Subdomain Solutions Research Articles

Discontinuous Galerkin Isogeometric Analysis of elliptic problems on segmentations with non-matching interfaces

In this paper, we develop a discontinuous Galerkin Isogeometric Analysis method for solving elliptic problems on decompositions of the computational domain into volumetric patches with non-matching parametrized interfaces. We specially focus on high order numerical solutions for complex gap regions and extend ideas from our previous work on simple gap regions. For the communication of the numerical solution between the subdomains, which are separated by the gap region, discontinuous Galerkin numerical fluxes are constructed taking into account the diametrically opposite points on the boundary of the gap. Due to lack of information on the behavior of the solution in the gap region, the fluxes coming from the interior of the gap are approximated by Taylor expansions with respect to the adjacent subdomain solutions. We follow the same ideas of our previous work and show a priori error estimates in the dG-norm, with respect to the mesh size and the gap distance. Numerical examples, performed for two-, three- and even four-dimensional computational domains, demonstrate the robustness of the proposed numerical method and validate the estimates predicted by the theory.

Point Equation, Line Equation, Plane Equation etc and Point Solution, Line Solution, Plane Solution etc —Expanding Concepts of Equation and Solution with Neutrosophy and Quad-stage Method

The concepts of equations and solutions are constantly developed and expanded. With Neutrosophy and Quad-stage method, this paper attempts to expand the concepts of equations and solutions in the way of re- ferring to the concepts of domain of function, the geome- try elements included in domain of function, and the like; and discusses point equation, line equation, plane equa- tion, solid equation, sub-domain equation, whole-domain equation, and the like; as well as point solution, line solu- tion, plane solution, solid solution, sub-domain solution, whole-domain solution, and the like. Where: the point so- lutions may be the solutions of point equation, line equa- tion, plane equation, and the like; similarly, the line solu- tions may be the solutions of point equation, line equa- tion, plane equation, and the like; and so on. This paper focuses on discussing the single point method to deter- mine "point solution". Also, the concepts of equations and solutions are con- stantly developed and expanded. From the historical per- spective, these developments and expansions are mainly processed for the complexity of variables, functional rela- tionships, operation methods, and the like. For example, from elementary mathematical equations develop and ex- pand into secondary mathematical equations, and advanced mathematical equations. Again, from algebra equations develop and expand into geometry equations, trigonomet- ric equations, differential equations, integral equations, and the like. With Neutrosophy and Quad-stage method, this paper considers another thought, and attempts to expand the con- cepts of equations and solutions in the way of referring to the concepts of domain of function, the geometry elements included in domain of function, and the like; and discusses point equation, line equation, plane equation, solid equa- tion, sub-domain equation, whole-domain equation, and the like; as well as point solution, line solution, plane solu- tion, solid solution, sub-domain solution, whole-domain solution, and the like. Where: the point solutions may be the solutions of point equation, line equation, plane equa- tion, and the like; similarly, the line solutions may be the solutions of point equation, line equation, plane equation, and the like; and so on.

Point Equation, Line Equation, Plane Equation etc and Point Solution, Line Solution, Plane Solution etc. - Expanding Concepts of Equation and Solution with Neutrosophy and Quad-stage Method

The concepts of equations and solutions are constantly developed and expanded. With Neutrosophy and Quad-stage method, this paper attempts to expand the concepts of equations and solutions in the way of referring to the concepts of domain of function, the geometry elements included in domain of function, and the like; and discusses equation, line equation, plane equation, solid equation, sub-domain equation, whole-domain equation, and the like; as well as solution, line solution, plane solution, solid solution, sub-domain solution, whole-domain solution, and the like. Where: the solutions may be the solutions of equation, line equation, plane equation, and the like; similarly, the line solutions may be the solutions of equation, line equation, plane equation, and the like; and so on. This paper focuses on discussing the single method to determine point solution. Key words: Neutrosophy, Quad-stage, equation, line equation, plane equation, solution, line solution, plane solution, single method

Asynchronous Parallelization of a CFD Solver

A Navier-Stokes equations solver is parallelized to run on a cluster of computers using the domain decomposition method. Two approaches of communication and computation are investigated, namely, synchronous and asynchronous methods. Asynchronous communication between subdomains is not commonly used in CFD codes; however, it has a potential to alleviate scaling bottlenecks incurred due to processors having to wait for each other at designated synchronization points. A common way to avoid this idle time is to overlap asynchronous communication with computation. For this to work, however, there must be something useful and independent a processor can do while waiting for messages to arrive. We investigate an alternative approach of computation, namely, conducting asynchronous iterations to improve local subdomain solution while communication is in progress. An in-house CFD code is parallelized using message passing interface (MPI), and scalability tests are conducted that suggest asynchronous iterations are a viable way of parallelizing CFD code.

Fault Resilient Domain Decomposition Preconditioner for PDEs

The move towards extreme-scale computing platforms challenges scientific simulations in many ways. Given the recent tendencies in computer architecture development, one needs to reformulate legacy codes in order to cope with large amounts of communication, system faults, and requirements of low-memory usage per core. In this work, we develop a novel framework for solving PDEs via domain decomposition that reformulates the solution as a state of knowledge with a probabilistic interpretation. Such reformulation allows resiliency with respect to potential faults without having to apply fault detection, avoids unnecessary communication, and is generally well-suited for rigorous uncertainty quantification studies that target improvements of predictive fidelity of scientific models. We demonstrate our algorithm for one-dimensional PDE examples where artificial faults have been implemented as bit flips in the binary representation of subdomain solutions.

2-D Analytical Subdomain Model of a Slotted PMSM With Shielding Cylinder

Due to their high efficiency and power density, permanent magnet synchronous machines (PMSMs) operating at high speed have recently gained a lot of attention. The development of high-speed PMSMs requires a good understanding of the physics related to the operation of such machines. Moreover, accurate modeling tools are needed when designing high-speed electrical machines. In this paper, the subdomain modeling technique is used to analytically compute the magnetic field of an electrical machine. The idea behind this technique is to divide the machine in a number of subdomains, in which the problem is simplified. The solutions in the different subdomains are then linked by imposing physical boundary conditions. The described model immediately considers the slotting effect and the eddy-current reaction field of a shielding cylinder (SC). The SC is a conductive sleeve, which is wrapped around the magnets. Its goal is to reduce the rotor losses at high-speed operation. This paper starts by introducing the applied modeling technique and the studied machine. Second, the basics of the model and its development are discussed. Finally, the results are compared with results of a finite-element (FE) model. A very good agreement between the proposed model and the FE model is observed. This implies that the developed model is indeed a powerful modeling tool for high-speed PMSMs. Moreover, it provides great insight in the machine's physics as well.

Partitioned treatment of uncertainty in coupled domain problems: A separated representation approach

This work is concerned with the propagation of uncertainty across coupled domain problems with high-dimensional random inputs. A stochastic model reduction approach based on low-rank separated representations is proposed for the partitioned treatment of the uncertainty space. The construction of the coupled domain solution is achieved though a sequence of approximations with respect to the dimensionality of the random inputs associated with each individual sub-domain and not the combined dimensionality, hence drastically reducing the overall computational cost. The coupling between the sub-domain solutions is done via the classical finite element tearing and interconnecting (FETI) method, thus providing a well suited framework for parallel computing. Two high-dimensional stochastic problems, a 2D elliptic PDE with random diffusion coefficient and a stochastic linear elasticity problem, have been considered to study the performance and accuracy of the proposed stochastic coupling approach.

Algorithm intended for space-energy parallelization in solving the criticality problem and implemented in the LUCKY_C multiprocessor program

The paper presents basic formulas for parallel solution in energy groups and spatial subdomains for solving the criticality problem of radiation transport on multiprocessor computers.

Design and implementation of a multiscale mixed method based on a nonoverlapping domain decomposition procedure

We use a nonoverlapping iterative domain decomposition procedure based on the Robin interface condition to develop a new multiscale mixed method to compute the velocity field in heterogeneous porous media. Hybridized mixed finite elements are used for the spatial discretization of the equations. We define local, multiscale mixed basis functions to represent the discrete solutions in subdomains. Appropriate subspaces of the vector space spanned by these basis functions can be considered in the numerical approximations of heterogeneous porous media flow problems. The balance between numerical accuracy and numerical efficiency is determined by the choice of these subspaces. A detailed description of the numerical method is presented. Following that, numerical experiments are discussed to illustrate the important features of the new procedure and its comparison to the traditional fine grid simulations.

A Posteriori Error Estimates for the Generalized Overlapping Domain Decomposition Methods

A posteriori error estimates for the generalized overlapping domain decomposition method (GODDM) (i.e., with Robin boundary conditions on the interfaces), for second order boundary value problems, are derived. We show that the error estimate in the continuous case depends on the differences of the traces of the subdomain solutions on the interfaces. After discretization of the domain by finite elements we use the techniques of the residual a posteriori error analysis to get an a posteriori error estimate for the discrete solutions on subdomains. The results of some numerical experiments are presented to support the theory.

Multidomain decomposition approach to time-dependent density functional calculations

A multidomain decomposition method is presented for time-dependent density functional calculations. The multidomain decomposition approach divides the coordinate space into subdomains where local solutions are generated. The subdomain solutions are used as basis states to describe the time propagation of electronic wave functions in the whole system. The applicability and flexibility of the approach are demonstrated on various test systems including the calculation of the photoabsorption cross section of a graphene ribbon and a light-harvesting triad.

Novel generalized- [formula omitted] methods for interfield parallel integration of heterogeneous structural dynamic systems

A novel partitioned algorithm able to solve ODEs arising from transient structural dynamics is presented. The spatial domain is partitioned into a set of disconnected subdomains owing to computational or physical considerations, and continuity conditions of velocity at the interface are modelled using a dual Schur formulation, where Lagrange multipliers represent reaction forces. Interface equations along with subdomain equations lead to a system of DAEs for which an interfield parallel procedure is developed. The algorithm solves interface Lagrange multipliers, which are subsequently used to advance the solution in subdomains. The proposed coupling algorithm that enables arbitrary Generalized- α schemes to be coupled with different time steps in each subdomain is an extension of a method originally proposed by Pegon and Magonette. Thus, subcycling permitting to deal also with stiff and nonstiff subsystems is allowed. In detail, the paper presents the convergence analysis of the novel interfield parallel scheme for linear single- and two-degrees-of-freedom systems because a multi-degrees-of-freedom system is too difficult for a mathematical treatment. However, the insight gained from the analysis of these coupled problems and the consequent conclusions are confirmed by means of numerical experiments on a four-degrees-of-freedom system.

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.

An analytic investigation into the behavior of Kepler’s equation1

Kepler’s Equation is solved over the entire range of elliptic and parabolic motion. Global solution accuracy is maintained by defining four M-e plane solution sub-domains. Three computational methods are concatenated in each sub-domain to solve the problem. Perturbation solutions obtain the basic solutions in each sub-domain. Accelerated series convergence is achieved by transforming the perturbation solutions with Pade approximations. A variable-order Newton extrapolation method is used to further transform the Pade results to yield thirteen digits of precision over the entire range of elliptic and parabolic motion. Only four trigonometric evaluations are required for half of the M-e plane. The remaining half of the M-e plane, including the parabolic special case, only requires a cube root and four trigonometric evaluations. Along the M∼0 and e∼0.7 boundary, a closed-form solution is obtained for a pair of complex-valued roots that possess inflection points that limit the radius of convergence of series approximations for Kepler’s equation. The proposed algorithm effectively provides a closed-form solution for Kepler’s equation, in the sense that the solution accuracy exceeds any projected operational mission need.

Modeling atmospheric chemistry and transport with dynamic adaptive resolution

We discuss an adaptive resolution system for modeling regional air pollution based on the chemical transport model STEM. The grid adaptivity is implemented using the generic adaptive mesh refinement tool Paramesh, which enables the grid management operations while harnessing the power of parallel computers. The computational algorithm is based on a decomposition of the domain, with the solution in different subdomains being computed with different spatial resolutions. Various refinement criteria that adaptively control the fine grid placement are analyzed to maximize the solution accuracy while maintaining an acceptable computational cost. Numerical experiments in a large-scale parallel setting (~0.5 billion variables) confirm that adaptive resolution, based on a well-chosen refinement criterion, leads to the decrease in spatial error with an acceptable increase in computational time. Fully dynamic grid adaptivity for air quality models is relatively new. We extend previous work on chemical and transport modeling by using dynamically adaptive grid resolution. Advantages and shortcomings of the present approach are also discussed.

A second-order continuity domain decomposition technique based on integrated Chebyshev polynomials for two-dimensional elliptic problems

This paper presents a second-order continuity non-overlapping domain decomposition (DD) technique for numerically solving second-order elliptic problems in two-dimensional space. The proposed DD technique uses integrated Chebyshev polynomials to represent the solution in subdomains. The constants of integration are utilized to impose continuity of the second-order normal derivative of the solution at the interior points of subdomain interfaces. To also achieve a C 2 function at the intersection of interfaces, two additional unknowns are introduced at each intersection point. Numerical results show that the present DD method yields a higher level of accuracy than conventional DD techniques based on differentiated Chebyshev polynomials.

Interior maximum norm estimates for finite element discretizations of the Stokes equations

Interior estimates are proved in the L ∞ norm for stable finite element discretizations of the Stokes equations on translation invariant meshes. These estimates yield information about the quality of the finite element solution in subdomains a positive distance from the boundary. While they have been established for second-order elliptic problems, these interior, or local, maximum norm estimates for the Stokes equations are new. By applying finite differenciation methods on a translation invariant mesh, we obtain optimal convergence rates in the mesh size h in the maximum norm. These results can be used for analyzing superconvergence in finite element methods for the Stokes equations.

Computation of viscous incompressible flow using pressure correction method on unstructured Chimera grid

In this paper, a pressure correction algorithm for computing incompressible flows is modified and implemented on unstructured Chimera grid. Schwarz method is used to couple the solutions of different sub-domains. A new interpolation to ensure consistency between primary variables and auxiliary variables is proposed. Other important issues such as global mass conservation and order of accuracy in the interpolations are also discussed. Two numerical simulations are successfully performed. They include one steady case, the lid-driven cavity and one unsteady case, the flow around a circular cylinder. The results demonstrate a very good performance of the proposed scheme on unstructured Chimera grids. It prevents the decoupling of pressure field in the overlapping region and requires only little modification to the existing unstructured Navier–Stokes (NS) solver. The numerical experiments show the reliability and potential of this method in applying to practical problems.

Application of the additive Schwarz method to large scale Poisson problems

AbstractThis paper presents an application of the additive Schwarz method to large scale Poisson problems on parallel computers. Domain decomposition in rectangular blocks with matching grids on a structured rectangular mesh has been used together with a stepwise approximation to approximate sloping sides and complicated geometric features. A seven‐point stencil based on central difference scheme has been used for the discretization of the Laplacian for both interior and boundary grid points, and this results in a symmetric linear algebraic system for any type of boundary conditions. The preconditioned conjugate gradient method has been used as an accelerator for the additive Schwarz method, and three different methods have been assessed for the solution of subdomain problems. Numerical experiments have been performed to determine the most suitable set of subdomain solvers and the optimal accuracy of subdomain solutions; to assess the effect of different decompositions of the problem domain; and to evaluate the parallel performance of the additive Schwarz preconditioner. Application to a practical problem involving complicated geometry is presented which establishes the efficiency and robustness of the method. Copyright © 2004 John Wiley & Sons, Ltd.

Accounting for Solution Singularity in Analysis of Normal Wave Diffraction in a Flat Waveguide

A method to account for a singularity in the derivatives of a solution for a problem of normal wave diffraction at angle border points in a flat fluid-filled waveguide is suggested. The basic approach is to subdivide the solution into two terms one of which contains the singularity and satisfies the boundary conditions on the waveguide's surfaces, while the other is a regular one. The method of analysis suggested here enables one to obtain an approximate solution that will contain the singularity in an explicit form. The report presents an estimation of the method efficiency based on the criteria of balance of power fluxes and on a discrepancy between matched partial solutions in subdomains.