger grid and can be solved or rendered separately by different processing nodes. Rendering of each sub‐domain can result in images which are then composited in a back‐to‐front or front‐to‐back manner. This scenario is useful when visualization is used concurrently with the simulation. However, the irregularity of boundaries may prohibit the correct image composition due to a visibility anomaly between the sub‐domains. In this paper, we present an algorithm based on object‐space partitioning to resolve this problem. To accelerate the partitioning process, two techniques are introduced. First, an image‐space partition representation is employed for fast assignment of data points to correct partitions. Secondly, a k‐d tree is used to subdivide the view‐space adaptively according to the complexity of the surface. This view‐space partition provides a trade‐off between performance and accuracy of the rendered image. Large gains in performance can be achieved with only small losses of accuracy. Two examples of curvilinear grids of different complexity are used to demonstrate the effectiveness of this scheme. Copyright © 2005 John Wiley & Sons, Ltd.     

Solution-Domain-Decomposition Method for Heat Transfer Problem Using Parallel Distributed Computing

Solution-domain-decomposition (SDD) method is formulated for solving heat transfer problem and generalized for solving multi-domain problem. A generalized algorithm is suggested for parallel and distributing computation. Chebyshev expansion on the dependent variables is used for pseudospectral approximation of the governing equation in this study. Linear superposition principle is adapted to incorporate the interactions between the subdomains. By effective subdivision of computational domain, significant computational efficiency and computational memory savings are accomplished without losing spectral accuracy of the solution. Owing to independent characteristics of the subdomains. the scheme is well suited for multi-processor machines. Convergence study reveals that spectra! accuracy is still conserved for the multi-domain calculation. The calculation domain is divided up to 8 subdomains and calculation is distributed up to independent CPUs. Significant speed-up ratio is obtained by distributing the subtasks through the network.     

Multidomain WENO Finite Difference Method with Interpolation at Subdomain Interfaces

High order finite difference WENO methods have the advantage of simpler coding and smaller computational cost for multi-dimensional problems, compared with finite volume WENO methods of the same order of accuracy. However a main restriction is that conservative finite difference methods of third and higher order of accuracy can only be used on uniform rectangular or smooth curvilinear meshes. In order to overcome this difficulty, in this paper we develop a multidomain high order WENO finite difference method which uses an interpolation procedure at the subdomain interfaces. A simple Lagrange interpolation procedure is implemented and compared to a WENO interpolation procedure. Extensive numerical examples are shown to indicate the effectiveness of each procedure, including the measurement of conservation errors, orders of accuracy, essentially non-oscillatory properties at the domain interfaces, and robustness for problems containing strong shocks and complex geometry. Our numerical experiments have shown that the simple and efficient Lagrange interpolation suffices for the subdomain interface treatment in the multidomain WENO finite difference method, to retain essential conservation, full high order of accuracy, essentially non-oscillatory properties at the domain interfaces even for strong shocks, and robustness for problems containing strong shocks and complex geometry. The method developed in this paper can be used to solve problems in relatively complex geometry at a much smaller CPU cost than the finite volume version of the same method for the same accuracy. The method can also be used for high order finite difference ENO schemes and an example is given to demonstrate a similar result as that for the WENO schemes.     

Stokes外问题的一种重叠型区域分解法

本文讨论了平面无界区域上Stokes问题的重叠型区域分解法.利用混合元方法求解内子区域问题得到速度和压力,再用Poisson积分公式解出外子区域的速度和压力,如此交替迭代克服区域无界性并按原始变量求出原问题的数值解.根据投影理论证明重叠型区域分解法的几何收敛性.最后给出数值例子.     

Subdomain solutions of complex variable boundary element method

In this paper, a new subdomain solution of the boundary element method based on complex variable fundamental solutions for non-homogeneous materials is developed. Being different from the conventional BEM, subdomains in the method presented can be produced by considering not only the properties of materials, but also the geometry and correspondent boundary conditions of the problem. The formulation may be combined with other complex variable fundamental solutions to provide higher accuracy and better efficiency. The coupling formulations are given in matrix form, and the numerical procedure is described. the advantages and high efficiency of the present method are demonstrated by two numerical examples.