In this paper,we generalize the direct method of lines for linear elasticity problems of composite materials in star-shaped domains and consider its application to inverse elasticity problems.We assume that the bounda...In this paper,we generalize the direct method of lines for linear elasticity problems of composite materials in star-shaped domains and consider its application to inverse elasticity problems.We assume that the boundary of the star-shaped domain can be described by an explicit C 1 parametric curve in the polar coordinate.We introduce the curvilinear coordinate,in which the irregular star-shaped domain is converted to a regular semi-infinite strip.The equations of linear elasticity are discretized with respect to the angular variable and we solve the resulting semidiscrete approximation analytically using a direct method.The eigenvalues of the semi-discrete approximation converge quickly to the true eigenvalues of the elliptic operator,which helps capture the singularities naturally.Moreover,an optimal error estimate of our method is given.For the inverse elasticity problems,we determine the Lam´e coefficients from measurement data by minimizing a regularized energy functional.We apply the direct method of lines as the forward solver in order to cope with the irregularity of the domain and possible singularities in the forward solutions.Several numerical examples are presented to show the effectiveness and accuracy of our method for both forward and inverse elasticity problems of composite materials.展开更多
In this paper,we analyse the stochastic collocation method for a linear Schr¨odinger equation with random inputs,where the randomness appears in the potential and initial data and is assumed to be dependent on a ...In this paper,we analyse the stochastic collocation method for a linear Schr¨odinger equation with random inputs,where the randomness appears in the potential and initial data and is assumed to be dependent on a random variable.We focus on the convergence rate with respect to the number of collocation points.Based on the interpolation theories,the convergence rate depends on the regularity of the solution with respect to the random variable.Hence,we investigate the dependence of the stochastic regularity of the solution on that of the random potential and initial data.We provide sufficient conditions on the random potential and initial data to ensure the smoothness of the solution and the spectral convergence.Finally,numerical results are presented to support our analysis.展开更多
基金This work was partially supported by the NSFC Projects No.12025104,11871298,81930119.
文摘In this paper,we generalize the direct method of lines for linear elasticity problems of composite materials in star-shaped domains and consider its application to inverse elasticity problems.We assume that the boundary of the star-shaped domain can be described by an explicit C 1 parametric curve in the polar coordinate.We introduce the curvilinear coordinate,in which the irregular star-shaped domain is converted to a regular semi-infinite strip.The equations of linear elasticity are discretized with respect to the angular variable and we solve the resulting semidiscrete approximation analytically using a direct method.The eigenvalues of the semi-discrete approximation converge quickly to the true eigenvalues of the elliptic operator,which helps capture the singularities naturally.Moreover,an optimal error estimate of our method is given.For the inverse elasticity problems,we determine the Lam´e coefficients from measurement data by minimizing a regularized energy functional.We apply the direct method of lines as the forward solver in order to cope with the irregularity of the domain and possible singularities in the forward solutions.Several numerical examples are presented to show the effectiveness and accuracy of our method for both forward and inverse elasticity problems of composite materials.
基金supported by the National Key Research and Development Plan of China No.2017YFC0601801 and NSFC Project No.11871298.
文摘In this paper,we analyse the stochastic collocation method for a linear Schr¨odinger equation with random inputs,where the randomness appears in the potential and initial data and is assumed to be dependent on a random variable.We focus on the convergence rate with respect to the number of collocation points.Based on the interpolation theories,the convergence rate depends on the regularity of the solution with respect to the random variable.Hence,we investigate the dependence of the stochastic regularity of the solution on that of the random potential and initial data.We provide sufficient conditions on the random potential and initial data to ensure the smoothness of the solution and the spectral convergence.Finally,numerical results are presented to support our analysis.
