A finite element (FE) model, that explicitly discretizes a single 3D spherulite is proposed. A spherulite is a two-phase microstructure consisting of amorphous and crystalline regions. Crystalline regions, that grow f...
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ISBN:
(纸本)9783319399294;9783319399270
A finite element (FE) model, that explicitly discretizes a single 3D spherulite is proposed. A spherulite is a two-phase microstructure consisting of amorphous and crystalline regions. Crystalline regions, that grow from a central nucleus in the form of lamellae, have particular lattice orientations. In the FE analyses, 8-chain and crystal viscoplasticity constitutive models are employed. Stress-strain distributions and slip system activities in the spherulite microstructure are studied and found to be in good agreement with the literature. Influences of the crystallinity ratio on the yield stress and the initial Young's modulus are also investigated.
The Vlasov-Poisson equation models the evolution of a plasma in an external or self-consistent electric field. The model consists of an advection equation in six dimensional phase space coupled to Poisson's equati...
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ISBN:
(数字)9783319282626
ISBN:
(纸本)9783319282626;9783319282602
The Vlasov-Poisson equation models the evolution of a plasma in an external or self-consistent electric field. The model consists of an advection equation in six dimensional phase space coupled to Poisson's equation. Due to the high dimensionality and the development of small structures the numerical solution is quite challenging. For two or four dimensional Vlasov problems, semi-Lagrangian solvers have been successfully applied. Introducing a sparse grid, the number of grid points can be reduced in higher dimensions. In this paper, we present a semi-Lagrangian Vlasov-Poisson solver on a tensor product of two sparse grids. In order to defeat the problem of poor representation of Gaussians on the sparse grid, we introduce a multiplicative delta-f method and separate a Gaussian part that is then handled analytically. In the semi-Lagrangian setting, we have to evaluate the hierarchical surplus on each mesh point. This interpolation step is quite expensive on a sparse grid due to the global nature of the basis functions. In our method, we use an operator splitting so that the advection steps boil down to a number of one dimensional interpolation problems. With this structure in mind we devise an evaluation algorithm with constant instead of logarithmic complexity per grid point. Results are shown for standard test cases and in four dimensional phase space the results are compared to a full-grid solution and a solution on the four dimensional sparse grid.
The reduced reliability of next generation exascale systems means that the resiliency properties of a numerical algorithm will become an important factor in both the choice of algorithm, and in its analysis. The multi...
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ISBN:
(纸本)9783319399294;9783319399270
The reduced reliability of next generation exascale systems means that the resiliency properties of a numerical algorithm will become an important factor in both the choice of algorithm, and in its analysis. The multigrid algorithm is the workhorse for the distributed solution of linear systems but little is known about its resiliency properties and convergence behavior in a fault- prone environment. In the current work, we propose a probabilistic model for the effect of faults involving random diagonal matrices. We summarize results of the theoretical analysis of the model for the rate of convergence of fault-prone multigrid methods which show that the standard multigrid method will not be resilient. Finally, we present a modification of the standard multigrid algorithm that will be resilient.
In this paper, a Riemannian BFGS method is defined for minimizing a smooth function on a Riemannian manifold endowed with a retraction and a vector transport. The method is based on a Riemannian generalization of a ca...
ISBN:
(纸本)9783319399294;9783319399270
In this paper, a Riemannian BFGS method is defined for minimizing a smooth function on a Riemannian manifold endowed with a retraction and a vector transport. The method is based on a Riemannian generalization of a cautious update and a weak line search condition. It is shown that, the Riemannian BFGS method converges (i) globally to a stationary point without assuming that the objective function is convex and (ii) superlinearly to a nondegenerate minimizer. The weak line search condition removes completely the need to consider the differentiated retraction. The joint diagonalization problem is used to demonstrate the performance of the algorithm with various parameters, line search conditions, and pairs of retraction and vector transport.
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