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A priori verification method for curl-conforming basis functions in simplices

MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2025年第1期48卷 144-154页

作者： Amor-Martin, Adrian Garcia-Castillo, Luis E. Univ Carlos III Madrid Signal Theory & Commun Dept Leganes 28911 Madrid Spain

The construction of (hierarchical) curl-conforming basis functions has been a hot topic in the last decades in the finite element community. Especially, functions applied to simplices have been quite popular after the work by N & eacute;d & eacute;lec in 1980. Many mixed-order and full-order families have been provided in the last years, but sometimes, it is difficult to assess if they belong to the original space proposed by N & eacute;d & eacute;lec (especially when orthogonalization procedures are applied). Here, a tool to determine if a family of basis functions belongs to the N & eacute;d & eacute;lec space is provided. Since affine coordinates are the most frequent choice for simplices, particularities about its use with this kind of coordinates are detailed. A detailed survey of existing families is provided, and the practical application of the tool to a representative set of these families is discussed. The tool is also available for the community in a public repository.

关键词： affine coordinates basis functions electromagnetics finite element method verification

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A Computationally Efficient Technique Using Characteristic basis functions and Large Block Sizes

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IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS 2024年第4期23卷 1341-1345页

作者： Garcia, Eliseo Delgado, Carlos Catedra, Felipe Univ Alcala Dept Automat Alcala De Henares 28801 Spain Univ Alcala Dept Ciencias Comp Alcala De Henares 28801 Spain

A technique for the reduction of the CPU-time in the solution of complex electromagnetic problems using the characteristic basis function method is presented. This approach allows the analysis of electrically large cases iteratively using the multilevel fast multipole method to account for the matrix-vector products. It defines a new procedure for the fast computation of the characteristic basis functions and the reduced matrix. Some representative test cases show that the approach provides accurate results for complex bodies with a noticeable efficiency improvement.

关键词： Couplings Sparse matrices Impedance Computational efficiency Method of moments Iterative methods Geometry basis functions electromagnetic analysis mom- ent methods sparse matrices

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Isogeometric Analysis Using basis functions of Splines Spaces over Hierarchical T-meshes with High Level Differences

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COMMUNICATIONS IN MATHEMATICS AND STATISTICS 2025年第2期13卷 403-430页

作者： Liu, Jingjing Deng, Fang Ma, Huanhuan Deng, Jiansong Hefei Normal Univ Sch Math & Stat Hefei 230601 Anhui Peoples R China Univ Sci & Technol China Sch Math Sci Hefei 230026 Anhui Peoples R China North China Univ Water Resources & Elect Power Sch Math & Stat Zhengzhou 450045 Henan Peoples R China

By using the method of space mapping, basis functions of biquadratic polynomial spline spaces over the hierarchical T-meshes without limitation for level difference can be constructed. In this paper, the basis functions defined over hierarchical T-meshes with high level differences are adopted for the application in the isogeometric analysis problems with rapidly changing local features. Without subdividing redundant cells to ensure the level difference of the adjacent cells, the refinement becomes more local, and fewer cells are subdivided for each refinement of the hierarchical T-mesh. Therefore, the dimension of the biquadratic polynomial spline space over the hierarchical T-mesh can be reduced, the superfluous control points or coefficients can be avoided, and the quantity of calculations can be decreased. Numerical examples show that these basis functions can work well on physical domains with different boundaries for the application in IGA.

关键词： Spline spaces over hierarchical T-meshes Space mapping basis functions Isogeometric analysis

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Solution of planar elastic stress problems using stress basis functions

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MATHEMATICS AND MECHANICS OF SOLIDS 2024年第8期29卷 1528-1563页

作者： Tiwari, Sankalp Chatterjee, Anindya Indian Inst Technol Hyderabad Dept Mech & Aerosp Engn Sangareddy India Indian Inst Technol Kanpur Dept Mech Engn Kanpur India Indian Inst Technol Hyderabad Dept Mech & Aerosp Engn Sangareddy 502284 Telangana India

The use of global displacement basis functions to solve boundary-value problems in linear elasticity is well established. No prior work uses a global stress tensor basis for such solutions. We present two such methods for solving stress problems in linear elasticity. In both methods, we split the sought stress sigma into two parts, where neither part is required to satisfy strain compatibility. The first part, sigma(p) , is any stress in equilibrium with the loading. The second part, sigma(h) is a self-equilibrated stress field on the unloaded body. In both methods, sigma(h) is expanded using tensor-valued global stress basis functions developed elsewhere. In the first method, the coefficients in the expansion are found by minimizing the strain energy based on the well-known complementary energy principle. For the second method, which is restricted to planar homogeneous isotropic bodies, we show that we merely need to minimize the squared L-2 norm of the trace of stress. For demonstration, we solve nine stress problems involving sharp corners, multiple-connectedness, non-zero net force and/or moment on an internal hole, body force, discontinuous surface traction, material inhomogeneity, and anisotropy. The first method presents a new application of a known principle. The second method presents a hitherto unreported principle, to the best of our knowledge.

关键词： Stress boundary value problems linear elasticity complementary energy residual stresses variational principles basis functions

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Fast pivoting gait generation by model predictive control designed with basis functions

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ADVANCED ROBOTICS 2022年第15期36卷 735-749页

作者： Zhang, Ang Wan, Weiwei Harada, Kensuke Osaka Univ Grad Sch Engn Sci Osaka Japan

Pivoting gait is an efficient way for robots to manipulate a heavy object. Although we can cope with the contact constraints of the pivoting gait by using the model predictive control (MPC), systems with complex dynamics, including the pivoting gait, usually require long horizons in the MPC and it leads to a heavy computational load. To overcome this problem, we introduce basis functions to parameterize the free variables in the MPC and formulate an optimization problem with new decision variables, which are coefficients of the basis functions. We especially introduce multiple basis functions and compare their performances in generating the robotic pivoting gait. As a result, the most effective reduction in the dimension of the free variables is achieved by using the Laguerre basis function and the computational efficiency of the MPC is greatly improved. The simulation and experiments show that the time cost of the generation of pivoting gaits by the proposed method is remarkably reduced and the generated pivoting gaits are feasible and robust where a dual-arm robot successfully manipulates a toy piano by the pivoting gait.

关键词： Nonprehensile manipulation model predictive control pivoting manipulation basis functions

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On the Choice of Regression basis functions and Machine Learning

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VESTNIK ST PETERSBURG UNIVERSITY-MATHEMATICS 2022年第1期55卷 7-15页

作者： Ermakov, S. M. Leora, S. N. St Petersburg State Univ St Petersburg 199034 Russia St Petersburg State Univ Econ St Petersburg 191023 Russia

As is known, regression-analysis tools are widely used in machine-learning problems to establish the relationship between the observed variables and to store information in a compact manner. Most often, a regression function is described by a linear combination of some given functions f(j)(X), j = 1, horizontal ellipsis , m, X is an element of D subset of R-s. If the observed data contain a random error, then the regression function reconstructed from the observations contains a random error and a systematic error depending on the selected functions f(j). This article indicates the possibility of an optimal, in the sense of a given functional metric, choice of f(j), if it is known that the true dependence obeys some functional equation. In some cases (a regular grid, s <= 2), close results can be obtained using a technique for random-process analysis. The numerical examples given in this work illustrate significantly broader opportunities for the assumed approach to regression problems.

关键词： regression analysis approximation basis functions operator method machine learning

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Solution of Volume Integral Equation Using the SWG-Edge Hybrid basis functions for Inhomogeneous Dielectric Objects With Multiboundary

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION 2021年第9期69卷 5812-5821页

作者： Zhang, Li-Ming Deng, A-Li Shandong Univ Sci & Technol Dept Elect Engn & Informat Technol Jinan 250031 Peoples R China Shandong Univ Sci & Technol Dept Basic Courses Jinan 250031 Peoples R China

A hybrid discretization scheme for solution of volume integral equation (VIE) by method of moments (MoM) for electromagnetic scattering from dielectric objects is proposed in this article. The Schaubert-Wilton-Glisson and edge (SWG-Edge) hybrid basis functions are used in this discretization scheme. According to the divergence-free condition of electric displacement vector, a kind of edge basis functions defined in elements including boundary faces which separate a dielectric object from the background is derived. As a result, we get a SWG-Edge hybrid basis set. Details for the calculation of the corresponding matrix elements for the edge basis and testing functions are presented. Numerical results show the validity and accuracy of the hybrid discretization scheme. Finally, the proposed method is used for efficient solution of VIE for inhomogeneous dielectric objects with multiboundary. It is shown that for multiboundary problems, the number of unknowns of the hybrid basis is only about 71% of the traditional SWG basis. This means that the memory for solution of VIE by the traditional SWG basis functions can be reduced by half. Therefore, the SWG-edge hybrid basis is much more efficient than the traditional SWG basis for solution of VIE with multiboundary problems.

关键词： Faces Dielectrics Method of moments Electromagnetic scattering Permittivity Nonhomogeneous media Integral equations basis functions electromagnetic scattering volume integral equation (VIE)

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Using the digamma function for basis functions in mesh-free computational methods

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ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS 2021年 131卷 218-227页

作者： Wilkins, Bryce D. Hromadka, Theodore V., II Carnegie Mellon Univ Sch Comp Sci 5000 Forbes Ave Pittsburgh PA 15213 USA US Mil Acad Dept Math Sci 646 Swift Rd West Point NY 10996 USA

We examine the utility of a new family of basis functions for use with the Complex Variable Boundary Element Method (CVBEM) and other mesh-free numerical methods for solving partial differential equations. The family of polygamma functions have found use in mathematics since as early as 1730 when James Stirling related the digamma function to the factorial function [1]. Now, we propose using the digamma function, as well as new variants of the digamma function, as basis functions for the CVBEM. This paper discusses technical aspects associated with using the digamma function as a CVBEM basis function. Then, we demonstrate the utility of the proposed basis function by applying it to a mixed boundary value problem of the Laplace type.

关键词： Complex variable boundary element method (CVBEM) basis functions Mesh-free methods Applied complex variables Potential flow

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Estimating basis functions in massive fields under the spatial mixed effects model

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STATISTICAL ANALYSIS AND DATA MINING 2021年第5期14卷 430-448页

作者： Pazdernik, Karl Maitra, Ranjan Pacific Northwest Natl Lab Natl Secur Directorate Richland WA USA North Carolina State Univ Dept Stat Raleigh NC USA Iowa State Univ Dept Stat Ames IA 50011 USA

Spatial prediction is commonly achieved under the assumption of a Gaussian random field by obtaining maximum likelihood estimates of parameters, and then using the kriging equations to arrive at predicted values. For massive datasets, fixed rank kriging using the expectation-maximization algorithm for estimation has been proposed as an alternative to the usual but computationally prohibitive kriging method. The method reduces computation cost of estimation by redefining the spatial process as a linear combination of basis functions and spatial random effects. A disadvantage of this method is that it imposes constraints on the relationship between the observed locations and the knots. We develop an alternative method that utilizes the spatial mixed effects model, but allows for additional flexibility by estimating the range of the spatial dependence between the observations and the knots via an alternating expectation conditional maximization algorithm. Experiments show that our methodology improves estimation without sacrificing prediction accuracy while also minimizing the additional computational burden of extra parameter estimation. The methodology is applied to a temperature dataset archived by the United States National Climate Data Center, with improved results over previous methodology.

关键词： alternating expectation conditional maximization algorithm bandwidth basis functions fixed rank kriging maximum likelihood estimation range parameter

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basis functions for residual stresses

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APPLIED MATHEMATICS AND COMPUTATION 2020年 386卷 125468-125468页

作者： Tiwari, Sankalp Chatterjee, Anindya Indian Inst Technol Dept Mech Engn Kanpur Uttar Pradesh India

We consider arbitrary preexisting residual stress states in arbitrarily shaped, unloaded bodies. These stresses must be self-equilibrating and traction free. Common treatments of the topic tend to focus on either the mechanical origins of the stress, or methods of stress measurement at certain locations. Here we take the stress field as given and consider the problem of approximating any such stress field, in a given body, as a linear combination of predetermined fields which can serve as a basis. We consider planar stress states in detail, and introduce an extremization problem that leads to a linear eigenvalue problem. Eigen-functions of that problem form an orthonormal basis for all possible residual stress states of sufficient smoothness. In numerical examples, convergence of the approximating stress fields is demonstrated in the L 2 norm for continuous stress fields as well as for a stress field with a simple discontinuity. Finally, we outline the extension of our theory to three dimensional bodies and states of stress. Our approach can be used to describe arbitrary preexisting residual stress states in arbitrarily shaped bodies using basis functions that are determined by the body geometry alone. (C) 2020 Elsevier Inc. All rights reserved.

关键词： Residual stresses basis functions Interpolation functions

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