The wave-based method （WBM） has been applied for the prediction of mid-frequency vibrations of fiat plates. The scaling factors, Gauss point selection rule and truncation rule are introduced to insure the wave model...The wave-based method （WBM） has been applied for the prediction of mid-frequency vibrations of fiat plates. The scaling factors, Gauss point selection rule and truncation rule are introduced to insure the wave model to converge. Numerical results show that the prediction tech- nique based on WBM is with higher accuracy and smaller computational effort than the one on FEM, which implies that this new technique on WBM can be applied to higher-frequency range.展开更多
This contribution is dedicated to the celebration of Rémi Abgrall’s accomplishments in Applied Mathematics and Scientific Computing during the conference“Essentially Hyperbolic Problems:Unconventional Numerics,...This contribution is dedicated to the celebration of Rémi Abgrall’s accomplishments in Applied Mathematics and Scientific Computing during the conference“Essentially Hyperbolic Problems:Unconventional Numerics,and Applications”.With respect to classical Finite Elements Methods,Trefftz methods are unconventional methods because of the way the basis functions are generated.Trefftz discontinuous Galerkin(TDG)methods have recently shown potential for numerical approximation of transport equations[6,26]with vectorial exponential modes.This paper focuses on a proof of the approximation properties of these exponential solutions.We show that vectorial exponential functions can achieve high order convergence.The fundamental part of the proof consists in proving that a certain rectangular matrix has maximal rank.展开更多
A wave number method （WNM） is proposed to deal with the two-dimensional coupled structural-acoustic problem. Based on an indirect Trefftz approach, the displacement and the pressure response are approximated respect...A wave number method （WNM） is proposed to deal with the two-dimensional coupled structural-acoustic problem. Based on an indirect Trefftz approach, the displacement and the pressure response are approximated respectively by a set of wave functions, which exactly satisfy the governing equations and are independent of the size of the coupled system. The wave functions comprise the exact solutions of the homogeneous part of the governing equations and some particular solution functions, which arise from the external excitation. The weighting coefficients of the wave functions can be obtained by enforcing the pressure approximation to satisfy the boundary conditions and it is performed by applying the weighted residual formulation. The example is computed by the WNM and the BEM. The results show that, the WNM can attain the same accuracy and convergence as the BEM with less degrees of freedom.展开更多
Based on the indirect Trefftz approach, a wave number method （WNM） is proposed to deal with three-dimensional steady-state acoustic problems. In the WNM, the dynamic pressure response variable is approximated by a s...Based on the indirect Trefftz approach, a wave number method （WNM） is proposed to deal with three-dimensional steady-state acoustic problems. In the WNM, the dynamic pressure response variable is approximated by a set of wave functions, which exactly satisfy the Helmholtz equation. The set of wave functions comprise the exact solutions of the homogeneous part of the governing equations and some particular solution functions. The unknown coefficients of the wave functions can be obtained by enforcing the pressure approximation to satisfy the boundary conditions. Compared with the boundary element method （BEM）, the WNM have a smaller system matrix, and is applicable to the radiation problems since the wave functions are independent of the domain size. A 3D acoustic cavity is exemplified to show the properties of the method. The results show that the wave number method is more efficient than the BEM, and it is fairly accurate.展开更多
This paper presents three boundary meshless methods for solving problems of steady-state and transient heat conduction in nonlinear functionally graded materials(FGMs).The three methods are,respectively,the method of ...This paper presents three boundary meshless methods for solving problems of steady-state and transient heat conduction in nonlinear functionally graded materials(FGMs).The three methods are,respectively,the method of fundamental solution(MFS),the boundary knot method(BKM),and the collocation Trefftz method(CTM)in conjunction with Kirchhoff transformation and various variable transformations.In the analysis,Laplace transform technique is employed to handle the time variable in transient heat conduction problem and the Stehfest numerical Laplace inversion is applied to retrieve the corresponding time-dependent solutions.The proposed MFS,BKM and CTM are mathematically simple,easyto-programming,meshless,highly accurate and integration-free.Three numerical examples of steady state and transient heat conduction in nonlinear FGMs are considered,and the results are compared with those from meshless local boundary integral equation method(LBIEM)and analytical solutions to demonstrate the effi-ciency of the present schemes.展开更多
The Trefftz-type boundary solution methods([1]) are applied in analysing moderately thick plate bending problems. A new type of locking problem caused by the overflow of Trefftz functions has been found and a so-calle...The Trefftz-type boundary solution methods([1]) are applied in analysing moderately thick plate bending problems. A new type of locking problem caused by the overflow of Trefftz functions has been found and a so-called variable-reducing procedure for eliminating such a phenomenon is also proposed.展开更多
描叙了圆柱壳与声耦合系统建模技术的基本概念。根据WB法(Wave based method,简称WB法)的基本原理,将结构场变量和声压展开成波函数和特解函数的叠加。应用加权余量法将结构和声学边界条件以及结构与声耦合界面上的连续性条件转换成近...描叙了圆柱壳与声耦合系统建模技术的基本概念。根据WB法(Wave based method,简称WB法)的基本原理,将结构场变量和声压展开成波函数和特解函数的叠加。应用加权余量法将结构和声学边界条件以及结构与声耦合界面上的连续性条件转换成近似积分,推导出一组由代数方程表示的耦合模型。通过分析二维的算例表明这种预报技术计算量小且精度高,因此可应用到中频的分析。展开更多
In the solution domain, the inhomogeneous part of Poisson equation is approximated with the 5-order polynomial using Galerkin method, and the particular solution of the polynomial can be determined easily. Then, the s...In the solution domain, the inhomogeneous part of Poisson equation is approximated with the 5-order polynomial using Galerkin method, and the particular solution of the polynomial can be determined easily. Then, the solution of the Poisson equation is approximated by superposition of the particular solution and the Tcomplete functions related to the Laplace equation. Unknown parameters are determined by Galerkin method, so that the approximate solution is to satisfy the boundary conditions. Comparison with analogous results of others numerical method, the two calculating examples of the paper indicate that the accuracy of the method is very high, which also has a very fast convergence rate.展开更多
In this paper we propose a novel two-stage method to solve the threedimensional Poisson equation in an arbitrary bounded domain enclosed by a smooth boundary.The solution is decomposed into a particular solution and a...In this paper we propose a novel two-stage method to solve the threedimensional Poisson equation in an arbitrary bounded domain enclosed by a smooth boundary.The solution is decomposed into a particular solution and a homogeneous solution.In the first stage a multiple-scale polynomial method(MSPM)is used to approximate the forcing term and then the formula of Tsai et al.[Tsai,Cheng,and Chen(2009)]is used to obtain the corresponding closed-form solution for each polynomial term.Then in the second stage we use a multiple/scale/direction Trefftz method(MSDTM)to find the solution of Laplace equation,of which the directions are uniformly distributed on a unit circle 1,and the scales are determined a priori by the collocation points on boundary.Two examples of 3D data interpolation,and several numerical examples of direct and inverse Cauchy problems in complex domain confirm the efficiency of the MSPM and the MSDTM.展开更多
In this paper,the equal-norm multiple-scale Trefftz method combined with the implicit Lie-group scheme is applied to solve the two-dimensional nonlinear sloshing problem with baffles.When considering solving sloshing ...In this paper,the equal-norm multiple-scale Trefftz method combined with the implicit Lie-group scheme is applied to solve the two-dimensional nonlinear sloshing problem with baffles.When considering solving sloshing problems with baffles by using boundary integral methods,degenerate geometry and problems of numerical instability are inevitable.To avoid numerical instability,the multiple-scale characteristic lengths are introduced into T-complete basis functions to efficiently govern the high-order oscillation disturbance.Again,the numerical noise propagation at each time step is eliminated by the vector regularization method and the group-preserving scheme.A weighting factor of the group-preserving scheme is introduced into a linear system and then used in the initial and boundary value problems(IBVPs)at each time step.More importantly,the parameters of the algorithm,namely,the T-complete function,dissipation factor,and time step,can obtain a linear relationship.The boundary noise interference and energy conservation are successfully overcome,and the accuracy of the boundary value problem is also improved.Finally,benchmark cases are used to verify the correctness of the numerical algorithm.The numerical results show that this algorithm is efficient and stable for nonlinear two-dimensional sloshing problems with baffles.展开更多
基金Project supported by the National Natural Science Foundation of China (No.10472035).
文摘The wave-based method （WBM） has been applied for the prediction of mid-frequency vibrations of fiat plates. The scaling factors, Gauss point selection rule and truncation rule are introduced to insure the wave model to converge. Numerical results show that the prediction tech- nique based on WBM is with higher accuracy and smaller computational effort than the one on FEM, which implies that this new technique on WBM can be applied to higher-frequency range.
文摘This contribution is dedicated to the celebration of Rémi Abgrall’s accomplishments in Applied Mathematics and Scientific Computing during the conference“Essentially Hyperbolic Problems:Unconventional Numerics,and Applications”.With respect to classical Finite Elements Methods,Trefftz methods are unconventional methods because of the way the basis functions are generated.Trefftz discontinuous Galerkin(TDG)methods have recently shown potential for numerical approximation of transport equations[6,26]with vectorial exponential modes.This paper focuses on a proof of the approximation properties of these exponential solutions.We show that vectorial exponential functions can achieve high order convergence.The fundamental part of the proof consists in proving that a certain rectangular matrix has maximal rank.
基金Project supported by the National Natural Science Foundation of China (No.10472035).
文摘A wave number method （WNM） is proposed to deal with the two-dimensional coupled structural-acoustic problem. Based on an indirect Trefftz approach, the displacement and the pressure response are approximated respectively by a set of wave functions, which exactly satisfy the governing equations and are independent of the size of the coupled system. The wave functions comprise the exact solutions of the homogeneous part of the governing equations and some particular solution functions, which arise from the external excitation. The weighting coefficients of the wave functions can be obtained by enforcing the pressure approximation to satisfy the boundary conditions and it is performed by applying the weighted residual formulation. The example is computed by the WNM and the BEM. The results show that, the WNM can attain the same accuracy and convergence as the BEM with less degrees of freedom.
文摘Based on the indirect Trefftz approach, a wave number method （WNM） is proposed to deal with three-dimensional steady-state acoustic problems. In the WNM, the dynamic pressure response variable is approximated by a set of wave functions, which exactly satisfy the Helmholtz equation. The set of wave functions comprise the exact solutions of the homogeneous part of the governing equations and some particular solution functions. The unknown coefficients of the wave functions can be obtained by enforcing the pressure approximation to satisfy the boundary conditions. Compared with the boundary element method （BEM）, the WNM have a smaller system matrix, and is applicable to the radiation problems since the wave functions are independent of the domain size. A 3D acoustic cavity is exemplified to show the properties of the method. The results show that the wave number method is more efficient than the BEM, and it is fairly accurate.
文摘This paper presents three boundary meshless methods for solving problems of steady-state and transient heat conduction in nonlinear functionally graded materials(FGMs).The three methods are,respectively,the method of fundamental solution(MFS),the boundary knot method(BKM),and the collocation Trefftz method(CTM)in conjunction with Kirchhoff transformation and various variable transformations.In the analysis,Laplace transform technique is employed to handle the time variable in transient heat conduction problem and the Stehfest numerical Laplace inversion is applied to retrieve the corresponding time-dependent solutions.The proposed MFS,BKM and CTM are mathematically simple,easyto-programming,meshless,highly accurate and integration-free.Three numerical examples of steady state and transient heat conduction in nonlinear FGMs are considered,and the results are compared with those from meshless local boundary integral equation method(LBIEM)and analytical solutions to demonstrate the effi-ciency of the present schemes.
基金National Natural Science Foundation of China(No.19872019)Solid Mechanics Open Research laboratory of Tongji University
文摘The Trefftz-type boundary solution methods([1]) are applied in analysing moderately thick plate bending problems. A new type of locking problem caused by the overflow of Trefftz functions has been found and a so-called variable-reducing procedure for eliminating such a phenomenon is also proposed.
文摘描叙了圆柱壳与声耦合系统建模技术的基本概念。根据WB法(Wave based method,简称WB法)的基本原理,将结构场变量和声压展开成波函数和特解函数的叠加。应用加权余量法将结构和声学边界条件以及结构与声耦合界面上的连续性条件转换成近似积分,推导出一组由代数方程表示的耦合模型。通过分析二维的算例表明这种预报技术计算量小且精度高,因此可应用到中频的分析。
文摘In the solution domain, the inhomogeneous part of Poisson equation is approximated with the 5-order polynomial using Galerkin method, and the particular solution of the polynomial can be determined easily. Then, the solution of the Poisson equation is approximated by superposition of the particular solution and the Tcomplete functions related to the Laplace equation. Unknown parameters are determined by Galerkin method, so that the approximate solution is to satisfy the boundary conditions. Comparison with analogous results of others numerical method, the two calculating examples of the paper indicate that the accuracy of the method is very high, which also has a very fast convergence rate.
基金The work described in this paper was supported by the Thousand Talents Plan of China(Grant No.A1211010)the Fundamental Research Funds for the Central Universities(Grant nos.2017B656X14,2017B05714)+1 种基金the Postgraduate Research&Practice Innovation Program of Jiangsu Province(Grant No.KYCX17_0487)the Natural Science Foundation of Shandong Province of China(Grant No.ZR2017BA003).
文摘In this paper we propose a novel two-stage method to solve the threedimensional Poisson equation in an arbitrary bounded domain enclosed by a smooth boundary.The solution is decomposed into a particular solution and a homogeneous solution.In the first stage a multiple-scale polynomial method(MSPM)is used to approximate the forcing term and then the formula of Tsai et al.[Tsai,Cheng,and Chen(2009)]is used to obtain the corresponding closed-form solution for each polynomial term.Then in the second stage we use a multiple/scale/direction Trefftz method(MSDTM)to find the solution of Laplace equation,of which the directions are uniformly distributed on a unit circle 1,and the scales are determined a priori by the collocation points on boundary.Two examples of 3D data interpolation,and several numerical examples of direct and inverse Cauchy problems in complex domain confirm the efficiency of the MSPM and the MSDTM.
基金The second author greatly appreciates the financial support provided by the Ministry of Science and Technology,Taiwan,ROC,under Contract No.MOST 108-2221-E-019-015.
文摘In this paper,the equal-norm multiple-scale Trefftz method combined with the implicit Lie-group scheme is applied to solve the two-dimensional nonlinear sloshing problem with baffles.When considering solving sloshing problems with baffles by using boundary integral methods,degenerate geometry and problems of numerical instability are inevitable.To avoid numerical instability,the multiple-scale characteristic lengths are introduced into T-complete basis functions to efficiently govern the high-order oscillation disturbance.Again,the numerical noise propagation at each time step is eliminated by the vector regularization method and the group-preserving scheme.A weighting factor of the group-preserving scheme is introduced into a linear system and then used in the initial and boundary value problems(IBVPs)at each time step.More importantly,the parameters of the algorithm,namely,the T-complete function,dissipation factor,and time step,can obtain a linear relationship.The boundary noise interference and energy conservation are successfully overcome,and the accuracy of the boundary value problem is also improved.Finally,benchmark cases are used to verify the correctness of the numerical algorithm.The numerical results show that this algorithm is efficient and stable for nonlinear two-dimensional sloshing problems with baffles.
