Trefftz methods are finite element-type schemes whose test and trial functions are (locally) solutions of the targeted differential equation. They are particularly popular for time-harmonic wave problems, as their tri...
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ISBN:
(纸本)9783319416403;9783319416380
Trefftz methods are finite element-type schemes whose test and trial functions are (locally) solutions of the targeted differential equation. They are particularly popular for time-harmonic wave problems, as their trial spaces contain oscillating basis functions and may achieve better approximation properties than classical piecewise-polynomial spaces. We review the construction and properties of several Trefftz variational formulations developed for the Helmholtz equation, including least squares, discontinuous Galerkin, ultra weak variational formulation, variational theory of complex rays and wave based methods. The most common discrete Trefftz spaces used for this equation employ generalised harmonic polynomials (circular and spherical waves), plane and evanescent waves, fundamental solutions and multipoles as basis functions;we describe theoretical and computational aspects of these spaces, focusing in particular on their approximation properties. One of the most promising, but not yet well developed, features of Trefftz methods is the use of adaptivity in the choice of the propagation directions for the basis functions. The main difficulties encountered in the implementation are the assembly and the ill-conditioning of linear systems, we briefly survey some strategies that have been proposed to cope with these problems.
This paper presents results of computational studies of the evolution law for the constrained mean curvature flow. The considered motion law originates in the theory of phase transitions in crystalline materials. It d...
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ISBN:
(纸本)9783319399294;9783319399270
This paper presents results of computational studies of the evolution law for the constrained mean curvature flow. The considered motion law originates in the theory of phase transitions in crystalline materials. It describes the evolution of closed embedded curves with constant enclosed area. In the paper, the motion law is treated by the parametric method, which leads into the system of degenerate parabolic equations for the parametric description of the curve. This system is numerically solved by means of the flowing finite volume method enhanced by tangential redistribution. Qualitative and quantitative results of computational studies are presented.
This paper presents a novel adaptive-sparse polynomial dimensional decomposition (PDD) method for stochastic design optimization of complex systems. The method entails an adaptive-sparse PDD approximation of a high-di...
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ISBN:
(数字)9783319282626
ISBN:
(纸本)9783319282626;9783319282602
This paper presents a novel adaptive-sparse polynomial dimensional decomposition (PDD) method for stochastic design optimization of complex systems. The method entails an adaptive-sparse PDD approximation of a high-dimensional stochastic response for statistical moment and reliability analyses;a novel integration of the adaptive-sparse PDD approximation and score functions for estimating the first-order design sensitivities of the statistical moments and failure probability;and standard gradient-based optimization algorithms. New analytical formulae are presented for the design sensitivities that are simultaneously determined along with the moments or the failure probability. Numerical results stemming from mathematical functions indicate that the new method provides more computationally efficient design solutions than the existing methods. Finally, stochastic shape optimization of a jet engine bracket with 79 variables was performed, demonstrating the power of the new method to tackle practical engineering problems.
We present a numerical method for the solution of integro-differential equations describing motion of an incompressible viscoelastic fluid with memory. In particular, the system of equations consists of the momentum c...
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ISBN:
(纸本)9783319399294;9783319399270
We present a numerical method for the solution of integro-differential equations describing motion of an incompressible viscoelastic fluid with memory. In particular, the system of equations consists of the momentum conservation equation with the Cauchy stress tensor divided in a viscous and an elastic parts which depend non-linearly on the symmetric part of velocity gradient and nonlinearly on the past values of the Finger strain tensor, respectively. The momentum conservation equation is completed with system of equations that describes relation between the velocity gradient and the Finger strain tensor. The method is based on a discontinuous Galerkin method in the spatial variables and the BDF methods in the time variables.
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