When parametric functions are used to blend 3D surfaces, geometric continuity of displacements and derivatives until to the surface boundary must be satisfied. By the traditional blending techniques, however, arbitrar...When parametric functions are used to blend 3D surfaces, geometric continuity of displacements and derivatives until to the surface boundary must be satisfied. By the traditional blending techniques, however, arbitrariness of the solutions arises to cause a difficulty in choosing a suitable blending surface. Hence to explore new blending techniques is necessary to construct good surfaces so as to satisfy engineering requirements. In this paper, a blending surface is described as a flexibly elastic plate both in partial differential equations and in their variational equations, thus to lead to a unique solution in a sense of the minimal global surface curvature. Boundary penalty finite element methods (BP-FEMs) with and without approximate integration are proposed to handle the complicated constraints along the blending boundary. Not only have the optimal convergence rate O(h(2)) of second order generalized derivatives of the solutions in the solution domain been obtained, but also the high convergence rate O(h(4)) of the tangent boundary condition of the solutions can be achieved, where h is the maximal boundary length of rectangular elements used. Moreover, useful guidance in computation is discovered to deal with interpolation and approximation in the boundary penalty integrals. A numerical example is also provided to verify perfectly the main theoretical analysis made. This paper yields a framework of mathematical modelling, numerical techniques and error analysis to the general and complicated blending problems.展开更多
In this paper,we consider high order multi-domain penalty spectral Galerkin methods for the approximation of hyperbolic conservation laws.This formulation has a penalty parameter which can vary in space and time,allow...In this paper,we consider high order multi-domain penalty spectral Galerkin methods for the approximation of hyperbolic conservation laws.This formulation has a penalty parameter which can vary in space and time,allowing for flexibility in the penalty formulation.This flexibility is particularly advantageous for problems with an inhomogeneous mesh.We show that the discontinuous Galerkin method is equivalent to the multi-domain spectral penalty Galerkin method with a particular value of the penalty parameter.The penalty parameter has an effect on both the accuracy and stability of the method.We examine the numerical issues which arise in the implementation of high order multi-domain penalty spectral Galerkin methods.The coefficient truncation method is proposed to prevent the rapid error growth due to round-off errors when high order polynomials are used.Finally,we show that an inconsistent evaluation of the integrals in the penalty method may lead to growth of errors.Numerical examples for linear and nonlinear problems are presented.展开更多
文摘When parametric functions are used to blend 3D surfaces, geometric continuity of displacements and derivatives until to the surface boundary must be satisfied. By the traditional blending techniques, however, arbitrariness of the solutions arises to cause a difficulty in choosing a suitable blending surface. Hence to explore new blending techniques is necessary to construct good surfaces so as to satisfy engineering requirements. In this paper, a blending surface is described as a flexibly elastic plate both in partial differential equations and in their variational equations, thus to lead to a unique solution in a sense of the minimal global surface curvature. Boundary penalty finite element methods (BP-FEMs) with and without approximate integration are proposed to handle the complicated constraints along the blending boundary. Not only have the optimal convergence rate O(h(2)) of second order generalized derivatives of the solutions in the solution domain been obtained, but also the high convergence rate O(h(4)) of the tangent boundary condition of the solutions can be achieved, where h is the maximal boundary length of rectangular elements used. Moreover, useful guidance in computation is discovered to deal with interpolation and approximation in the boundary penalty integrals. A numerical example is also provided to verify perfectly the main theoretical analysis made. This paper yields a framework of mathematical modelling, numerical techniques and error analysis to the general and complicated blending problems.
基金The work of both authors has been supported by the NSF under Grant No.DMS-0608844.
文摘In this paper,we consider high order multi-domain penalty spectral Galerkin methods for the approximation of hyperbolic conservation laws.This formulation has a penalty parameter which can vary in space and time,allowing for flexibility in the penalty formulation.This flexibility is particularly advantageous for problems with an inhomogeneous mesh.We show that the discontinuous Galerkin method is equivalent to the multi-domain spectral penalty Galerkin method with a particular value of the penalty parameter.The penalty parameter has an effect on both the accuracy and stability of the method.We examine the numerical issues which arise in the implementation of high order multi-domain penalty spectral Galerkin methods.The coefficient truncation method is proposed to prevent the rapid error growth due to round-off errors when high order polynomials are used.Finally,we show that an inconsistent evaluation of the integrals in the penalty method may lead to growth of errors.Numerical examples for linear and nonlinear problems are presented.
