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1	Enhancing the scaled boundary finite element method Vu, Thu Hang January 2006 (has links) [Truncated abstract] The scaled boundary finite element method is a novel computational method developed by Wolf and Song which reduces partial differential equations to a set of ordinary linear differential equations. The method, which is semi-analytical, is suitable for solving linear elliptic, parabolic and hyperbolic partial differential equations. The method has proved to be very efficient in solving various types of problems, including problems of potential flow and diffusion. The method out performs the finite element method when solving unbounded domain problems and problems involving stress singularities and discontinuities. The scaled boundary finite element method involves solution of a quadratic eigenproblem, the computational expense of which increases rapidly as the number of degrees of freedom increases. Consequently, to a greater extent than the finite element method, it is desirable to obtain solutions at a specified level of accuracy while using the minimum number of degrees of freedom necessary. In previous work, no systematic study had been performed so far into the use of elements of higher order, and no consideration made of p adaptivity. . . The primal problem is solved normally using the basic scaled boundary finite element method. The dual problem is solved by the new technique using the fundamental solution. A guaranteed upper error bound based on the Cauchy-Schwarz inequality is derived. A iv goal-oriented p-hierarchical adaptive procedure is proposed and implemented efficiently in the scaled boundary finite element method. Engineering mathematics Boundary element methods Finite element method P-hierarchical adaptivity Projection-based interpolation technique Concentrated load analysis
2	Development of High-order CENO Finite-volume Schemes with Block-based Adaptive Mesh Refinement (AMR) Ivan, Lucian 31 August 2011 (has links) A high-order central essentially non-oscillatory (CENO) finite-volume scheme in combination with a block-based adaptive mesh refinement (AMR) algorithm is proposed for solution of hyperbolic and elliptic systems of conservation laws on body- fitted multi-block mesh. The spatial discretization of the hyperbolic (inviscid) terms is based on a hybrid solution reconstruction procedure that combines an unlimited high-order k-exact least-squares reconstruction technique following from a fixed central stencil with a monotonicity preserving limited piecewise linear reconstruction algorithm. The limited reconstruction is applied to computational cells with under-resolved solution content and the unlimited k-exact reconstruction procedure is used for cells in which the solution is fully resolved. Switching in the hybrid procedure is determined by a solution smoothness indicator. The hybrid approach avoids the complexity associated with other ENO schemes that require reconstruction on multiple stencils and therefore, would seem very well suited for extension to unstructured meshes. The high-order elliptic (viscous) fluxes are computed based on a k-order accurate average gradient derived from a (k+1)-order accurate reconstruction. A novel h-refinement criterion based on the solution smoothness indicator is used to direct the steady and unsteady refinement of the AMR mesh. The predictive capabilities of the proposed high-order AMR scheme are demonstrated for the Euler and Navier-Stokes equations governing two-dimensional compressible gaseous flows as well as for advection-diffusion problems characterized by the full range of Peclet numbers, Pe. The ability of the scheme to accurately represent solutions with smooth extrema and yet robustly handle under-resolved and/or non-smooth solution content (i.e., shocks and other discontinuities) is shown for a range of problems. Moreover, the ability to perform mesh refinement in regions of smooth but under-resolved and/or non-smooth solution content to achieve the desired resolution is also demonstrated. computational fluid dynamics finite-volume method high-order scheme adaptive mesh refinement (AMR) ENO scheme essentially non-oscillatory CENO body-fitted multi-block mesh high-order spatial discretization numerical method hyperbolic and elliptic PDE fixed stencil reconstruction piecewise linear reconstruction smoothness indicator Euler and Navier-Stokes equations advection-diffusion equation smooth and non-smooth solution content k-exact least-squares reconstruction hybrid numerical scheme compressible gaseous flows refinement criteria high-performance parallel computing solution monotonicity enforcement large-eddy simulation (LES) interpolation technique structured grid inviscid and viscous flow TVD scheme h-p-adaptation high-order boundary conditions complex curved geometries 0538
3	Development of High-order CENO Finite-volume Schemes with Block-based Adaptive Mesh Refinement (AMR) Ivan, Lucian 31 August 2011 (has links) A high-order central essentially non-oscillatory (CENO) finite-volume scheme in combination with a block-based adaptive mesh refinement (AMR) algorithm is proposed for solution of hyperbolic and elliptic systems of conservation laws on body- fitted multi-block mesh. The spatial discretization of the hyperbolic (inviscid) terms is based on a hybrid solution reconstruction procedure that combines an unlimited high-order k-exact least-squares reconstruction technique following from a fixed central stencil with a monotonicity preserving limited piecewise linear reconstruction algorithm. The limited reconstruction is applied to computational cells with under-resolved solution content and the unlimited k-exact reconstruction procedure is used for cells in which the solution is fully resolved. Switching in the hybrid procedure is determined by a solution smoothness indicator. The hybrid approach avoids the complexity associated with other ENO schemes that require reconstruction on multiple stencils and therefore, would seem very well suited for extension to unstructured meshes. The high-order elliptic (viscous) fluxes are computed based on a k-order accurate average gradient derived from a (k+1)-order accurate reconstruction. A novel h-refinement criterion based on the solution smoothness indicator is used to direct the steady and unsteady refinement of the AMR mesh. The predictive capabilities of the proposed high-order AMR scheme are demonstrated for the Euler and Navier-Stokes equations governing two-dimensional compressible gaseous flows as well as for advection-diffusion problems characterized by the full range of Peclet numbers, Pe. The ability of the scheme to accurately represent solutions with smooth extrema and yet robustly handle under-resolved and/or non-smooth solution content (i.e., shocks and other discontinuities) is shown for a range of problems. Moreover, the ability to perform mesh refinement in regions of smooth but under-resolved and/or non-smooth solution content to achieve the desired resolution is also demonstrated. computational fluid dynamics finite-volume method high-order scheme adaptive mesh refinement (AMR) ENO scheme essentially non-oscillatory CENO body-fitted multi-block mesh high-order spatial discretization numerical method hyperbolic and elliptic PDE fixed stencil reconstruction piecewise linear reconstruction smoothness indicator Euler and Navier-Stokes equations advection-diffusion equation smooth and non-smooth solution content k-exact least-squares reconstruction hybrid numerical scheme compressible gaseous flows refinement criteria high-performance parallel computing solution monotonicity enforcement large-eddy simulation (LES) interpolation technique structured grid inviscid and viscous flow TVD scheme h-p-adaptation high-order boundary conditions complex curved geometries 0538

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