Locking-free HDG methods for Reissner-Mindlin plates equations on polygonal meshes

We present and analyze a new hybridizable discontinuous Galerkin method (HDG) for the Reissner-Mindlin plate bending system. Our method is based on the formulation utilizing Helmholtz Decomposition. Then the system is decomposed into three problems: two trivial Poisson problems and a perturbed saddle-point problem. We apply HDG scheme for these three problems fully. This scheme yields the optimal convergence rate ($(k+1)$th order in the $\mathrm{L}^2$ norm) which is uniform with respect to plate thickness (locking-free) on general meshes. We further analyze the matrix properties and precondition the new finite element system. Numerical experiments are presented to confirm our theoretical analysis

A variational multiscale stabilized finite element formulation for Reissner–Mindlin plates and Timoshenko beams

The theories for thick plates and beams, namely Reissner–Mindlin’s and Timoshenko’s theories, are well known to suffer numerical locking when approximated using the standard Galerkin finite element method for small thicknesses. This occurs when the same interpolations are used for displacement and rotations, reason for which stabilization becomes necessary. To overcome this problem, a Variational Multiscale stabilization method is analyzed in this paper. In this framework, two different approaches are presented: the Algebraic Sub-Grid Scale formulation and the Orthogonal Sub-Grid Scale formulation. Stability and convergence is proved for both approaches, explaining why the latter performs much better. Although the numerical examples show that the Algebraic Sub-Grid Scale approach is in some cases able to overcome the numerical locking, it is highly sensitive to stabilization parameters and presents difficulties to converge optimally with respect to the element size in the L 2 norm. In this regard, the Orthogonal Sub-Grid Scale approach, which considers the space of the sub-grid scales to be orthogonal to the finite element space, is shown to be stable and optimally convergent independently of the thickness of the solid. The final formulation is similar to approaches developed previously, thus justifying them in the frame of the Variational Multiscale concept.This work was supported by Vicerrectoría de Investigación, Chile, Desarrollo e Innovación (VRIDEI) of the Univeridad de Santiago de Chile, and the National Agency for Research and Development (ANID) Doctorado Becas Chile/2019 - 72200128 of the Government of Chile. R. Codina acknowledges the support received from the ICREA Acadèmia Research Program of the Catalan Government, Spain .Peer ReviewedPostprint (published version

First order least squares method with weakly imposed boundary condition for convection dominated diffusion problems

We present and analyze a first order least squares method for convection dominated diffusion problems, which provides robust L2 a priori error estimate for the scalar variable even if the given data f in L2 space. The novel theoretical approach is to rewrite the method in the framework of discontinuous Petrov - Galerkin (DPG) method, and then show numerical stability by using a key equation discovered by J. Gopalakrishnan and W. Qiu [Math. Comp. 83(2014), pp. 537-552]. This new approach gives an alternative way to do numerical analysis for least squares methods for a large class of differential equations. We also show that the condition number of the global matrix is independent of the diffusion coefficient. A key feature of the method is that there is no stabilization parameter chosen empirically. In addition, Dirichlet boundary condition is weakly imposed. Numerical experiments verify our theoretical results and, in particular, show our way of weakly imposing Dirichlet boundary condition is essential to the design of least squares methods - numerical solutions on subdomains away from interior layers or boundary layers have remarkable accuracy even on coarse meshes, which are unstructured quasi-uniform

Least-Squares FEM: Literature Review

During the last years the interest in least squares finite element methods (LSFEM) has grown continuously. Least squares finite element methods offer some advantages over the widely used Galerkin variational principle. One reason is the ability to cope with first order differential operators without special treatment as required by the Galerkin FEM. The other reason comes from the numerical point of view, where the LSFEM leads to symmetric positive definite matrices which can be solved very efficiently under some conditions. This report gives an overview about the recent literature which appeared in the field of least squares finite element methods and summarises the essential results and facts about the LSFEM.Während der letzten Jahre hat das Interesse an Least Squares Finite Element Methoden (LSFEM) stetig zugenommen. Least Squares Finite Element Methoden bieten einige Vorteile gegenüber dem etablierten Galerkin Variationsansatz. So können Differentialoperatoren erster Ordnung ohne besondere numerische Techniken, wie z.B. Stabilisierung, direkt behandelt werden. Ein anderer Grund für den Einsatz der LSFEM liegt in den entstehenden algebraischen Gleichungssystemen, die immer symmetrisch positiv definit sind und unter bestimmten Vorraussetzungen eine effiziente Lösung ermöglichen.Dieser Bericht gibt einen Überblick über die aktuelle Literatur zur LSFEM und faßt die entscheidenden Ergebnisse zusammen

Hp-spectral Methods for Structural Mechanics and Fluid Dynamics Problems

We consider the usage of higher order spectral element methods for the solution of problems in structures and fluid mechanics areas. In structures applications we study different beam theories, with mixed and displacement based formulations, consider the analysis of plates subject to external loadings, and large deformation analysis of beams with continuum based formulations. Higher order methods alleviate the problems of locking that have plagued finite element method applications to structures, and also provide for spectral accuracy of the solutions. For applications in computational fluid dynamics areas we consider the driven cavity problem with least squares based finite element methods. In the context of higher order methods, efficient techniques need to be devised for the solution of the resulting algebraic systems of equations and we explore the usage of element by element bi-orthogonal conjugate gradient solvers for solving problems effectively along with domain decomposition algorithms for fluid problems. In the context of least squares finite element methods we also explore the usage of Multigrid techniques to obtain faster convergence of the the solutions for the problems of interest. Applications of the traditional Lagrange based finite element methods with the Penalty finite element method are presented for modelling porous media flow problems. Finally, we explore applications to some CFD problems namely, the flow past a cylinder and forward facing step

Computational Engineering

The focus of this Computational Engineering Workshop was on the mathematical foundation of state-of-the-art and emerging finite element methods in engineering analysis. The 52 participants included mathematicians and engineers with shared interest on discontinuous Galerkin or Petrov-Galerkin methods and other generalized nonconforming or mixed finite element methods

Hp-spectral Methods for Structural Mechanics and Fluid Dynamics Problems

We consider the usage of higher order spectral element methods for the solution of problems in structures and fluid mechanics areas. In structures applications we study different beam theories, with mixed and displacement based formulations, consider the analysis of plates subject to external loadings, and large deformation analysis of beams with continuum based formulations. Higher order methods alleviate the problems of locking that have plagued finite element method applications to structures, and also provide for spectral accuracy of the solutions. For applications in computational fluid dynamics areas we consider the driven cavity problem with least squares based finite element methods. In the context of higher order methods, efficient techniques need to be devised for the solution of the resulting algebraic systems of equations and we explore the usage of element by element bi-orthogonal conjugate gradient solvers for solving problems effectively along with domain decomposition algorithms for fluid problems. In the context of least squares finite element methods we also explore the usage of Multigrid techniques to obtain faster convergence of the the solutions for the problems of interest. Applications of the traditional Lagrange based finite element methods with the Penalty finite element method are presented for modelling porous media flow problems. Finally, we explore applications to some CFD problems namely, the flow past a cylinder and forward facing step

Schur complement preconditioning for elliptic systems of partial differential equations

One successful approach in the design of solution methods for saddle-point problems requires the efficient solution of the associated Schur complement problem. In the case of problems arising from partial differential equations the factorization of the symbol of the operator can often suggest useful approximations for this problem. In this work we examine examples of preconditioners for regular elliptic systems of partial differential equations based on the Schur complement of the symbol of the operator and highlight the possibilities and some of the difficulties one may encounter with this approach

A hybrid discrete exterior calculus and finite difference method for anelastic convection in spherical shells

The present work develops, verifies, and benchmarks a hybrid discrete exterior calculus and finite difference (DEC-FD) method for density-stratified thermal convection in spherical shells. Discrete exterior calculus (DEC) is notable for its coordinate independence and structure preservation properties. The hybrid DEC-FD method for Boussinesq convection has been developed by Mantravadi et al. (Mantravadi, B., Jagad, P., & Samtaney, R. (2023). A hybrid discrete exterior calculus and finite difference method for Boussinesq convection in spherical shells. Journal of Computational Physics, 491, 112397). Motivated by astrophysics problems, we extend this method assuming anelastic convection, which retains density stratification; this has been widely used for decades to understand thermal convection in stars and giant planets. In the present work, the governing equations are splitted into surface and radial components and discrete anelastic equations are derived by replacing spherical surface operators with DEC and radial operators with FD operators. The novel feature of this work is the discretization of anelastic equations with the DEC-FD method and the assessment of a hybrid solver for density-stratified thermal convection in spherical shells. The discretized anelastic equations are verified using the method of manufactured solution (MMS). We performed a series of three-dimensional convection simulations in a spherical shell geometry and examined the effect of density ratio on convective flow structures and energy dynamics. The present observations are in agreement with the benchmark models.Comment: 32 pages, 13 figure