A quasi-dual Lagrange multiplier space for serendipity mortar finite elements in 3D

Domain decomposition techniques provide a flexible tool for the numerical approximation of partial differential equations. Here, we consider mortar techniques for quadratic finite elements in 3D with different Lagrange multiplier spaces. In particular, we focus on Lagrange multiplier spaces which yield optimal discretization schemes and a locally supported basis for the associated constrained mortar spaces in case of hexahedral triangulations. As a result, standard efficient iterative solvers as multigrid methods can be easily adapted to the nonconforming situation. We present the discretization errors in different norms for linear and quadratic mortar finite elements with different Lagrange multiplier spaces. Numerical results illustrate the performance of our approach

Natural preconditioners for saddle point systems

The solution of quadratic or locally quadratic extremum problems subject to linear(ized) constraints gives rise to linear systems in saddle point form. This is true whether in the continuous or discrete setting, so saddle point systems arising from discretization of partial differential equation problems such as those describing electromagnetic problems or incompressible flow lead to equations with this structure as does, for example, the widely used sequential quadratic programming approach to nonlinear optimization.\ud This article concerns iterative solution methods for these problems and in particular shows how the problem formulation leads to natural preconditioners which guarantee rapid convergence of the relevant iterative methods. These preconditioners are related to the original extremum problem and their effectiveness -- in terms of rapidity of convergence -- is established here via a proof of general bounds on the eigenvalues of the preconditioned saddle point matrix on which iteration convergence depends

Convergence analysis of an unfitted mesh semi-implicit coupling scheme for incompressible fluid-structure interaction

International audienceThe complete numerical analysis of time splitting schemes which avoid strong coupling has rarely been addressed in the literature of unfitted mesh methods for incompressible fluid-structure interaction. In this paper, an error analysis of the semi-implicit scheme recently reported in [Int. J. Numer. Methods Eng., 2021; 122:5384--5408] is performed for a linear fluid-structure interaction system. The analysis shows that, under a hyperbolic-CFL condition, the leading term in the energy error scales as $O(h^{r-1/2})$, where $r=1, 2$ stands for the extrapolation order of the solid velocity in the viscous fluid substep. The theoretical findings are illustrated via some numerical experiments which show, in particular, that the considered method avoids the spatial non-uniformity issues of standard loosely coupled schemes and that it delivers practically the same accuracy as the strongly coupled scheme

SOLID-SHELL FINITE ELEMENT MODELS FOR EXPLICIT SIMULATIONS OF CRACK PROPAGATION IN THIN STRUCTURES

Crack propagation in thin shell structures due to cutting is conveniently simulated using explicit finite element approaches, in view of the high nonlinearity of the problem. Solidshell elements are usually preferred for the discretization in the presence of complex material behavior and degradation phenomena such as delamination, since they allow for a correct representation of the thickness geometry. However, in solid-shell elements the small thickness leads to a very high maximum eigenfrequency, which imply very small stable time-steps. A new selective mass scaling technique is proposed to increase the time-step size without affecting accuracy. New ”directional” cohesive interface elements are used in conjunction with selective mass scaling to account for the interaction with a sharp blade in cutting processes of thin ductile shells

Numerical simulation of solid deformation driven by creeping flow using an immersed finite element method

An immersed finite element method for solid–fluid interaction is presented with application focus on highly deformable elastic bodies in a Stokes flow environment. The method is based on a global balance equation which combines the solid and fluid momentum balances, the fluid mass balance and, in weak form, the interface conditions. By means of an Updated Lagrangian description for finite elasticity, only one analysis mesh is used, where the solid particles are backtracked in order to preserve the deformation history. The method results in a full coupling of the solid-fluid system which is solved by an exact Newton method. The location of the material interface is captured by a signed distance function and updated according to the computed displacement increments and the help of an explicit surface parameterisation; no body-fitted volume meshes are needed. Special emphasis is placed on the accurate integration of finite elements traversed by the interface and the related numerical stability of the shape function basis. A number of applications for compressible Neo-Hookean solids subject to creeping flow are presented, motivated by microfluidic experimentation in mechanobiology

Mixed finite element approximation of porous media flows

The reliable simulation of flow in fractured porous media is a key aspect in the decision making process of stakeholders within politics and the geosciences, for example when assessing the suitability of burial sites for storage of high–level radioactive waste. This thesis aims to tackle the challenge that is the accurate simulation of these flows and does so via three computational developments. That is, suitable models for porous media flow with fractures; obtaining rigorous and reliable estimates of errors generated through these models; and the accurate simulation of the times–of–flight for particles transported by groundwater within the porous medium. Firstly, an expansion procedure for fractures in porous media is developed so that physical fluid laws are still retained when tracking particles across fracture–bulk interfaces. Moreover, the second contribution of this work is the utilisation of the dual–weighted–residual method to define suitable elementwise indicators for generic quantities of interest. The third contribution of this thesis is the attainment of accurate simulations of travel times for particles in porous media, achieved through linearising the functional representing the time–of–flight; in practice, numerical examples, including one inspired by the Sellafield site in Cumbria, UK, validate the performance of the proposed error estimator, and hence are useful in the safety assessment of storage facilities intended for radioactive waste

Mixed finite element approximation of porous media flows

The reliable simulation of flow in fractured porous media is a key aspect in the decision making process of stakeholders within politics and the geosciences, for example when assessing the suitability of burial sites for storage of high–level radioactive waste. This thesis aims to tackle the challenge that is the accurate simulation of these flows and does so via three computational developments. That is, suitable models for porous media flow with fractures; obtaining rigorous and reliable estimates of errors generated through these models; and the accurate simulation of the times–of–flight for particles transported by groundwater within the porous medium. Firstly, an expansion procedure for fractures in porous media is developed so that physical fluid laws are still retained when tracking particles across fracture–bulk interfaces. Moreover, the second contribution of this work is the utilisation of the dual–weighted–residual method to define suitable elementwise indicators for generic quantities of interest. The third contribution of this thesis is the attainment of accurate simulations of travel times for particles in porous media, achieved through linearising the functional representing the time–of–flight; in practice, numerical examples, including one inspired by the Sellafield site in Cumbria, UK, validate the performance of the proposed error estimator, and hence are useful in the safety assessment of storage facilities intended for radioactive waste

Penalty-free Nitsche method for interface problems in computational mechanics

Nitsche’s method is a penalty-based method to enforce weakly the boundary conditions in the finite element method. In this thesis, we consider a penalty-free version of Nitsche’s method, we prove its stability and convergence in various frameworks. The idea of the penalty-free method comes from the nonsymmetric version of the Nitsche’s method where the penalty parameter has been set to zero; it can be seen as a Lagrange multiplier method, where the Lagrange multiplier has been replaced by the boundary fluxes of the discrete elliptic operator. The main observation is that although coercivity fails, inf-sup stability can be proven. The study focuses on compressible and incompressible elasticity. An unfitted framework is considered when the computational mesh does not fit with the physical domain (fictitious domain method). The penalty-free Nitsche’s method is also used to enforce the coupling for interface problems when the mesh fits the interface (nonconforming domain decomposition) or not (unfitted domain decomposition). Fluid structure interaction is also investigated, a new fully discrete implicit scheme is introduced