Summary
The first superconvergent hybridisable discontinuous Galerkin method for linear elastic problems capable of using the same degree of approximation for both the primal and mixed variables is ...presented. The key feature of the method is the strong imposition of the symmetry of the stress tensor by means of the well known and extensively used Voigt notation, circumventing the use of complex mathematical concepts to enforce the symmetry of the stress tensor either weakly or strongly. A novel procedure to construct element by element a superconvergent postprocessed displacement is proposed. Contrary to other hybridisable discontinuous Galerkin formulations, the methodology proposed here is able to produce a superconvergent displacement field for low‐order approximations. The resulting method is robust and locking‐free in the nearly incompressible limit. An extensive set of numerical examples is utilised to provide evidence of the optimality of the method and its superconvergent properties in two and three dimensions and for different element types.
Summary
This work proposes a novel finite volume paradigm, ie, the face‐centred finite volume (FCFV) method. Contrary to the popular vertex and cell‐centred finite volume methods, the novel FCFV ...defines the solution on the mesh faces (edges in two dimensions) to construct locally conservative numerical schemes. The idea of the FCFV method stems from a hybridisable discontinuous Galerkin formulation with constant degree of approximation, and thus inheriting the convergence properties of the classical hybridisable discontinuous Galerkin. The resulting FCFV features a global problem in terms of a piecewise constant function defined on the faces of the mesh. The solution and its gradient in each element are then recovered by solving a set of independent element‐by‐element problems. The mathematical formulation of FCFV for Poisson and Stokes equation is derived, and numerical evidence of optimal convergence in two dimensions and three dimensions is provided. Numerical examples are presented to illustrate the accuracy, efficiency, and robustness of the proposed methodology. The results show that, contrary to other finite volume methods, the accuracy of the FCFV method is not sensitive to mesh distortion and stretching. In addition, the FCFV method shows its better performance, accuracy, and robustness using simplicial elements, facilitating its application to problems involving complex geometries in three dimensions.
•A new finite volume paradigm is proposed for linear elastic problems.•Optimal convergence for displacement and stress is obtained without reconstruction.•The accuracy of the solution is not ...sensitive to mesh distortion.•The proposed method does not show volumetric locking in the incompressible limit.•Examples show the optimal approximation properties and demonstrate the efficiency.
A face-centred finite volume (FCFV) method is proposed for linear elastostatic problems. The FCFV is a mixed hybrid formulation, featuring a system of first-order equations, that defines the unknowns on the faces (edges in two dimensions) of the mesh cells. The symmetry of the stress tensor is strongly enforced using the well-known Voigt notation and the displacement and stress fields inside each cell are obtained by means of explicit formulas. The resulting FCFV method is robust and locking-free in the nearly incompressible limit. Numerical experiments in two and three dimensions show optimal convergence of the displacement and the stress fields without any reconstruction. Moreover, the accuracy of the FCFV method is not sensitive to mesh distortion and stretching. Classical benchmark tests including Kirch’s plate and Cook’s membrane problems in two dimensions as well as three dimensional problems involving shear phenomenons, pressurised thin shells and complex geometries are presented to show the capability and potential of the proposed methodology.
This work proposes a superconvergent hybridizable discontinuous Galerkin (HDG) method for the approximation of the Cauchy formulation of the Stokes equation using same degree of polynomials for the ...primal and mixed variables. The novel formulation relies on the well-known Voigt notation to strongly enforce the symmetry of the stress tensor. The proposed strategy introduces several advantages with respect to the existing HDG formulations. First, it remedies the suboptimal behavior experienced by the classical HDG method for formulations involving the symmetric part of the gradient of the primal variable. The optimal convergence of the mixed variable is retrieved and an element-by-element postprocess procedure leads to a superconvergent velocity field, even for low-order approximations. Second, no additional enrichment of the discrete spaces is required and a gain in computational efficiency follows from reducing the quantity of stored information and the size of the local problems. Eventually, the novel formulation naturally imposes physical tractions on the Neumann boundary. Numerical validation of the optimality of the method and its superconvergent properties is performed in 2D and 3D using meshes of different element types.
The NURBS-enhanced finite element method (NEFEM) combined with a hybridisable discontinuous Galerkin (HDG) approach is presented for the first time. The proposed technique completely eliminates the ...uncertainty induced by a polynomial approximation of curved boundaries that is common within an isoparametric approach and, compared to other DG methods, provides a significant reduction in number of degrees of freedom. In addition, by exploiting the ability of HDG to compute a postprocessed solution and by using a local a priori error estimate valid for elliptic problems, an inexpensive, reliable and computable error estimator is devised. The proposed methodology is used to solve Stokes flow problems using automatic degree adaptation. Particular attention is paid to the importance of an accurate boundary representation when changing the degree of approximation in curved elements. Several strategies are compared and the superiority and reliability of HDG-NEFEM with degree adaptation is illustrated.
The computational cost of parametric studies currently represents the major limitation to the application of simulation-based engineering techniques in a daily industrial environment. This work ...presents the first nonintrusive implementation of the proper generalised decomposition (PGD) in OpenFOAM, for the approximation of parametrised laminar incompressible Navier–Stokes equations. The key feature of this approach is the seamless integration of a reduced order model (ROM) in the framework of an industrially validated computational fluid dynamics software. This is of special importance in an industrial environment because in the online phase of the PGD ROM the description of the flow for a specific set of parameters is obtained simply via interpolation of the generalised solution, without the need of any extra solution step. On the one hand, the spatial problems arising from the PGD separation of the unknowns are treated using the classical solution strategies of OpenFOAM, namely the semi-implicit method for pressure linked equations (SIMPLE) algorithm. On the other hand, the parametric iteration is solved via a collocation approach. The resulting ROM is applied to several benchmark tests of laminar incompressible Navier–Stokes flows, in two and three dimensions, with different parameters affecting the flow features. Eventually, the capability of the proposed strategy to treat industrial problems is verified by applying the methodology to a parametrised flow control in a realistic geometry of interest for the automotive industry.
•Proper generalised decomposition for parametrised incompressible Navier–Stokes flows.•Nonintrusive implementation of the PGD in the industrially validated solver OpenFOAM.•Seamless exploitation of OpenFOAM SIMPLE algorithm for the spatial step of the PGD.•Offline computation of a reduced basis with no a priori knowledge of the solution.•Inexpensive online characterisation of the flow via interpolation of the PGD solution.•Application to flow control problems of interest for the automotive industry.
•Now dual time stepping approach for the HDG formulation of convection-diffusion problems.•Proof of existence of steady-state solutions in dual time.•Proof of critical an optimal dual time ...steps.•Numerical examples demonstrate the performance of the method.•Fact convergence achieved of convection dominated problems.
This work presents, for the first time, a dual time stepping (DTS) approach to solve the global system of equations that appears in the hybridisable discontinuous Galerkin (HDG) formulation of convection-diffusion problems. A proof of the existence and uniqueness of the steady state solution of the HDG global problem with DTS is presented. The stability limit of the DTS approach is derived using a von Neumann analysis, leading to a closed form expression for the critical dual time step. An optimal choice for the dual time step, producing the maximum damping for all the frequencies, is also derived. Steady and transient convection-diffusion problems are considered to demonstrate the performance of the proposed DTS approach, with particular emphasis on convection dominated problems. Two simple approaches to accelerate the convergence of the DTS approach are also considered and three different time marching approaches for the dual time are compared.
In this work we present the Cartesian grid discontinuous Galerkin (cgDG) finite element method, a novel numerical technique that combines the high accuracy and efficiency of a high‐order ...discontinuous Galerkin discretization with the simplicity and hierarchical structure of a geometry‐independent Cartesian mesh. The elements that intersect the boundary of the physical domain require special treatment in order to minimize their effect on the performance of the algorithm. We considered the exact representation of the geometry for the boundary of the domain avoiding any nonphysical artifacts. We also define a stabilization procedure that eliminates the step size restriction of the time marching scheme due to extreme cut patterns. The unstable degrees of freedom are eliminated and the supporting regions of their shape functions are reassigned to neighboring elements. A subdomain matching algorithm and an a posterior enrichment strategy are presented. Combining these techniques we obtain a final spatial discretization that preserves stability and accuracy of the standard body‐fitted discretization. The method is validated through a series of numerical tests and it is successfully applied to the solution of problems of interest in the context of electromagnetic scattering with increasing complexity.
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