24 research outputs found
Discontinuous Galerkin approximations in computational mechanics: hybridization, exact geometry and degree adaptivity
Discontinuous Galerkin (DG) discretizations with exact representation of the geometry and local polynomial degree adaptivity are revisited. Hybridization techniques are employed to reduce the computational cost of DG approximations and devise the hybridizable discontinuous Galerkin (HDG) method. Exact geometry described by non-uniform rational B-splines (NURBS) is integrated into HDG using the framework of the NURBS-enhanced finite element method (NEFEM). Moreover, optimal convergence and superconvergence properties of HDG-Voigt formulation in presence of symmetric second-order tensors are exploited to construct inexpensive error indicators and drive degree adaptive procedures. Applications involving the numerical simulation of problems in electrostatics, linear elasticity and incompressible viscous flows are presented. Moreover, this is done for both high-order HDG approximations and the lowest-order framework of face-centered finite volumes (FCFV).Peer ReviewedPostprint (author's final draft
A priori and a posteriori error analysis of an unfitted HDG method for semi-linear elliptic problems
We present a priori and a posteriori error analysis of a high order
hybridizable discontinuous Galerkin (HDG) method applied to a semi-linear
elliptic problem posed on a piecewise curved, non polygonal domain. We
approximate by a polygonal subdomain and propose an HDG
discretization, which is shown to be optimal under mild assumptions related to
the non-linear source term and the distance between the boundaries of the
polygonal subdomain and the true domain . Moreover, a local
non-linear post-processing of the scalar unknown is proposed and shown to
provide an additional order of convergence. A reliable and locally efficient a
posteriori error estimator that takes into account the error in the
approximation of the boundary data of is also provided
HDGlab: An Open-Source Implementation of the Hybridisable Discontinuous Galerkin Method in MATLAB
This paper presents HDGlab, an open source MATLAB implementation of the hybridisable discontinuous Galerkin (HDG) method. The main goal is to provide a detailed description of both the HDG method for elliptic problems and its implementation available in HDGlab. Ultimately, this is expected to make this relatively new advanced discretisation method more accessible to the computational engineering community. HDGlab presents some features not available in other implementations of the HDG method that can be found in the free domain. First, it implements high-order polynomial shape functions up to degree nine, with both equally-spaced and Fekete nodal distributions. Second, it supports curved isoparametric simplicial elements in two and three dimensions. Third, it supports non-uniform degree polynomial approximations and it provides a flexible structure to devise degree adaptivity strategies. Finally, an interface with the open-source high-order mesh generator Gmsh is provided to facilitate its application to practical engineering problems
A Cut Finite Element Method with Boundary Value Correction
In this contribution we develop a cut finite element method with boundary value correction of the type originally proposed by Bramble, Dupont, and Thomee. The cut finite element method is a fictitious domain method with Nitsche type enforcement of Dirichlet conditions together with stabilization of the elements at the boundary which is stable and enjoy optimal order approximation properties. A computational difficulty is, however, the geometric computations related to quadrature on the cut elements which must be accurate enough to achieve higher order approximation. With boundary value correction we may use only a piecewise linear approximation of the boundary, which is very convenient in a cut finite element method, and still obtain optimal order convergence. The boundary value correction is a modified Nitsche formulation involving a Taylor expansion in the normal direction compensating for the approximation of the boundary. Key to the analysis is a consistent stabilization term which enables us to prove stability of the method and a priori error estimates with explicit dependence on the meshsize and distance between the exact and approximate boundary
High-fidelity surrogate models for parametric shape design in microfluidics
Nowadays, the main computational bottleneck in computer-assisted industrial design procedures
is the necessity of testing multiple parameter settings for the same problem. Material properties,
boundary conditions or geometry may have a relevant influence on the solution of those
problems. Consequently, the effects of changes in these quantities on the numerical solution
need to be accurately estimated. That leads to significantly time-consuming multi-query
procedures during decision-making processes. Microfluidics is one of the many fields affected by this issue, especially in the context of the design of robotic devices inspired by natural
microswimmers. Reduced-order modelling procedures are commonly employed to reduce the
computational burden of such parametric studies with multiple parameters. Moreover, highfidelity simulation techniques play a crucial role in the accurate approximation of the flow
features appearing in complex geometries. This thesis proposes a coupled methodology based
on the high-order hybridisable discontinuous Galerkin (HDG) method and the proper generalized decomposition (PGD) technique. Geometrically parametrised Stokes equations are solved exploiting the innovative HDG-PGD framework. On the one hand, the parameters describing the geometry of the domain act as extra-coordinates and PGD is employed to construct a separated approximation of the solution. On the other hand, HDG mixed formulation allows separating exactly the terms introduced by the parametric mapping into products of functions depending either on the spatial or on the parametric unknowns. Convergence results validate the methodology and more realistic test cases, inspired by microswimmer devices involving variable geometries, show the potential of the proposed HDG-PGD framework in parametric shape design. The PGD-based surrogate models are also utilised to construct separated response surfaces for the drag force. A comparison between response surfaces obtained through the apriori and the a posteriori PGD is exposed. A critical analysis of the two techniques is presented reporting advantages and drawbacks of both in terms of computational costs and accuracy.Actualmente, el principal obstáculo en los procesos de diseño industrial computarizado es la necesidad de examinar múltiples parámetros para el mismo problema. Las propiedades de los materiales, las condiciones de contorno o la geometrÃa pueden tener una influencia relevante en la solución de esos problemas. Por lo tanto, es necesario estimar con precisión los efectos de las variaciones de esas cantidades en la solución numérica. Esto da origen a procedimientos de consultas múltiples que requieren considerable tiempo durante los procesos de toma de decisión. La microfluÃdica es uno de los varios campos afectados por esta problemática, especialmente en el contexto del diseño de dispositivos robóticos inspirados en los micronadadores naturales. Generalmente se recurre a procedimientos de reducción de orden de modelo para reducir la complejidad computacional de estos estudios paramétricos basados en múltiples parámetros. Además, los esquemas de alto orden son fundamentales para la aproximación precisa de las particularidades de los flujos que aparecen en las geometrÃas complejas. Esta tesis propone una metodologÃa acoplada basada en el método de Galerkin discontinuo hibridizable de alto orden (HDG) y la técnica de descomposición propia generalizada (PGD). Las ecuaciones de Stokes geométricamente parametrizadas se resuelven empleando el innovador método HDG-PGD. Por un lado, los parámetros que describen la geometrÃa del dominio actúan como extra-coordinadas y la PGD permite construir una aproximación separada de la solución. Por otra parte, la formulación mixta de HDG admite la separación exacta de los términos introducidos por la descripción paramétrica del dominio en productos de funciones dependientes de las incógnitas espaciales o paramétricas. Los resultados de convergencia validan la metodologÃa y estudios de casos más realistas, inspirados en los dispositivos de micronatación con geometrÃas variables, muestran el potencial del marco propuesto de HDG-PGD en el diseño de formas parametrizadas. Los modelos reducidos basados en la PGD también permiten construir superficies de respuesta separadas para la fuerza de arrastre. Se realiza una comparación entre las superficies de respuesta obtenidas mediante la PGD a priori y a posteriori. Se exponen una análisis crÃtica de las dos técnicas reportando las ventajas y desventajas de ambas en términos de costes computacionales y precisión
A new approach to handle curved meshes in the Hybrid High-Order method
The hybrid high-order method is a modern numerical framework for the
approximation of elliptic PDEs. We present here an extension of the hybrid
high-order method to meshes possessing curved edges/faces. Such an extension
allows us to enforce boundary conditions exactly on curved domains, and capture
curved geometries that appear internally in the domain e.g. discontinuities in
a diffusion coefficient. The method makes use of non-polynomial functions on
the curved faces and does not require any mappings between reference
elements/faces. Such an approach does not require the faces to be polynomial,
and has a strict upper bound on the number of degrees of freedom on a curved
face for a given polynomial degree. Moreover, this approach of enriching the
space of unknowns on the curved faces with non-polynomial functions should
extend naturally to other polytopal methods. We show the method to be stable
and consistent on curved meshes and derive optimal error estimates in L2 and
energy norms. We present numerical examples of the method on a domain with
curved boundary, and for a diffusion problem such that the diffusion tensor is
discontinuous along a curved arc.Comment: 23 pages, 9 figures, 2 table