2,423 research outputs found
Solving high-order partial differential equations with indirect radial basis function networks
This paper reports a new numerical method based on radial basis function networks (RBFNs) for solving high-order partial differential equations (PDEs). The variables and their derivatives in the governing equations are represented by integrated RBFNs. The use of integration in constructing neural networks allows the straightforward implementation of multiple boundary conditions and the accurate approximation of high-order derivatives. The proposed RBFN method is verified successfully through the solution of thin-plate bending and viscous flow problems which are governed by biharmonic equations. For thermally driven cavity flows, the solutions are obtained up to a high Rayleigh number
A Method for Geometry Optimization in a Simple Model of Two-Dimensional Heat Transfer
This investigation is motivated by the problem of optimal design of cooling
elements in modern battery systems. We consider a simple model of
two-dimensional steady-state heat conduction described by elliptic partial
differential equations and involving a one-dimensional cooling element
represented by a contour on which interface boundary conditions are specified.
The problem consists in finding an optimal shape of the cooling element which
will ensure that the solution in a given region is close (in the least squares
sense) to some prescribed target distribution. We formulate this problem as
PDE-constrained optimization and the locally optimal contour shapes are found
using a gradient-based descent algorithm in which the Sobolev shape gradients
are obtained using methods of the shape-differential calculus. The main novelty
of this work is an accurate and efficient approach to the evaluation of the
shape gradients based on a boundary-integral formulation which exploits certain
analytical properties of the solution and does not require grids adapted to the
contour. This approach is thoroughly validated and optimization results
obtained in different test problems exhibit nontrivial shapes of the computed
optimal contours.Comment: Accepted for publication in "SIAM Journal on Scientific Computing"
(31 pages, 9 figures
Hybrid Chebyshev Polynomial Scheme for the Numerical Solution of Partial Differential Equations
In the numerical solution of partial differential equations (PDEs), it is common to find situations where the best choice is to use more than one method to arrive at an accurate solution. In this dissertation, hybrid Chebyshev polynomial scheme (HCPS) is proposed which is applied in two-step approach and one-step approach. In the two-step approach, first, Chebyshev polynomials are used to approximate a particular solution of a PDE. Chebyshev nodes which are the roots of Chebyshev polynomials are used in the polynomial interpolation due to its spectral convergence. Then, the resulting homogeneous equation is solved by boundary type methods including the method of fundamental solution (MFS) and the equilibrated collocation Trefftz method. However, this scheme can be applied to solve PDEs with constant coefficients only. So, for solving a wide variety of PDEs, one-step hybrid Chebyshev polynomial scheme is proposed. This approach combines two matrix systems of two-step approach into a single matrix system. The solution is approximated by the sum of particular solution and homogeneous solution. The Laplacian or biharmonic operator is kept on the left hand side and all the other terms are moved to the right hand side and treated as the forcing term. Various boundary value problems governed by the Poisson equation in two and three dimensions are considered for the numerical experiments. HCPS is also applied to solve an inhomogeneous Cauchy-Navier equations of elasticity in two dimensions. Numerical results show that HCPS is direct, easy to implement, and highly accurate
A fully-coupled discontinuous Galerkin method for two-phase flow in porous media with discontinuous capillary pressure
In this paper we formulate and test numerically a fully-coupled discontinuous
Galerkin (DG) method for incompressible two-phase flow with discontinuous
capillary pressure. The spatial discretization uses the symmetric interior
penalty DG formulation with weighted averages and is based on a wetting-phase
potential / capillary potential formulation of the two-phase flow system. After
discretizing in time with diagonally implicit Runge-Kutta schemes the resulting
systems of nonlinear algebraic equations are solved with Newton's method and
the arising systems of linear equations are solved efficiently and in parallel
with an algebraic multigrid method. The new scheme is investigated for various
test problems from the literature and is also compared to a cell-centered
finite volume scheme in terms of accuracy and time to solution. We find that
the method is accurate, robust and efficient. In particular no post-processing
of the DG velocity field is necessary in contrast to results reported by
several authors for decoupled schemes. Moreover, the solver scales well in
parallel and three-dimensional problems with up to nearly 100 million degrees
of freedom per time step have been computed on 1000 processors
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