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Zhang, Luxin, Pascal Germain, Yacine Kessaci, and Christophe Biernacki. "Interpretable Domain Adaptation for Hidden Subdomain Alignment in the Context of Pre-trained Source Models." Proceedings of the AAAI Conference on Artificial Intelligence 36, no. 8 (June 28, 2022): 9057–65. http://dx.doi.org/10.1609/aaai.v36i8.20890.
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Domain adaptation aims to leverage source domain knowledge to predict target domain labels. Most domain adaptation methods tackle a single-source, single-target scenario, whereas source and target domain data can often be subdivided into data from different distributions in real-life applications (e.g., when the distribution of the collected data changes with time). However, such subdomains are rarely given and should be discovered automatically. To this end, some recent domain adaptation works seek separations of hidden subdomains, w.r.t. a known or fixed number of subdomains. In contrast, this paper introduces a new subdomain combination method that leverages a variable number of subdomains. Precisely, we propose to use an inter-subdomain divergence maximization criterion to exploit hidden subdomains. Besides, our proposition stands in a target-to-source domain adaptation scenario, where one exploits a pre-trained source model as a black box; thus, the proposed method is model-agnostic. By providing interpretability at two complementary levels (transformation and subdomain levels), our method can also be easily interpreted by practitioners with or without machine learning backgrounds. Experimental results over two fraud detection datasets demonstrate the efficiency of our method.
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Harbi, A., and M. Boulbrachene. "Maximum Norm Analysis of a Nonmatching Grids Method for Nonlinear Elliptic PDES." Journal of Applied Mathematics 2011 (2011): 1–18. http://dx.doi.org/10.1155/2011/605140.
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We provide a maximum norm analysis of a finite element Schwarz alternating method for a nonlinear elliptic PDE on two overlapping subdomains with nonmatching grids. We consider a domain which is the union of two overlapping subdomains where each subdomain has its own independently generated grid. The two meshes being mutually independent on the overlap region, a triangle belonging to one triangulation does not necessarily belong to the other one. Under a Lipschitz asssumption on the nonlinearity, we establish, on each subdomain, an optimalL∞error estimate between the discrete Schwarz sequence and the exact solution of the PDE.
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Dryja, Maksymilian, Juan Galvis, and Marcus Sarkis. "A Deluxe FETI-DP Preconditioner for a Composite Finite Element and DG Method." Computational Methods in Applied Mathematics 15, no. 4 (October 1, 2015): 465–82. http://dx.doi.org/10.1515/cmam-2015-0025.
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AbstractIn this paper, we present and analyze a FETI-DP solver with deluxe scaling for a Nitsche-type discretization [Comput. Methods Appl. Math. 3 (2003), 76–85], [SIAM J. Numer. Anal. 49 (2011), 1761–1787] based on a discontinuous Galerkin (DG) method for elliptic two-dimensional problems with discontinuous coefficients and non-matching meshes only across subdomains. We establish a condition number estimate for the preconditioned linear system which is scalable with respect to the number of subdomains, is quasi-optimal polylogarithmic with respect to subdomain mesh size, and is independent of coefficient discontinuities and ratio of mesh sizes across subdomain interfaces. Numerical experiments support the theory and show that the deluxe scaling improves significantly the performance over classical scaling.
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Journal, Baghdad Science. "Solving Two-Points Singular Boundary Value Problem Using Hermite Interpolation." Baghdad Science Journal 12, no. 4 (December 6, 2015): 826–32. http://dx.doi.org/10.21123/bsj.12.4.826-832.
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In this paper, we have been used the Hermite interpolation method to solve second order regular boundary value problems for singular ordinary differential equations. The suggest method applied after divided the domain into many subdomains then used Hermite interpolation on each subdomain, the solution of the equation is equal to summation of the solution in each subdomain. Finally, we gave many examples to illustrate the suggested method and its efficiency.
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Zhang, Di, Yuan Wei, Baoqiang Wang, and Shulin Liu. "Scale adaptive subdomain matching network for bearing fault diagnosis." Measurement Science and Technology 33, no. 2 (December 8, 2021): 025006. http://dx.doi.org/10.1088/1361-6501/ac3627.
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Abstract The wide application of transfer learning technology can effectively solve the problem of the difference between data collection and actual application equipment of traditional intelligent fault diagnosis methods in the practical application process. However, the difference in subdomain space and the serious imbalance of data samples in the process of simultaneous transfer restricts the deep transfer learning technology to the engineering application of high-precision diagnosis. In order to solve the problem of subdomain matching with different subspaces and unbalanced data samples, in this paper we study the subdomain adaptive method and propose a scale adaptive subdomain matching (SASM) method. The SASM method divides the global feature space according to the sample labels, and features with the same label will be divided into the same sub-feature space. Using the edge distribution of the sample and the category weight of the label, the SASM method can effectively optimize the feature distribution of the same subdomain and the weight distribution of different subdomains. Based on the establishment of a clearer internal structure of features, the field adaptation effect is improved, and the matching ability is enhanced when the sample is unevenly distributed. At the same time, the SASM network (SASMN) method for unsupervised bearing fault diagnosis is constructed and validated by experiments. The results indicate that SASMN can effectively optimize the subdomain adaptive effect, and the diagnostic accuracy of the target domain data set is significantly higher than the other three currently popular domain adaptive fault diagnosis methods.
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PANASENKO, G. P. "METHOD OF ASYMPTOTIC PARTIAL DECOMPOSITION OF DOMAIN." Mathematical Models and Methods in Applied Sciences 08, no. 01 (February 1998): 139–56. http://dx.doi.org/10.1142/s021820259800007x.
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A new method of partial decomposition of a domain is proposed for partial differential equations, depending on a small parameter. It is based on the information about the structure of the asymptotic solution in different parts of the domain. The principal idea of the method is to extract the subdomain of singular behavior of the solution and to simplify the problem in the subdomain of regular behavior of the solution. The special interface conditions are imposed on the common boundary of these partially decomposed subdomains. If, for example, the domain depends on the small parameter and some parts of the domain change their dimension after the passage to the limit, then the proposed method reduces the initial problem to the system of equations posed in the domains of different dimensions with the special interface conditions.
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Huang, Peiqi, Jinru Chen, and Mingchao Cai. "A Mortar Method Using Nonconforming and Mixed Finite Elements for the Coupled Stokes-Darcy Model." Advances in Applied Mathematics and Mechanics 9, no. 3 (January 17, 2017): 596–620. http://dx.doi.org/10.4208/aamm.2016.m1397.
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AbstractIn this work, we study numerical methods for a coupled fluid-porous media flow model. The model consists of Stokes equations and Darcy's equations in two neighboring subdomains, coupling together through certain interface conditions. The weak form for the coupled model is of saddle point type. A mortar finite element method is proposed to approximate the weak form of the coupled problem. In our method, nonconforming Crouzeix-Raviart elements are applied in the fluid subdomain and the lowest order Raviart-Thomas elements are applied in the porous media subdomain; Meshes in different subdomains are allowed to be nonmatching on the common interface; Interface conditions are weakly imposed via adding constraint in the definition of the finite element space. The well-posedness of the discrete problem and the optimal error estimate for the proposed method are established. Numerical experiments are also given to confirm the theoretical results.
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Gander, Martin J., and Hui Zhang. "Schwarz methods by domain truncation." Acta Numerica 31 (May 2022): 1–134. http://dx.doi.org/10.1017/s0962492922000034.
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Schwarz methods use a decomposition of the computational domain into subdomains and need to impose boundary conditions on the subdomain boundaries. In domain truncation one restricts the unbounded domain to a bounded computational domain and must also put boundary conditions on the computational domain boundaries. In both fields there are vast bodies of literature and research is very active and ongoing. It turns out to be fruitful to think of the domain decomposition in Schwarz methods as a truncation of the domain onto subdomains. Seminal precursors of this fundamental idea are papers by Hagstrom, Tewarson and Jazcilevich (1988), Després (1990) and Lions (1990). The first truly optimal Schwarz method that converges in a finite number of steps was proposed by Nataf (1993), and used precisely transparent boundary conditions as transmission conditions between subdomains. Approximating these transparent boundary conditions for fast convergence of Schwarz methods led to the development of optimized Schwarz methods – a name that has become common for Schwarz methods based on domain truncation. Compared to classical Schwarz methods, which use simple Dirichlet transmission conditions and have been successfully used in a wide range of applications, optimized Schwarz methods are much less well understood, mainly due to their more sophisticated transmission conditions.A key application of Schwarz methods with such sophisticated transmission conditions turned out to be time-harmonic wave propagation problems, because classical Schwarz methods simply do not work in this case. The past decade has given us many new Schwarz methods based on domain truncation. One review from an algorithmic perspective (Gander and Zhang 2019) showed the equivalence of many of these new methods to optimized Schwarz methods. The analysis of optimized Schwarz methods, however, is lagging behind their algorithmic development. The general abstract Schwarz framework cannot be used for the analysis of these methods, and thus there are many open theoretical questions about their convergence. Just as for practical multigrid methods, Fourier analysis has been instrumental for understanding the convergence of optimized Schwarz methods and for tuning their transmission conditions. Similar to local Fourier mode analysis in multigrid, the unbounded two-subdomain case is used as a model for Fourier analysis of optimized Schwarz methods due to its simplicity. Many aspects of the actual situation, e.g. boundary conditions of the original problem and the number of subdomains, were thus neglected in the unbounded two-subdomain analysis. While this gave important insight, new phenomena beyond the unbounded two-subdomain models were discovered.This present situation is the motivation for our survey: to give a comprehensive review and precise exploration of convergence behaviours of optimized Schwarz methods based on Fourier analysis, taking into account the original boundary conditions, many-subdomain decompositions and layered media. We consider as our model problem the operator $-\Delta + \eta $ in the diffusive case $\eta>0$ (screened Laplace equation) or the oscillatory case $\eta <0$ (Helmholtz equation), in order to show the fundamental difference in behaviour of Schwarz solvers for these problems. The transmission conditions we study include the lowest-order absorbing conditions (Robin), and also more advanced perfectly matched layers (PMLs), both developed first for domain truncation. Our intensive work over the last two years on this review has led to several new results presented here for the first time: in the bounded two-subdomain analysis for the Helmholtz equation, we see strong influence of the original boundary conditions imposed on the global problem on the convergence factor of the Schwarz methods, and the asymptotic convergence factors with small overlap can differ from the unbounded two-subdomain analysis. In the many-subdomain analysis, we find the scaling with the number of subdomains, e.g. when the subdomain size is fixed, robust convergence of the double-sweep Schwarz method for the free-space wave problem, either with fixed overlap and zeroth-order Taylor conditions or with a logarithmically growing PML, and we find that Schwarz methods with PMLs work like smoothers that converge faster for higher Fourier frequencies; in particular, for the free-space wave problem, plane waves (in the error) passing through interfaces at a right angle converge more slowly. In addition to our main focus on analysis in Sections 2 and 3, we start in Section 1 with an expository historical introduction to Schwarz methods, and in Section 4 we give a brief interpretation of the recently proposed optimal Schwarz methods for decompositions with cross-points from the viewpoint of transmission conditions. We conclude in Section 5 with a summary of open research problems. In Appendix A we provide a Matlab program for a block LU form of an optimal Schwarz method with cross-points, and in Appendix B we give the Maple program for the two-subdomain Fourier analysis.
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Boulaaras, Salah, Mohammed Cherif Bahi, Mohamed Haiour, and Abderrahman Zarai. "The maximum norm analysis of a nonmatching grids method for a class of parabolic equation with nonlinear source terms." Boletim da Sociedade Paranaense de Matemática 38, no. 4 (March 10, 2019): 157–74. http://dx.doi.org/10.5269/bspm.v38i4.40272.
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Motivated by the idea which has been introduced by M. Haiour and S.Boulaaras (Proc. Indian Acad. Sci. (Math. Sci.) Vol. 121,No. 4, November 2011,pp.481--493), We provide a maximum norm analysis of Euler scheme combined with finite element Schwarz alternating method for a class of parabolic equation with nonlinear source terms on two overlapping subdomains with nonmatching grids. We consider a domain which is the union of two overlapping subdomains where each subdomain has its own independently generated grid. The two meshes being mutually independent on the overlap region, a triangle belonging to one triangulation does not necessarily belong to the other one. Under a stability analysis on the theta scheme which given by our work in (App. Math. Comp., 217, 6443--6450 (2011).), we establish, on each subdomain, an optimal asymptotic behavior between the discrete Schwarz sequence and the asymptotic solution of parabolic differential equations.
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Akçali, I. D., and G. Dittrich. "Path generation by subdomain method." Mechanism and Machine Theory 24, no. 1 (January 1989): 45–52. http://dx.doi.org/10.1016/0094-114x(89)90082-7.
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Agrawal, Ajay K., S. Krishnan, and Tah-teh Yang. "Use of Subdomains for Inverse Problems in Branching Flow Passages." Journal of Fluids Engineering 115, no. 2 (June 1, 1993): 227–32. http://dx.doi.org/10.1115/1.2910128.
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For inverse problems in complex flow passages, a calculation procedure based on a multizone Navier-Stokes method was developed. A heuristic approach was employed to derive wall shape corrections from the wall pressure error. Only two subdomains sharing a row of control volumes were used. The grid work in the common region was identical for both subdomains. The flow solver, inverse calculation procedure, multizone Navier-Stokes method and subdomain inverse calculation procedure were validated independently against experimental data or numerical predictions. Then, the subdomain inverse calculation method was used to determine the wall shape of the main duct of a branching flow passage. A slightly adverse pressure gradient was prescribed downstream of the sidebranch. Inverse calculations resulted in a curved wall diffuser for which the wall pressure distribution matched the design (prescribed) wall pressure distribution. The present method was illustrated for laminar, incompressible flows in branching passages. However, the method presented is flexible and can be extended for turbulent flows in multiply connected domains.
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Brackstone, M., and A. S. Deakin. "Approximations of Time Series." ISRN Applied Mathematics 2011 (September 22, 2011): 1–10. http://dx.doi.org/10.5402/2011/321683.
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A method is proposed to approximate the main features or patterns including interventions that may occur in a time series. Collision data from the Ontario Ministry of Transportation illustrate the approach using monthly collision counts from police reports over a 10-year period from 1990 to 1999. The domain of the time series is partitioned into nonoverlapping subdomains. The major condition on the approximation requires that the series and the approximation have the same average value over each subdomain. To obtain a smooth approximation, based on the second difference of the series, a few iterations are necessary since an iteration over one subdomain is affected by the previous iteration over the adjacent subdomains.
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Arshad, Muhammad, Madiha Sana, and Muhammad Mustahsan. "Multiblock Mortar Mixed Approach for Second Order Parabolic Problems." Mathematics 7, no. 4 (April 2, 2019): 325. http://dx.doi.org/10.3390/math7040325.
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In this paper, the multiblock mortar mixed approximation of second order parabolic partial differential equations is considered. In this method, the simulation domain is decomposed into the non-overlapping subdomains (blocks), and a physically-meaningful boundary condition is set on the mortar interface between the blocks. The governing equations hold locally on each subdomain region. The local problems on blocks are coupled by introducing a special approximation space on the interfaces of neighboring subdomains. Each block is locally covered by an independent grid and the standard mixed finite element method is applied to solve the local problem. The unique solvability of the discrete problem is shown, and optimal order convergence rates are established for the approximate velocity and pressure on the subdomain. Furthermore, an error estimate for the interface pressure in mortar space is presented. The numerical experiments are presented to validate the efficiency of the method.
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Zhao, Shengkai, and Matthew J. Yedlin. "Multidomain Chebyshev spectral method for 3-D dc resistivity modeling." GEOPHYSICS 61, no. 6 (November 1996): 1616–23. http://dx.doi.org/10.1190/1.1444080.
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We use the multidomain Chebyshev spectral method to solve the 3-D forward direct current (dc) resistivity problem. We divided the whole domain into a number of subdomains and approximate the potential function by a separate set of Chebyshev polynomials in each subdomain. At an interface point, we require that both the potential and the flux be continuous. Numerical results show that for the same accuracy the multidomain Chebyshev spectral method is 2 to 260 times faster than the finite‐difference method.
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Geng, Huihui, Xueyi Zhang, Shilong Yan, Yufeng Zhang, Lei Wang, Yutong Han, and Wei Wang. "Magnetic Field Analysis of an Inner-Mounted Permanent Magnet Synchronous Motor for New Energy Vehicles." Energies 15, no. 11 (June 1, 2022): 4074. http://dx.doi.org/10.3390/en15114074.
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The motor is an important component that affects the output performance of new energy vehicles (using new energy sources such as electric energy and hydrogen fuel energy to drive the motor and provide kinetic energy). Motors with high power and low noise can effectively improve the dynamic performance, passability and smoothness of new energy vehicles and bring a comfortable experience to driver and passengers. The magnetic field analytical model of the inner-mounted permanent magnet synchronous motor (IPMSM) is studied to improve its output quality. The motor is divided into four subdomains: the stator slot subdomain, the stator slot notch subdomain, the air-gap subdomain, and the permanent magnet (PM) subdomain. The general solution of the vector magnetic potential of each subdomain is solved, and the expression of magnetic flux density of each subdomain is derived. Meanwhile, the analytical model of the non-uniform air gap is established according to the uniform air-gap model. The model’s accuracy is verified by finite element analysis and prototype tests. The results show that the calculation results of the analytical model are effective. The model can be applied to predict the no-load back electromotive force (EMF) and cogging torque of the motor under different main air gaps. It also provides an effective and fast analysis method for the design and optimization of IPMSM for new energy vehicles.
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BERRONE, S., and L. EMMEL. "A REALIZATION OF A WAVELET GALERKIN METHOD ON NONTRIVIAL DOMAINS." Mathematical Models and Methods in Applied Sciences 12, no. 11 (November 2002): 1525–54. http://dx.doi.org/10.1142/s0218202502002227.
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In this paper we describe a realization of the Wavelet Element Method (WEM), for numerically solving second-order elliptic PDEs on fairly general domains. We describe in a detailed form the construction of biorthogonal wavelet bases on these domains. The domain of interest is split into subdomains and mapped to the unit reference cube. The bases obtained on each subdomain are matched to obtain continuous global wavelet bases. Suitable [Formula: see text] data structures for an efficient implementation of the wavelet Galerkin method are described.
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Chu, Fuyun, Lihua Wang, and Zheng Zhong. "Finite subdomain radial basis collocation method." Computational Mechanics 54, no. 2 (February 13, 2014): 235–54. http://dx.doi.org/10.1007/s00466-014-0981-9.
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Moutafis, Byron E., Christos K. Filelis-Papadopoulos, and George A. Gravvanis. "Parallel Multiprojection Preconditioned Methods Based on Subspace Compression." Mathematical Problems in Engineering 2017 (2017): 1–11. http://dx.doi.org/10.1155/2017/2580820.
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During the last decades, the continuous expansion of supercomputing infrastructures necessitates the design of scalable and robust parallel numerical methods for solving large sparse linear systems. A new approach for the additive projection parallel preconditioned iterative method based on semiaggregation and a subspace compression technique, for general sparse linear systems, is presented. The subspace compression technique utilizes a subdomain adjacency matrix and breadth first search to discover and aggregate subdomains to limit the average size of the local linear systems, resulting in reduced memory requirements. The depth of aggregation is controlled by a user defined parameter. The local coefficient matrices use the aggregates computed during the formation of the subdomain adjacency matrix in order to avoid recomputation and improve performance. Moreover, the rows and columns corresponding to the newly formed aggregates are ordered last to further reduce fill-in during the factorization of the local coefficient matrices. Furthermore, the method is based on nonoverlapping domain decomposition in conjunction with algebraic graph partitioning techniques for separating the subdomains. Finally, the applicability and implementation issues are discussed and numerical results along with comparative results are presented.
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Ren, Zhengyong, Thomas Kalscheuer, Stewart Greenhalgh, and Hansruedi Maurer. "A finite-element-based domain-decomposition approach for plane wave 3D electromagnetic modeling." GEOPHYSICS 79, no. 6 (November 1, 2014): E255—E268. http://dx.doi.org/10.1190/geo2013-0376.1.
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We developed a novel parallel domain-decomposition approach for 3D large-scale electromagnetic induction modeling in the earth. We used the edge-based finite-element method and unstructured meshes. Unstructured meshes were divided into sets of nonoverlapping subdomains. We used the curl-curl electric field equation to carry out the analysis. In each subdomain, the electric field was discretized by first-order vector shape functions along the edges of tetrahedral elements. The tangential components of the magnetic field on the interfaces of the subdomains were defined as a set of Lagrange multipliers. The unknown Lagrange multipliers were solved from an interface problem defined on the interfaces of the subdomains. With the availability of the Lagrange multipliers, the electric field values in each subdomain were solved independently. Three synthetic examples were evaluated to verify our code. Excellent agreement with previously published solutions was obtained. Synthetic examples revealed that our domain decomposition technique is scalable with respect to the number of subdomains and robust with regard to frequency and the heterogeneous distribution of material parameters, i.e., electric conductivity, electric permittivity, and magnetic permeability.
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Li, An-qi, Cheng-you Yin, Qian-qian Zhang, and Yong-ji Gan. "Research on Fast Algorithm of Radio Wave Propagation in Low-Lossy Obstacles Environment." International Journal of Antennas and Propagation 2021 (April 20, 2021): 1–13. http://dx.doi.org/10.1155/2021/5519586.
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To predict the propagation of radio waves in the environment of dielectric ground and dielectric obstacles, a new two-way parabolic equation (2W-PE) method based on the domain decomposition principle and surface impedance boundary conditions (SIBC) is proposed. First, we decompose the obstacle area into different subdomains and derive the SIBC in each subdomain in detail; then, the discrete hybrid Fourier transform (DMFT) in the upper subdomain and finite difference (FD) algorithm in the lower subdomain is used to solve 2W-PE combined with SIBC, respectively. After that, we explain the algorithm steps in the process of calculating the total field, compared with the traditional 2W-PE, and then finally introduce the method of moments (MoM) combined with the enhanced discrete complex image (E-DCIM) method for accuracy verification of the new 2W-PE algorithm. The simulation results show that no matter how the obstacle medium parameters change, the results of 2W-PE method proposed in this paper and MoM are always in good agreement, which proves the accuracy of 2W-PE and its superiority in speed. Therefore, this paper provides a reliable and efficient method for solving the problem of radio wave propagation in the presence of obstacles, especially in the case of low-lossy obstacles.
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Kwan, Yuen-Yick. "Additive Schwarz Preconditioners with Minimal Overlap for Triangular Spectral Elements." Communications in Computational Physics 13, no. 2 (February 2013): 411–27. http://dx.doi.org/10.4208/cicp.080711.160212a.
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AbstractThe additive Schwarz preconditioner with minimal overlap is extended to triangular spectral elements (TSEM). The method is a generalization of the corresponding method in tensorial quadrilateral spectral elements (QSEM). The proposed preconditioners are based on partitioning the domain into overlapping subdomains, solving local problems on these subdomains and solving an additional coarse problem associated with the subdomain mesh. The results of numerical experiments show that the proposed preconditioner are robust with respect to the number of elements and are more efficient than the preconditioners with generous overlaps.
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Huang, Li Na, Ming Xin Xue, Hao Dong, and Bo Yang. "A Subdomain Method for the Aeroacoustic Simulation of a Generic Side View Mirror." Applied Mechanics and Materials 437 (October 2013): 321–24. http://dx.doi.org/10.4028/www.scientific.net/amm.437.321.
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The aerodynamic noise caused by the flow field around a generic side view mirror (SVM) was simulated using a subdomain large eddy simulation (LES) method. In this method, the LES solution could be run only in the subdomain, which can be the flow field near the SVM. The subdomain LES results show good agreement with the cited experimental data in some related works. With the principal advantage of saving CFD cell numbers, the subdomain LES method would be a perspective way to simulate the aerodynamic noise of complex geometries such as the real automobiles.
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Sivak, Sergey, Mihail Royak, Ilya Stupakov, Aleksandr Aleksashin, and Ekaterina Voznjuk. "The implementation of the boundary element method to the Helmholtz equation of acoustics." Information and Control Systems, no. 2 (April 29, 2021): 13–19. http://dx.doi.org/10.31799/1684-8853-2021-2-13-19.
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Introduction: To solve the Helmholtz equation is important for the branches of engineering that require the simulation of wave phenomenon. Numerical methods allow effectiveness’ enhancing of the related computations. Methods: To find a numerical solution of the Helmholtz equation one may apply the boundary element method. Only the surface mesh constructed for the boundary of the three-dimensional domain of interest must be supplied to make the computations possible. This method’s trait makes it possible toconduct numerical experiments in the regions which are external in relation to some Euclidian three-dimensional subdomain bounded in the three-dimensional space. The later also provides the opportunity of not using additional geometric techniques to consider the infinitely distant boundary. However, it’s only possible to use the boundary element methods either for the homogeneous domains or for the domains composed out of adjacent homogeneous subdomains. Results: The implementation of the boundary elementmethod was committed in the program complex named Quasar. The discrepancy between the analytic solution approximation and the numerical results computed through the boundary element method for internal and external boundary value problems was analyzed. The results computed via the finite element method for the model boundary value problems are also provided for the purpose of the comparative analysis done between these two approaches. Practical relevance: The method gives an opportunityto solve the Helmholtz equation in an unbounded region which is a significant advantage over the numerical methods requiring the volume discretization of computational domains in general and over the finite element method in particular. Discussion: It is planned to make a coupling of the two methods for the purpose of providing the opportunity to conduct the computations in the complex regions with unbounded homogeneous subdomain and subdomains with substantial inhomogeneity inside.
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Wu, F., Q. Gao, and W. X. Zhong. "Subdomain Precise Integration Method for Periodic Structures." Shock and Vibration 2014 (2014): 1–11. http://dx.doi.org/10.1155/2014/657589.
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A subdomain precise integration method is developed for the dynamical responses of periodic structures comprising many identical structural cells. The proposed method is based on the precise integration method, the subdomain scheme, and the repeatability of the periodic structures. In the proposed method, each structural cell is seen as a super element that is solved using the precise integration method, considering the repeatability of the structural cells. The computational efforts and the memory size of the proposed method are reduced, while high computational accuracy is achieved. Therefore, the proposed method is particularly suitable to solve the dynamical responses of periodic structures. Two numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method through comparison with the Newmark and Runge-Kutta methods.
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Vabishchevich, P. N. "Subdomain solution decomposition method for nonstationary problems." Journal of Computational Physics 472 (January 2023): 111679. http://dx.doi.org/10.1016/j.jcp.2022.111679.
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Yang, Dinghui, Xijun He, Xiao Ma, Yanjie Zhou, and Jingshuang Li. "An optimal nearly analytic discrete-weighted Runge-Kutta discontinuous Galerkin hybrid method for acoustic wavefield modeling." GEOPHYSICS 81, no. 5 (September 2016): T251—T263. http://dx.doi.org/10.1190/geo2015-0686.1.
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The newly developed optimal nearly analytic discrete (ONAD) and the weighted Runge-Kutta discontinuous Galerkin (WRKDG) methods can effectively suppress the numerical dispersion caused by discretizing wave equations, but it is difficult for ONAD to implement on flexible meshes, whereas the WRKDG has high computational cost for wavefield simulations. We have developed a new hybrid algorithm by combining the ONAD method with the WRKDG method. In this hybrid algorithm, the computational domain was split into several subdomains, in which the subdomain for the ONAD method used regular Cartesian grids, whereas the subdomain for the WRKDG method used triangular grids. The hybrid method was at least third-order spatially accurate. We have applied the proposed method to simulate the scalar wavefields for different models, including a homogeneous model, a rough topography model, a fracture model, and a cave model. The numerical results found that the hybrid method can deal with complicated geometrical structures, effectively suppress numerical dispersion, and provide accurate seismic wavefields. Numerical examples proved that our hybrid method can significantly reduce the CPU time and save storage requirement for the tested models. This implies that the hybrid method is especially suitable for the simulation of waves propagating in complex media.
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MAGOULÈS, FRÉDÉRIC, FRANÇOIS-XAVIER ROUX, and LAURENT SERIES. "ALGEBRAIC WAY TO DERIVE ABSORBING BOUNDARY CONDITIONS FOR THE HELMHOLTZ EQUATION." Journal of Computational Acoustics 13, no. 03 (September 2005): 433–54. http://dx.doi.org/10.1142/s0218396x05002827.
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This paper is dedicated to the optimal convergence properties of a domain decomposition method involving two-Lagrange multipliers at the interface between the subdomains and additional augmented interface operators. Most methods for optimizing these augmented interface operators are based on the discretization of continuous approximations of the optimal transparent operators.1–5 Such approach is strongly linked to the continuous equation, and to the discretization scheme. At the discrete level, the optimal transparent operator can be proved to be equal to the Schur complement of the outer subdomain. Our idea consists of approximating directly the Schur complement matrix with purely algebraic techniques involving local condensation of the subdomain degree of freedom on small patch defined on the interface between the subdomains. The main advantage of such approach is that it is much more easy to implement in existing code without any information on the geometry of the interface and of the finite element formulation used. Such technique leads to a so-called "black box" for the users. Convergence results and parallel efficiency of this new and original algebraic optimization technique of the interface operators are presented for acoustics applications.
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Li, Hao, Pierre Ladevèze, and Hervé Riou. "Hybrid Finite Element Method and Variational Theory of Complex Rays for Helmholtz Problems." Journal of Computational Acoustics 24, no. 04 (December 2016): 1650015. http://dx.doi.org/10.1142/s0218396x16500156.
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In this paper, we consider the Weak Trefftz Discontinuous Galerkin (WTDG) method, which enables one to use at the same time the Finite Element Method (FEM) or Variational Theory of Complex Rays (VTCR) discretizations (polynoms and waves), for vibration problems. It has already been developed such that the FEM and the VTCR can be used in different adjacent subdomains in the same problem. Here, it is revisited and extended in order to allow one to use the two discretizations in the same subdomain, at the same time. Numerical examples illustrate the performances of such an approach.
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Hessari, Peyman, Sang Dong Kim, and Byeong-Chun Shin. "Numerical Solution for Elliptic Interface Problems Using Spectral Element Collocation Method." Abstract and Applied Analysis 2014 (2014): 1–11. http://dx.doi.org/10.1155/2014/780769.
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The aim of this paper is to solve an elliptic interface problem with a discontinuous coefficient and a singular source term by the spectral collocation method. First, we develop an algorithm for the elliptic interface problem defined in a rectangular domain with a line interface. By using the Gordon-Hall transformation, we generalize it to a domain with a curve boundary and a curve interface. The spectral element collocation method is then employed to complex geometries; that is, we decompose the domain into some nonoverlaping subdomains and the spectral collocation solution is sought in each subdomain. We give some numerical experiments to show efficiency of our algorithm and its spectral convergence.
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Liu, F. L., and K. M. Liew. "Vibration Analysis of Discontinuous Mindlin Plates by Differential Quadrature Element Method." Journal of Vibration and Acoustics 121, no. 2 (April 1, 1999): 204–8. http://dx.doi.org/10.1115/1.2893965.
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A new numerical technique, the differential quadrature element method (DQEM), has been developed for solving the free vibration of the discontinuous Mindlin plate in this paper. By the DQEM, the complex plate domain is decomposed into small simple continuous subdomains (elements) and the differential quadrature method (DQM) is applied to each continuous subdomain to solve the problems. The detailed formulations for the DQEM and the connection conditions between each element are presented. Several numerical examples are analyzed to demonstrate the accuracy and applicability of this new method to the free vibration analysis of the Mindlin plate with various discontinuities which are not solvable directly using the differential quadrature method.
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Chen, Qing, Baoqing Liu, and Qikui Du. "A D-N Alternating Algorithm for Solving 3D Exterior Helmholtz Problems." Mathematical Problems in Engineering 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/418426.
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The nonoverlapping domain decomposition method, which is based on the natural boundary reduction, is applied to solve the exterior Helmholtz problem over a three-dimensional domain. The basic idea is to introduce a spherical artificial boundary; the original unbounded domain is changed into a bounded subdomain and a typical unbounded region; then, a Dirichlet-Nuemann (D-N) alternating method is presented; the finite element method and natural boundary element methods are alternately applied to solve the problems in the bounded subdomain and the typical unbounded subdomain. The convergence of the D-N alternating algorithm and its discretization are studied. Some numerical experiments are presented to show the performance of this method.
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Zhang, Mei, and Peng Cheng Zhai. "Spline Subdomain Approach to Thermal Conduction Problem for Composite with Rectangular Shape Particle." Advanced Materials Research 152-153 (October 2010): 454–58. http://dx.doi.org/10.4028/www.scientific.net/amr.152-153.454.
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In this paper the spline subdomain approach is applied to the 2D simulations of the temperature distributions for composites containing a single rectangular particle with an interfacial thermal resistance at the interface between the particle and matrix. The bicubic B-splines are used to construct the trial functions for the approximations of the potential fields of composites. Applying the weighted residual point collocation method inside each subdomain and also on the boundaries between different subdomains, a system of linear algebraic equations is set up to determine the unknowns of the trial functions. The temperature distributions both inside the rectangular particle and along the interfaces under different interfacial contact conditions can be simulated approximately. Numerical results which are compared with the available solutions obtained by FEM method illustrate the accuracy and suitability of the present approach for steady-state conduction. Even in the adjacent areas of corners in the rectangular particle, the simulation results are also satisfactory.
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Badea, Lori. "On the Convergence of the Damped Additive Schwarz Methods and the Subdomain Coloring." Mathematical and Computational Applications 27, no. 4 (July 13, 2022): 59. http://dx.doi.org/10.3390/mca27040059.
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In this paper, we consider that the subdomains of the domain decomposition are colored such that the subdomains with the same color do not intersect and introduce and analyze the convergence of a damped additive Schwarz method related to such a subdomain coloring for the resolution of variational inequalities and equations. In this damped method, a single damping value is associated with all the subdomains having the same color. We first make this analysis both for variational inequalities and, as a special case, for equations in an abstract framework. By introducing an assumption on the decomposition of the convex set of the variational inequality, we theoretically analyze in a reflexive Banach space the convergence of the damped additive Schwarz method. The introduced assumption contains a constant C0, and we explicitly write the expression of the convergence rates, depending on the number of colors and the constant C0, and find the values of the damping constants which minimize them. For problems in the finite element spaces, we write the constant C0 as a function of the overlap parameter of the domain decomposition and the number of colors of the subdomains. We show that, for a fixed overlap parameter, the convergence rate, as a function of the number of subdomains has an upper limit which depends only on the number of the colors of the subdomains. Obviously, this limit is independent of the total number of subdomains. Numerical results are in agreement with the theoretical ones. They have been performed for an elasto-plastic problem to verify the theoretical predictions concerning the choice of the damping parameter, the dependence of the convergence on the overlap parameter and on the number of subdomains.
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Jaysaval, Piyoosh, Daniil Shantsev, and Sébastien de la Kethulle de Ryhove. "Fast multimodel finite-difference controlled-source electromagnetic simulations based on a Schur complement approach." GEOPHYSICS 79, no. 6 (November 1, 2014): E315—E327. http://dx.doi.org/10.1190/geo2014-0043.1.
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We have developed an efficient numerical scheme for fast multimodel 3D electromagnetic simulations by applying a Schur complement approach to a frequency-domain finite-difference method. The scheme is based on direct solvers and developed with constrained inversion algorithms in view. Such algorithms normally need many forward modeling jobs with different resistivities for the target zone and/or background formation. We geometrically divide the computational domain into two subdomains: an anomalous subdomain, the resistivities of which were permitted to change, and a background subdomain, having fixed resistivities. The system matrix is partially factorized by precomputing a Schur complement to eliminate unknowns associated with the background subdomain. The Schur complement system is then solved to compute fields inside the anomalous subdomain. Finally, the background subdomain fields are computed using inexpensive local substitutions. For each successive simulation, only the relatively small Schur complement system has to be solved, which results in significant computational savings. We applied this approach to two moderately sized 3D problems in marine controlled-source electromagnetic modeling: (1) a deepwater model in which the resistivities of the seawater and the air layer were kept fixed and (2) a model in which focused inversion was performed in a scenario in which the resistivities of the background formation, the air layer, and the seawater were known. We found a significant reduction of the modeling time in inversion that depended on the relative sizes of the constrained and unconstrained volumes: the smaller the unconstrained volume, the larger the savings. Specifically, for a focused inversion of the Troll oil field in the North Sea, the gain amounted up to 80% of the total modeling time.
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Gravenkamp, Hauke, Albert A. Saputra, and Sascha Eisenträger. "Three-dimensional image-based modeling by combining SBFEM and transfinite element shape functions." Computational Mechanics 66, no. 4 (August 4, 2020): 911–30. http://dx.doi.org/10.1007/s00466-020-01884-4.
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Abstract The scaled boundary finite element method (SBFEM) has recently been employed as an efficient tool to model three-dimensional structures, in particular when the geometry is provided as a voxel-based image. To this end, an octree decomposition of the computational domain is deployed, and each cubic cell is treated as an SBFE subdomain. The surfaces of each subdomain are discretized in the finite element sense. We improve on this idea by combining the semi-analytical concept of the SBFEM with a particular class of transition elements on the subdomains’ surfaces. Thus, a triangulation of these surfaces as executed in previous works is avoided, and consequently, the number of surface elements and degrees of freedom is reduced. In addition, these discretizations allow coupling elements of arbitrary order such that local p-refinement can be achieved straightforwardly.
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Deputat, V., and P. Oja. "QUADRATIC SPLINE SUBDOMAIN METHOD FOR VOLTERRA INTEGRAL EQUATIONS." Mathematical Modelling and Analysis 10, no. 4 (December 31, 2005): 335–44. http://dx.doi.org/10.3846/13926292.2005.9637291.
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In our work we consider the step‐by‐step and nonlocal subdomain methods with quadratic splines. We prove that the first method is unstable. In the case of nonlocal method we replaced the first derivative condition by a not‐a‐knot boundary condition at the other end of the interval of integration. As a result, we get stability of this method. Main results about stability and convergence are based on the uniform boundedness of quadratic spline histopolation projections. The numerical tests given at the end support the theoretical results.
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Gadala, M. S., and S. G. Tang. "Subdomain solutions of complex variable boundary element method." Engineering with Computers 14, no. 4 (December 1998): 321–29. http://dx.doi.org/10.1007/bf01201763.
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Chen, Jiun-Shyan, Lihua Wang, Hsin-Yun Hu, and Sheng-Wei Chi. "Subdomain radial basis collocation method for heterogeneous media." International Journal for Numerical Methods in Engineering 80, no. 2 (October 8, 2009): 163–90. http://dx.doi.org/10.1002/nme.2624.
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Wang, Lihua, Jiun-Shyan Chen, and Hsin-Yun Hu. "Subdomain radial basis collocation method for fracture mechanics." International Journal for Numerical Methods in Engineering 83, no. 7 (March 9, 2010): 851–76. http://dx.doi.org/10.1002/nme.2860.
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Benlarbi, Hakima, and Ahmed-Salah Chibi. "A Posteriori Error Estimates for the Generalized Overlapping Domain Decomposition Methods." Journal of Applied Mathematics 2012 (2012): 1–15. http://dx.doi.org/10.1155/2012/947085.
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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 residuala posteriorierror analysis to get ana posteriorierror estimate for the discrete solutions on subdomains. The results of some numerical experiments are presented to support the theory.
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Fazli Malidareh, Babak, Seyed Abbas Hosseini, and Ebrahim Jabbari. "Discrete mixed subdomain least squares (DMSLS) meshless method with collocation points for modeling dam-break induced flows." Journal of Hydroinformatics 18, no. 4 (February 11, 2016): 702–23. http://dx.doi.org/10.2166/hydro.2016.116.
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This paper presents a new meshless numerical scheme to overcome the problem of shock waves and to apply boundary conditions in cases of dam-break flows in channels with constant and variable widths. The numerical program solves shallow water equations based on the discrete mixed subdomain least squares (DMSLS) meshless method with collocation points. The DMSLS meshless method is based on the minimization of a least squares functional defined as the weighted summation of the squared residuals of the governing equations over the entire domain and requiring the summation of residual function to be zero at collocation points in boundary subdomains. The collocated discrete subdomain meshless method is applied on the boundary, whereas the collocated discrete least squares meshless technique is applied to the interior domain. The meshless scheme extends for dam-break formulation of shallow water equations. The model is verified by comparing computed results with analytical and experimental data for constant and varying width channels. The developed model is also used to study one-dimensional dam-break problems involving different flow situations by considering changes to the channel width, a bumpy channel with various downstream boundary conditions, and the effects of bed friction and bed slope as source terms on wave propagation. The accuracy of the results is acceptable.
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Shojaei, Arman, Bijan Boroomand, and Farshid Mossaiby. "A simple meshless method for challenging engineering problems." Engineering Computations 32, no. 6 (August 3, 2015): 1567–600. http://dx.doi.org/10.1108/ec-06-2014-0131.
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Purpose – The purpose of this paper is to present a simple meshless solution method for challenging engineering problems such as those with high wave numbers or convection-diffusion ones with high Peclet number. The method uses a set of residual-free bases in a local form. Design/methodology/approach – The residual-free bases, called here as exponential basis functions, are found so that they satisfy the governing equations within each subdomain. The compatibility between the subdomains is weakly satisfied by enforcing the local approximation of the main state variables pass through the data at nodes surrounding the central node of the subdomain. The central state variable is first recovered from the approximation and then re-assigned to the central node to construct the associated equation. This leads to the least compatibility required in the solution, e.g. C0 continuity in Laplace problems. Findings – The authors shall show that one can solve a variety of problems with regular and irregular point distribution with high convergence rate. The authors demonstrate that this is impossible to achieve using finite element method. Problems with Laplace and Helmholtz operators as well as elasto-static problems are solved to demonstrate the effectiveness of the method. A convection-diffusion problem with high Peclet number and problems with high wave numbers are among the examples. In all cases, results with high rate of convergence are obtained with moderate number of nodes per cloud. Originality/value – The paper presents a simple meshless method which not only is capable of solving a variety of challenging engineering problems but also yields results with high convergence rate.
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Gilmore, Natalie, Daniel Mirman, and Swathi Kiran. "Young Adults With Acquired Brain Injury Show Longitudinal Improvements in Cognition After Intensive Cognitive Rehabilitation." Journal of Speech, Language, and Hearing Research 65, no. 4 (April 4, 2022): 1494–520. http://dx.doi.org/10.1044/2021_jslhr-21-00324.
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Purpose: The aim of this study was to assess the effect of an intensive cognitive and communication rehabilitation (ICCR) program on language and other cognitive performance in young adults with acquired brain injury (ABI). Method: Thirty young adults with chronic ABI participated in this study. Treatment participants ( n = 22) attended ICCR 6 hours/day, 4 days/week for at least one 12-week semester. Deferred treatment/usual care control participants ( n = 14) were evaluated before and after at least one 12-week semester. Pre- and postsemester standardized cognitive assessment items were assigned to subdomains. Between-groups and within-group generalized linear mixed-effects models assessed the effect of time point on overall item accuracy and differences by item subdomain. Subdomain analyses were adjusted for multiple comparisons. Results: Between-groups analyses revealed that treatment participants improved significantly faster over time than deferred treatment/usual care participants in overall item accuracy and specifically on items in the verbal expression subdomain. Investigating the three-way interaction between time point, group, and etiology revealed that the overall effects of the treatment were similar for individuals with nontraumatic and traumatic brain injuries. The treatment group showed an overall effect of treatment and significant gains over time in the verbal expression, written expression, memory, and problem solving subdomains. The control group did not significantly improve over time on overall item accuracy and showed significant subdomain-level gains in auditory comprehension, which did not survive correction. Conclusions: Sustaining an ABI in young adulthood can significantly disrupt key developmental milestones, such as attending college and launching a career. This study provides strong evidence that integrating impairment-based retraining of language and other cognitive skills with “real-world” application in academically focused activities promotes gains in underlying cognitive processes that are important for academic success as measured by standardized assessment items. These findings may prompt a revision to the current continuum of rehabilitative care for young adults with ABI. Supplemental Material: https://doi.org/10.23641/asha.19320068
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Chen, Qing, Baoqing Liu, and Qikui Du. "The Schwartz Alternating Algorithm for Solving 3D Exterior Helmholtz Problems." Mathematical Problems in Engineering 2018 (2018): 1–9. http://dx.doi.org/10.1155/2018/6029484.
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Based on the natural boundary reduction, an overlapping domain decomposition method is discussed for solving exterior Helmholtz problem over a three-dimensional (3D) domain. By introducing two different artificial boundaries, the original unbounded domain is divided into a bounded subdomain and a typical unbounded region, and a Schwartz alternating method is presented. The finite element method and natural boundary element method are alternately applied to solve the problems in the bounded subdomain and the typical unbounded subdomain. Moreover, the convergence of the Schwartz alternating algorithm is studied. Finally, some numerical experiments are presented to show the performance of this method.
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Soares, Delfim. "Coupled Numerical Methods to Analyze Interacting Acoustic-Dynamic Models by Multidomain Decomposition Techniques." Mathematical Problems in Engineering 2011 (2011): 1–28. http://dx.doi.org/10.1155/2011/245170.
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In this work, coupled numerical analysis of interacting acoustic and dynamic models is focused. In this context, several numerical methods, such as the finite difference method, the finite element method, the boundary element method, meshless methods, and so forth, are considered to model each subdomain of the coupled model, and multidomain decomposition techniques are applied to deal with the coupling relations. Two basic coupling algorithms are discussed here, namely the explicit direct coupling approach and the implicit iterative coupling approach, which are formulated based on explicit/implicit time-marching techniques. Completely independent spatial and temporal discretizations among the interacting subdomains are permitted, allowing optimal discretization for each sub-domain of the model to be considered. At the end of the paper, numerical results are presented, illustrating the performance and potentialities of the discussed methodologies.
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Belgacem, Faker Ben, Lawrence K. Chilton, and Padmanabhan Seshaiyer. "Non-conforming Computational Methods for Mixed Elasticity Problems." Computational Methods in Applied Mathematics 3, no. 1 (2003): 23–34. http://dx.doi.org/10.2478/cmam-2003-0003.
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AbstractIn this paper, we present a non-conforming hp computational modeling methodology for solving elasticity problems. We consider the incompressible elasticity model formulated as a mixed displacement-pressure problem on a global domain which is partitioned into several local subdomains. The approximation within each local subdomain is designed using div-stable hp-mixed finite elements. The displacement is computed in a mortared space while the pressure is not subjected to any constraints across the interfaces. Our computational results demonstrate that the non-conforming finite element method presented for the elasticity problem satisfies similar rates of convergence as the conforming finite element method, in the presence of various h-version and p-version discretizations.
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Dolean, Victorita, Martin J. Gander, and Erwin Veneros. "Asymptotic analysis of optimized Schwarz methods for maxwell’s equations with discontinuous coefficients." ESAIM: Mathematical Modelling and Numerical Analysis 52, no. 6 (November 2018): 2457–77. http://dx.doi.org/10.1051/m2an/2018041.
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Discretized time harmonic Maxwell’s equations are hard to solve by iterative methods, and the best currently available methods are based on domain decomposition and optimized transmission conditions. Optimized Schwarz methods were the first ones to use such transmission conditions, and this approach turned out to be so fundamentally important that it has been rediscovered over the last years under the name sweeping preconditioners, source transfer, single layer potential method and the method of polarized traces. We show here how one can optimize transmission conditions to take benefit from the jumps in the coefficients of the problem, when they are aligned with the subdomain interface, and obtain methods which converge for two subdomains in certain situations independently of the mesh size, which would not be possible without jumps in the coefficients.
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Martínez, Darío, Henar Herrero, and Francisco Pla. "2D Newton Schwarz Legendre Collocation Method for a Convection Problem." Mathematics 10, no. 19 (October 10, 2022): 3718. http://dx.doi.org/10.3390/math10193718.
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In this work, an alternate Schwarz domain decomposition method is proposed to solve a Rayleigh–Bénard problem. The problem is modeled with the incompressible Navier–Stokes equations coupled with a heat equation in a rectangular domain. The Boussinesq approximation is considered. The nonlinearity is solved with Newton’s method. Each iteration of Newton’s method is discretized with an alternating Schwarz scheme, and each Schwarz problem is solved with a Legendre collocation method. The original domain is divided into several subdomains in both directions of the plane. Legendre collocation meshes are coarse, so the problem in each subdomain is well conditioned, and the size of the total mesh can grow by increasing the number of subdomains. In this way, the ill conditioning of Legendre collocation is overcome. The present work achieves an efficient alternating Schwarz algorithm such that the number of subdomains can be increased indefinitely in both directions of the plane. The method has been validated with a benchmark with numerical solutions obtained with other methods and with real experiments. Thanks to this domain decomposition method, the aspect ratio and Rayleigh number can be increased considerably by adding subdomains. Rayleigh values near to the turbulent regime can be reached. Namely, the great advantage of this method is that we obtain solutions close to turbulence, or in domains with large aspect ratios, by solving systems of linear equations with well-conditioned matrices of maximum size one thousand. This is an advantage over other methods that require solving systems with huge matrices of the order of several million, usually with conditioning problems. The computational cost is comparable to other methods, and the code is parallelizable.
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Yang, Hyunik, and D. A. Hoeltzel. "Automatic Finite Element Mesh Generation Over Domains Comprised of Multiply Connected, Intersecting, Rigid Body-Movable Subdomains." Journal of Mechanical Design 114, no. 4 (December 1, 1992): 603–15. http://dx.doi.org/10.1115/1.2917050.
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An approach for the automatic generation and refinement of finite element meshes over nonconvex domains subdivided by multibody connected, rigid body-movable subdomains has been developed. The basis of this method relies in order on (1) the automatic insertion of nodes on the digitized boundaries and within the interiors of movable subdomains, (2) the generation of superelement meshes within the subdomains, (3) determination of intersection points between adjacent subdomains following their rigid body movement, (4) ensuring the satisfaction of interelement connectivity across subdomain boundaries, and (5) the interactive refinement of user-selectable subdomains using quadrilaterization for global refinement and triangularization for local refinement. The creation of a finite element mesh for an acetabular cup inserted in a human pelvis, which is representative of a mesh generated over complex, two-dimensional, multiply connected subdomains, as employed in an orthopedic total hip replacement, serves as a realistic application of this approach and demonstrates its utility for expeditiously performing finite element-based, parametric design studies.
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Shanazari, Kamal. "An Adaptive Domain Partitioning Technique for Meshfree-Type Methods." Journal of Applied Mathematics 2012 (2012): 1–13. http://dx.doi.org/10.1155/2012/817026.
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An overlapping domain partitioning based on adapting nodes is presented for the meshless-type methods. The decomposition of the domain is carried out based on the distribution of the nodes produced rather than the geometry of the problem. A set of adaptive nodes is first generated using the dimension reduction and equidistributing along the coordinate directions with respect to arc-length monitor. The domain is then partitioned in such a way that the same number of nodes are allocated to the subdomains. A radial basis function collocation method is applied to each subdomain followed by assembling the global solution from the subproblem's solutions. A generalized thin plate spline with sufficient smoothness is used as a basis function in the collocation method. Some numerical results will be presented to show the performance of the proposed method.