Regular Subdomain Research Articles

Decoupling numerical method based on deep neural network for nonlinear degenerate interface problems

Many practical problems, including modeling composite materials, nuclear waste disposal, oil reservoir simulations, and flows in porous medium, commonly involve interface problems. However, the solution to interface problems with discontinuous coefficients of PDEs using fully decoupled numerical methods is challenging. The main objective is to solve the interface problems with fully decoupled numerical methods. This paper proposes an efficient decoupled numerical method for solving degenerate interface problems with double singularities. First, we divide the whole domain into singular and regular subdomains. Then, we use the Deep Neural Network (DNN) to find the solution on the singular subdomain and approximate the solution on the regular subdomain using the finite difference method. The scheme combines the solutions of singular and regular subdomains, which is an exciting idea. The key to the new approach is to split nonlinear degenerate partial differential equations with an interface into two independent boundary value problems based on deep learning. In this way, the expansion of the solution on the singular domain does not contain undetermined parameters, and two independent boundary value problems can be solved with any well-known traditional numerical methods. The main advantage of the proposed scheme is that we not only get the order of convergence of the degenerate interface problems on the whole domain, but we also can calculate VERY BIG jump ratio (such as 1012:1 or 1:1012) for the interface problems including degenerate and non-degenerate cases. Finally, with examples, we demonstrate the efficiency and accuracy of methods for 1 and 2D problems. It is also interesting that the proposed method is valid for the interface problems with degenerate and non-degenerate cases, we show it with some examples.

A parallel scalable domain decomposition preconditioner for elastic crack simulation using XFEM

AbstractIn this article, a parallel overlapping domain decomposition preconditioner is proposed to solve the linear system of equations arising from the extended finite element discretization of elastic crack problems. The algorithm partitions the computational mesh into two types of subdomains: the regular subdomains and the crack tip subdomains based on the observation that the crack tips have a significant impact on the convergence of the iterative method while the impact of the crack lines is not that different from those of regular mesh points. The tip subdomains consist of mesh points at crack tips and all neighboring points where the branch enrichment functions are applied. The regular subdomains consist of all other mesh points, including those on the crack lines. To overcome the mismatch between the number of subdomains and the number of processor cores, the proposed method is divided into two steps: solve the crack tip problem and then the regular subdomain problem during each iteration. The proposed method was used to develop a parallel XFEM package which is able to test different types of iterative methods. To achieve good parallel efficiency, additional methods were introduced to reduce communication and to maintain the load balance between processors. Numerical experiments indicate that the proposed method significantly reduces the number of iterations and the total computation time compared to the classical methods. In addition, the method scales up to 8192 processor cores with over 70% parallel efficiency to solve problems with more than degrees of freedom.

A domain decomposition method for solving singularly perturbed parabolic reaction‐diffusion problems with time delay

We design and analyse a domain decomposition method for solving singularly perturbed parabolic reaction‐diffusion problems with time delay. Using the asymptotic behavior of the solution, we decompose the original domain of the problem into three overlapping subdomains, two of which are boundary layer subdomains and one is a regular subdomain. On each subdomain, we discretize the problem by the backward Euler scheme in the time direction and the central difference scheme in the spatial direction. The proposed method is shown to be uniformly convergent, having almost second order in space and first order in time. In addition, we prove that the proposed method converges much faster for small values of perturbation parameterAt the end, some numerical results are given in support of theoretical findings.

Analytical Calculation and Experimental Verification of Cogging Torque and Optimal Point in Permanent Magnet Synchronous Motors

In this paper, an exact analytical solution based on Fourier analysis is proposed to compute the cogging torque in a surface-mounted permanent magnet synchronous machine. The analytical solutions are derived by solving the field-governing equations in each simple and regular subdomain, i.e., permanent magnet (PM), air, slot opening, and slot, and then by applying the boundary conditions to the interfaces between these subdomains. Based on these solutions, the cogging torque is also determined analytically. In order to verify the analytical results of the proposed method, an experimental system was implemented on a servomotor and a torque sensor. All analytical results were extensively validated using nonlinear 2-D finite-element analysis and experimental results. Using the proposed method, we investigated the influence on the cogging torque, according to the machine parameters. Therefore, the proposed method should be very useful in the initial design and cogging torque optimization of PM machines.

Analytical prediction for electromagnetic performance of interior permanent magnet machines based on subdomain model

This paper presents an analytical model for the computation of the electromagnetic performance in interior permanent magnet (IPM) machines that accounts for the stator and the complex rotor structure. Using the subdomain method, we propose a simplified analytical model that considers the magnetic properties of the IPM machine. The analytical solutions are derived by solving the field-governing equations in each simple and regular subdomain, i.e., magnet, barrier, air gap, slot opening, and slot, and then applying the boundary conditions to the interfaces between these subdomains. The analytical model accurately accounts for the influence of the interaction between the slots, the relative recoil permeability of the magnets, and the boundary conditions. The magnetic field and electromagnetic performance obtained using the analytical method are compared with those obtained using finite element analysis. Finally, the analytical predictions are compared with the measured data in order to confirm the validity of the methods proposed in this paper.

Analysis of some injection bounds for Sobolev spaces by wavelet decomposition

We consider the Sobolev spaces H s ( Ω ) and H 0 s ( Ω ) and the Besov spaces B 2 , ∞ 1 / 2 ( Ω ) , where Ω is a sufficiently regular (see Lemma 2) subdomain of R 2 . It is well known that for the values of s ∈ [ 0 , 1 / 2 ) the two Sobolev spaces coincide, with equivalence of the norms, and that the inclusion B 2 , ∞ 1 / 2 ( Ω ) ⊂ H s ( Ω ) holds. The Note is concerned with the explicit analysis of the constants appearing in the continuity bounds for the injections H s ( Ω ) ↪ H 0 s ( Ω ) and B 2 , ∞ 1 / 2 ( Ω ) ↪ H s ( Ω ) and of their dependence on the regularity s of the spaces. The analysis is carried out by using the wavelet characterization of the corresponding norms.

Maximal subextensions of plurisubharmonic functions

In this paper we are concerned with the problem of local and global subextensions of (quasi-)plurisubharmonic functions from a subdomain of a compact Kahler manifold. We prove that a precise bound on the complex Monge-Ampere mass of the given function implies the existence of a subextension to a bigger regular subdomain or to the whole compact manifold. In some cases we show that the maximal subextension has a well defined complex Monge-Ampere measure and obtain precise estimates on this measure. Finally we give an example of a plurisubharmonic function with a well defined Monge-Ampere measure and the right bound on its Monge-Ampere mass on the unit ball in $\C^n$ for which the maximal subextension to the complex projective space $\mb P_n$ does not have a globally well defined complex Monge-Ampere measure.

An Accurate Subdomain Model for Magnetic Field Computation in Slotted Surface-Mounted Permanent-Magnet Machines

The paper presents an accurate analytical subdomain model for computation of the open-circuit magnetic field in surface-mounted permanent-magnet machines with any pole and slot combinations, including fractional slot machines, accounting for stator slotting effect. It is derived by solving the field governing equations in each simple and regular subdomain, i.e., magnet, air-gap and stator slots, and applying the boundary conditions to the interfaces between these subdomains. The model accurately accounts for the influence of interaction between slots, radial/parallel magnetization, internal/external rotor topologies, relative recoil permeability of magnets, and odd/even periodic boundary conditions. The back-electromotive force, electromagnetic torque, cogging torque, and unbalanced magnetic force are obtained based on the field model. The relationship between this accurate subdomain model and the conventional subdomain model, which is based on the simplified one slot per pole machine model, is also discussed. The investigation shows that the proposed accurate subdomain model has better accuracy than the subdomain model based on one slot/pole machine model. The finite element and experimental results validate the analytical prediction.

Numerical Simulation of 3-D Potential Problems by Regular Hybrid Boundary Node Method

The Regular Hybrid Boundary Node Method (RHBNM) is developed for solving three-dimensional potential problems. Formulations are developed and a general computer code written in C++. The RHBNM is formulated in terms of the domain and boundary variables. The domain variables are interpolated by classical fundamental solutions with the source points located outside the domain, and the boundary variables are interpolated by MLS approximation. The main idea is to retain the dimensionality advantages of the Boundary Element Method, and localize the integration domain to a regular sub-domain so that no mesh is needed for integration. All integrals can be easily evaluated over regularly shaped domains (in general, semi-sphere in the 3-D problem) and their boundaries. Numerical examples demonstrate that high convergence rate with mesh refinement and high accuracy with a small node number are achievable.

The regular hybrid boundary node method for three-dimensional linear elasticity

The Regular Hybrid Boundary Node Method (RHBNM) is developed in this paper for solving three-dimensional linear elasticity problems. Coupling modified functional with the Moving Least Squares (MLS) approximation, the RHBNM only requires discrete nodes constructed on the surface of a domain. Formulations and a general computer code of the RHBNM for 3D linear elasticity problems and the MLS interpolation on a generic surface have been developed. The RHBNM is formulated in terms of the domain and boundary variables. The domain variables are interpolated by classical fundamental solutions with the source points located outside the domain; and the boundary variables are interpolated by MLS approximation. The main idea is to retain the dimensionality advantages of the BNM, and localize the integration domain to a regular sub-domain, as in the MLBIE, such that no mesh is needed for integration. All integrals can be easily evaluated over regular shaped domains (in general, semi-sphere in the 3D problem) and their boundaries. Numerical examples for the solution of 3D elastostatic problems show that the high convergence rates with mesh refinement and the high accuracy with a small node number are achievable. The treatment of singularities and further integrations required for the computation of the unknown domain variables, as in the conventional BEM and BNM, can be avoided.

Global superconvergence of simplified hybrid combinations of the Ritz–Galerkin and FEMs for elliptic equations with singularities II. Lagrange elements and Adini's elements

To solve the elliptic problems with singularities, the simplified hybrid combinations of the Ritz–Galerkin method and the finite element method (RGM–FEM) are explored, which lead to the high global superconvergence rates on the entire solution domain. Let the solution domain be split into a singular subdomain involving a singular point, and a regular subdomain where the true solution is smooth enough. In the singular subdomain, the singular particular functions are chosen to be admissible functions. In the regular subdomain either the k-order Lagrange rectangles or Adini's elements are adapted. Along their common boundary, the simplified hybrid techniques are employed to couple two different numerical methods. It is proven in this paper that the global superconvergence rates, O( h k+3/2 ), on the entire domain can be achieved for k(⩾2)-order Lagrange rectangles, and that the global superconvergence rates O( h 3.5) for the Adini's elements. Numerical experiments are reported for the combinations of the Ritz–Galerkin and Adini's methods. This paper presents a development of [Z.C. Li, Computing 65 (2000) 27–44] in high accurate solutions for the general case of the Poisson problems on a polygonal domain S estimates for the Sobolev norm ∥·∥ 1, given in a much more general sense than known before, cf. [P.G. Ciarlet, J.L. Lions (Eds.) Finite Element Methods (Part 1), North-Holland, Amsterdam, 1991, pp. 17–351, 501–522; SIAM J. Sci. Statist. Comput. 11 (1990) 343; J. Comput. Appl. Math. 20 (1987) 341; Numer. Math. 63 (1992) 483; L. Wahlbin, Superconvergence in Galerkin Finite Element Methods, Springer, Berlin, 1995; Numer. Methods Partial Differential Equations 3 (1987) 65, 357].

Atmospheric pollution transport: the parallelization of a transport & chemistry code

We discuss a model for computing the chemical reactions occurring among air pollutants and predict their dispersion in the atmosphere. We consider a parallel implementation on a CRAY T3D, using up to 128 processors. A regular subdomain decomposition is considered to optimize inter-processor communications, while load balancing – the main technical difficulty to obtain a good efficiency – is achieved by re-assigning temporarily pieces of work to other processors, according to a global strategy. The performance of our implementation is described by an analytical relation whose validity is checked on realistic data sets. A speedup of 100 is obtained with 128 processors.

The Use of Pointwise Interpolation in Domain Decomposition Methods with Nonnested Meshes

In this paper, the author develops a new technique and a corresponding theory for Schwarz-type overlapping domain decomposition methods for solving large sparse linear systems which arise from finite element discretization of elliptic partial differential equations. The theory provides an optimal convergence of an additive Schwarz algorithm that is constructed with a nonnested coarse space and a not necessarily shape regular subdomain partitioning. The theory is also applicable to the graph partitioning algorithms recently developed by Cai and Saad [Overlapping domain decomposition algorithms for general sparse matrices, Preprint 93-027, Univ. of Minnesota] and Farhat and Lesoine [Internat. J. Numer. Methods Engrg., 36 (1993), pp. 745–764] for problems defined on unstructured meshes.

Three-dimensional crack and contact problems with a general geometric configuration

A new method, based on point set theory and properties of orthogonal functions, is developed for determining exact solutions to three-dimensional crack and contact problems with complicated geometric configurations (e.g. a star-convex domain) in an infinite linear elastic medium. The governing equation is a two-dimensional Fredholm integral equation of the first kind. The central idea of this method is the chain extension of an exact solution from a regular subdomain to an irregular entire domain. Examples are given illustrating how this solution procedure can be used to obtain exact closed form solutions for a general Hertz contact problem and various crack problems in an inhomogeneous isotropic medium with an elastic modulus which is a power function of depth.

Functions of Uniformly Bounded Characteristic on Riemann Surfaces

A characteristic function $T(D,w,f)$ of Shimizu and Ahlfors type for a function $f$ meromorphic in a Riemann surface $R$ is defined, where $D$ is a regular subdomain of $R$ containing a reference point $w \in R$. Next we suppose that $R$ has the Green functions. Letting $T(w,f) = {\lim _{D \uparrow R}}T(D,w,f)$, we define $f$ to be of uniformly bounded characteristic in $R$, $f \in {\text {UBC}}(R)$ in notation, if ${\sup _{w \in R}}T(w,f) < \infty$. We shall propose, among other results, some criteria for $f$ to be in ${\text {UBC}}(R)$ in various terms, namely, Green’s potentials, harmonic majorants, and counting functions. They reveal that ${\text {UBC}}(\Delta )$ for the unit disk $\Delta$ coincides precisely with that introduced in our former work. Many known facts on ${\text {UBC}}(\Delta )$ are extended to ${\text {UBC}}(R)$ by various methods. New proofs even for $R = \Delta$ are found. Some new facts, even for $\Delta$, are added.

Functions of uniformly bounded characteristic on Riemann surfaces

A characteristic function T ( D , w , f ) T(D,w,f) of Shimizu and Ahlfors type for a function f f meromorphic in a Riemann surface R R is defined, where D D is a regular subdomain of R R containing a reference point w ∈ R w \in R . Next we suppose that R R has the Green functions. Letting T ( w , f ) = lim D ↑ R T ( D , w , f ) T(w,f) = {\lim _{D \uparrow R}}T(D,w,f) , we define f f to be of uniformly bounded characteristic in R R , f ∈ UBC ( R ) f \in {\text {UBC}}(R) in notation, if sup w ∈ R T ( w , f ) &gt; ∞ {\sup _{w \in R}}T(w,f) &gt; \infty . We shall propose, among other results, some criteria for f f to be in UBC ( R ) {\text {UBC}}(R) in various terms, namely, Green’s potentials, harmonic majorants, and counting functions. They reveal that UBC ( Δ ) {\text {UBC}}(\Delta ) for the unit disk Δ \Delta coincides precisely with that introduced in our former work. Many known facts on UBC ( Δ ) {\text {UBC}}(\Delta ) are extended to UBC ( R ) {\text {UBC}}(R) by various methods. New proofs even for R = Δ R = \Delta are found. Some new facts, even for Δ \Delta , are added.