A Finite Element Splitting Extrapolation for Second Order Hyperbolic Equations

Splitting extrapolation is an efficient technique for solving large scale scientific and engineering problems in parallel. This article discusses a finite element splitting extrapolation for second order hyperbolic equations with time-dependent coefficients. This method possesses a higher degree of parallelism, less computational complexity, and more flexibility than Richardson extrapolation while achieving the same accuracy. By means of domain decomposition and isoparametric mapping, some grid parameters are chosen according to the problem. The multiparameter asymptotic expansion of the d-quadratic finite element error is also established. The splitting extrapolation formulas are developed from this expansion. An approximation with higher accuracy on a globally fine grid can be computed by solving a set of smaller discrete subproblems on different coarser grids in parallel. Some a posteriori error estimates are also provided. Numerical examples show that this method is efficient for solving discontinuous problems and nonlinear hyperbolic equations

Three-dimensional finite element analysis for high velocity impact

A finite element algorithm for solving unsteady, three-dimensional high velocity impact problems is presented. A computer program was developed based on the Eulerian hydroelasto-viscoplastic formulation and the utilization of the theorem of weak solutions. The equations solved consist of conservation of mass, momentum, and energy, equation of state, and appropriate constitutive equations. The solution technique is a time-dependent finite element analysis utilizing three-dimensional isoparametric elements, in conjunction with a generalized two-step time integration scheme. The developed code was demonstrated by solving one-dimensional as well as three-dimensional impact problems for both the inviscid hydrodynamic model and the hydroelasto-viscoplastic model

Geometric partial differential equations: Theory, numerics and applications

This workshop concentrated on partial differential equations involving stationary and evolving surfaces in which geometric quantities play a major role. Mutual interest in this emerging field stimulated the interaction between analysis, numerical solution, and applications

Splitting Extrapolation Method for Solving Second-order Parabolic Equations with Curved Boundaries by Using Domain Decomposition and D-quadratic Isoparametric Finite Elements

This article discusses a splitting extrapolation method for solving second-order parabolic equations with curved boundaries by using domain decomposition and d-quadratic isoparametric finite elements. This method possesses superconvergence, a high order of accuracy and a high degree of parallelism. First, we prove the multi-variable asymptotic expansion of fully discrete d-quadratic isoparametric finite element errors. Based on the expansion, we generate splitting extrapolation formulas. These formulas generate a numerical solution on a globally fine grid with higher accuracy by solving only a set of smaller discrete subproblems on different coarser grids. Therefore, a large-scale multidimensional problem with a curved boundary is turned into a set of smaller discrete subproblems on a polyhedron. Because these subproblems are independent of each other and have similar scales, our algorithm possesses a high degree of parallelism. In addition, this method is effective for solving discontinuous problems if we regard the interfaces of the problems as the interfaces of the initial domain decomposition. Our numerical results also show that the algorithm is effective for solving nonlinear parabolic equations

Adaptive low and high-order hybridized methods for unsteady incompressible flow simulations

Tesi en modalitat de cotutela: Universitat Politècnica de Catalunya i Università degli Studi di PaviaSimulations of incompressible flows are performed on a daily basis to solve problems of practical and industrial interest in several fields of engineering, including automotive, aeronautical, mechanical and biomedical applications. Although finite volume (FV) methods are still the preferred choice by the industry due to their efficiency and robustness, sensitivity to mesh quality and limited accuracy represent two main bottlenecks of these approaches. This is especially critical in the context of transient phenomena, in which FV methods show excessive numerical diffusion. In this context, there has been a growing interest towards high-order discretisation strategies in last decades. In this PhD thesis, a high-order adaptive hybidisable discontinuous Galerkin (HDG) method is proposed for the approximation of steady and unsteady laminar incompressible Navier-Stokes equations. Voigt notation for symmetric second-order tensors is exploited to devise an HDG method for the Cauchy formulation of the momentum equation with optimal convergence properties, even when low-order polynomial degrees of approximation are considered. In addition, a postprocessing strategy accounting for rigid translational and rotational modes is proposed to construct an element-by-element superconvergent velocity field. The discrepancy between the computed and postprocessed velocities is utilised to define a local error indicator to drive degree adaptivity procedures and accurately capture localised features of the flow. The resulting HDG solver is thus extended to the case of transient problems via high-order time integration schemes, namely the explicit singly diagonal implicit Runge-Kutta (ESDIRK) schemes. In this context, the embedded explicit step is exploited to define an inexpensive estimate of the temporal error to devise an efficient timestep control strategy. Finally, in order to efficiently solve the global problem arising from the HDG discretisation, a preconditioned iterative solver is proposed. This is critical in the context of high-order approximations in three-dimensional domains leading to large-scale problems, especially in transient simulations. A block diagonal preconditioner coupled with an inexpensive approximation of the Schur complement of the matrix is proposed to reduce the computational cost of the overall HDG solver. Extensive numerical validation of two and three-dimensional steady and unsteady benchmark tests of viscous laminar incompressible flows is performed to validate the proposed methodology.Simulaciones de flujo incompresible se emplean a diario para resolver problemas de interés práctico e industrial en varios campos de la ingeniería, p.ej. en aplicaciones automovilísticas, aeronáuticas, mecánicas y biomédicas. Aunque los métodos de volúmenes finitos (FV) siguen siendo la opción preferida por la industria debido a su eficiencia y robustez, la sensibilidad a la calidad de la malla y la baja precisión representan dos limitaciones importantes para estas técnicas. Estas limitaciones son todavía más críticas en el contexto de simulaciones de fenómenos transitorios, donde los FV están penalizados por su excesiva difusión numérica. En este contexto, las estrategias de discretización de alto orden han ganado una popularidad creciente en las últimas décadas para problemas transitorios dónde se necesitan soluciones precisas. Esta tesis propone un método de Galerkin discontinuo híbrido (HDG), de alto orden y adaptativo para la aproximación de las ecuaciones de Navier-Stokes incomprensible laminar, en el caso estacionario y transitorio en el entorno de aplicaciones ingenieriles. Para ello, la notación de Voigt para tensores simétricos de segundo orden (habituales en mecánica de los medios continuos) permite introducir un método HDG para la formulación de Cauchy de la ecuación de momento. La novedad de este resultado reside en la convergencia óptima alcanzada por el método, incluso para aproximaciones de orden polinómico bajo. Además, se desarrolla una estrategia de post-proceso local para construir elemento a elemento un campo de velocidad súper-convergente, tomando en cuenta los modos rígidos de traslación y rotación. La discrepancia entre el campo de velocidad calculado y el súper-convergente, obtenido a través del post-proceso, permite definir un indicador del error local. De esta forma, se desarrolla una estrategia para realizar adecuar elemento a elemento el grado de la aproximación polinómica y así mejorar la precisión adaptándose a las características localizadas del flujo. Seguidamente, se extiende el método HDG propuesto al tratamiento de problemas dependientes del tiempo. Más concretamente, se consideran los esquemas de integración temporal de alto orden explicit singly diagonal implicit Runge-Kutta (ESDIRK). En este contexto, se utiliza el paso explícito embedded para calcular una estimación computacionalmente eficiente del error temporal y definir una estrategia de adaptividad del paso de tiempo. Finalmente, se desarrolla un precondicionador adaptado a la estrategia HDG que acelera la convergencia del método iterativo empleado y, de esta forma, obtener resoluciones eficaces del problema global surgido de la discretización HDG. Es importante resaltar la importancia de una herramienta de resolución eficiente para problemas de gran escala en el contexto de aproximaciones de alto orden y en dominios tridimensionales. Estas herramientas se hacen aún más criticas en simulaciones transitorias. Más concretamente, se proponen un precondicionador diagonal por bloques y una aproximación eficiente del complemento Schur de la matriz para reducir el coste computacional del método HDG. Para validar la metodología propuesta, se realizan varias simulaciones numéricas de flujo incompresible laminar viscoso, para problemas estacionarios y transitorios, en dos y tres dimensiones.Postprint (published version

Finite element solution for elliptic partial differential equations

The contents of this thesis are a detailed study of the implementation of Finite Element method for solving linear and non-linear elliptic partial differential equations. It commences with a description and classification of partial differential equations, the related matrix and eigenvalue theory and the related matrix methods to solve the linear and non-linear systems of equations. In Chapter Three, we discuss the development of the, finite element method and its application with a full description of an orderly step-by-step process. In Chapter Four, we discuss the implementation of developing an efficient easy-to-use finite element program for the general two-dimensional problem along with the capability of handling problems for different domains and boundary conditions and with a fully automated mesh generation and refinement technique along with a description of generalised pre- and post-processors for the Finite Element Method. [Continues.

Applications of Isogeometric Analysis Coupled with Finite Volume Method

In this thesis, a combination of Isogeometric Analysis (IGA) and Finite Volume Method (FVM) on geometries parameterized by Non-Uniform Rational Basis Splines (NURBS) is explored with applications in fluid flow, heat transfer, and shape optimization. An IGA framework supplemented with FVM is created in MATLAB® to solve problems defined over single patch domains with mesh refinement by node insertion. Additionally, a second-order finite difference method is developed using non-orthogonal curvilinear coordinates and a numerical Jacobian of the NURBS geometry. The examples include fully developed laminar flow through ducts, potential flow around a tilted ellipse, transient heat conduction, linear advection-diffusion, and a basic shape optimization example using a particle swarm technique. The numerical results are compared among the methods and verified with available analytical solutions

Progress on unstructured-grid based high-order CFD method

Several new methods have been developed to meet the critical and diversified challenges in the state-of-art unstructured-grids based high-order methods for 3D real-world applications, including 1) parameter-free high-order generalized moment limiter for arbitrary mesh; 2) efficient line implicit method; 3) efficient quadrature-free SV method; 4) novel high-order mesh generation method for 3D hexahedral mesh. The parameter-free high-order generalized moment limiter does not need any user-specified free parameter to detect the discontinuities and exclude the smooth extrema. The present limiter has been designed to be naturally generic, compact, and efficient, which can be applied for arbitrary mesh and general unstructured-grids based high-order methods. The present low-storage line implicit BLU-SGS method significantly overcomes the anisotropy stiffness due to highly stretched wall grids in high Reynolds number flows. Improved robustness and up to 3 times of savings on CPU time have been demonstrated comparing with the cell BLU-SGS solver. This line implicit method preserves the favorable feature of high compactness from the cell BLU-SGS method, and can be programmed as a black box so as to be easily applied in general high-order methods. The quadrature-free SV method has improved the original SV method by replacing the large number of quadrature for face integrals in 3D case with many less nodal operations based on analytical shape functions. Finally for high-order unstructured mesh generation, the present novel and fully automatic algorithm guarantee to resolve the self-intersection problem for non-linear quadrilateral or hexahedral mesh with strong robustness. The algorithm also offers the advantage of correcting grid self-intersection without changing the basic aspect ratio of the original grids or degrading the original grid quality

Arbitrary high order discontinuous Galerkin methods for the shallow water and incompressible Navier-Stokes equations on unstructured staggered meshes

In this work we present a new class of well-balanced, arbitrary high order accurate semi-implicit discontinuous Galerkin methods for the solution of the shallow water and incompressible Navier-Stokes equations on staggered unstructured curved meshes. Isoparametric finite elements are used to take into account curved domain boundaries. Regarding two-dimensional shallow water equations, the discrete free surface elevation is defined on a primal triangular grid, while the discrete total height and the discrete velocity field are defined on an edge-based staggered dual grid. Similarly, for the two-dimensional incompressible Navier-Stokes case, the discrete pressure is defined on the main triangular grid and the velocity field is defined on the edge-based staggered grid. While staggered meshes are state of the art in classical finite difference approximations of the incompressible Navier-Stokes equations, their use in the context of high order DG schemes is novel and still quite rare. High order (better than second order) in time can be achieved by using a space-time finite element framework, where the basis and test functions are piecewise polynomials in both space and time. Formal substitution of the discrete momentum equation on the dual grid into the discrete continuity equation on the primary grid yields a very sparse system for the scalar pressure involving only the direct neighbor elements, so that it becomes a block four-point system in 2D and a block five-point system for 3D tetrahedral meshes. The resulting linear system is conveniently solved with a matrix-free GMRES algorithm. Note that the same space-time DG scheme on a collocated grid would lead to ten non-zero blocks per element in 2D and seventeen non-zero blocks in 3D, since substituting the discrete velocity into the discrete continuity equation on a collocated mesh would involve also neighbors of neighbors. From numerical experiments we find that our linear system is well-behaved and that the GMRES method converges quickly even without the use of any preconditioner, which is a unique feature in the context of high order implicit DG schemes. A very simple and efficient Picard iteration is then used in order to derive a space-time pressure correction algorithm that achieves also high order of accuracy in time, which is in general a non-trivial task in the context of high order discretizations for the incompressible Navier-Stokes equations. The special case of high order in space low order in time allows us to recover further regularity about the main linear system for the pressure, such as the symmetry and the positive semi-definiteness in the general case. This allows us to use a very fast linear solver such as the conjugate gradient (CG) method. The flexibility and accuracy of high order space-time DG methods on curved unstructured meshes allows to discretize even complex physical domains with very coarse grids in both space and time. We will further extend the previous method to three-dimensional incompressible Navier-Stokes system using a tetrahedral main grid and a corresponding face-based hexaxedral dual grid. The resulting dual mesh consists in non-standard 5-vertex hexahedral elements that cannot be represented using tensor products of one dimensional basis functions. Indeed a modal polynomial basis will be used for the dual mesh. This new family of numerical schemes is verified by solving a series of typical numerical test problems and by comparing the obtained numerical results with available exact analytical solutions or other numerical reference data. Furthermore, the comparison with available experimental results will be presented for incompressible Navier-Stokes equations

Nonlinear Structural Analysis

Nonlinear structural analysis techniques for engine structures and components are addressed. The finite element method and boundary element method are discussed in terms of stress and structural analyses of shells, plates, and laminates