Discrete weak duality of hybrid high-order methods for convex minimization problems

This paper derives a discrete dual problem for a prototypical hybrid high-order method for convex minimization problems. The discrete primal and dual problem satisfy a weak convex duality that leads to a priori error estimates with convergence rates under additional smoothness assumptions. This duality holds for general polytopal meshes and arbitrary polynomial degree of the discretization. A nouvelle postprocessing is proposed and allows for a~posteriori error estimates on simplicial meshes using primal-dual techniques. This motivates an adaptive mesh-refining algorithm, which performs superiorly compared to uniform mesh refinements

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

Numerical simulation for viscoplastic fluids via finite element methods

The design of efficient, robust and flexible numerical schemes to cope with nonlinear CFD problems has become the main nerve in the field of numerical simulation. This work has developed and analyzed the Newton-Multigrid process in the frame of monolithic approaches to solve stationary and nonstationary viscoplastic fluid problems. From the mathematical point of view, the viscoplastic problem exhibits several severe problems which might be arisen to draw the mathematical challenges. The major difficulty is the unbounded value of the viscosity which needs regularization. Several regularization techniques have been proposed to cope with this problem yet, while the accuracy is still not even close to be compared to the real model. Herein, two methods are used for the treatment of the non-differentiability, namely Bercovier-Engelman and modified bi-viscous models regularizations. To compute the solution at very small values of the regularization parameter which can be considered numerically as zero, we use the continuation technique. Other difficulties would be addressed in the circle of the nonlinearity, the solenoidal velocity field, as well as the convection dominated problem which are typically involved in the standard Navier-Stokes equation. The use of mixed higher order finite element methods for flow problems is advantageous, since one can partially avoid the addition of stabilization terms to handle for instance the lack of coercivity, the domination of the convective part as well as the incompressibility. In the case of mixed lower order finite element methods, edge oriented stabilization has been introduced to provide results in the case of the lack of coercivity and convection dominated problems. The main drawback of this stabilizer is to optimize or choose appropriately the free parameters to maintain high accuracy results from the scheme. Viscoplastic fluids are involved in many industrial applications which require numerical simulation to get a big mathematical insight and to predict the fluids behavior. The dependence of pressure on the viscoplastic constitutive law is confirmed as much as the dependence of velocity. Moreover, the behavior of the pressure is strongly related to the yield property for the unyielded regimes. In the case of a constant yield stress value together with the absence of the external densities, the field of pressure is prescribed by the null value wherever the null value of the deformation tensor is considered. Real life examples to prescribe the behavior of the viscoplastic fluids might be described in case of standard benchmarks: viscoplastic flow in channel, viscoplastic flow in a lid driven cavity and viscoplastic flow around a cylinder. In each case we confirm the experimental and theoretical results which are used to analyze viscoplastic problems for the physical behavior with respect to the unyielded regimes and the cessation of time

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

Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference

The 6th ECCOMAS Young Investigators Conference YIC2021 will take place from July 7th through 9th, 2021 at Universitat Politècnica de València, Spain. The main objective is to bring together in a relaxed environment young students, researchers and professors from all areas related with computational science and engineering, as in the previous YIC conferences series organized under the auspices of the European Community on Computational Methods in Applied Sciences (ECCOMAS). Participation of senior scientists sharing their knowledge and experience is thus critical for this event.YIC 2021 is organized at Universitat Politécnica de València by the Sociedad Española de Métodos Numéricos en Ingeniería (SEMNI) and the Sociedad Española de Matemática Aplicada (SEMA). It is promoted by the ECCOMAS.The main goal of the YIC 2021 conference is to provide a forum for presenting and discussing the current state-of-the-art achievements on Computational Methods and Applied Sciences,including theoretical models, numerical methods, algorithmic strategies and challenging engineering applications.Nadal Soriano, E.; Rodrigo Cardiel, C.; Martínez Casas, J. (2022). Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference. Editorial Universitat Politècnica de València. https://doi.org/10.4995/YIC2021.2021.15320EDITORIA

Efficient FEM solver for quasi-Newtonian flow problems with application to granular materials

This thesis is concerned with new numerical and algorithmic tools for flows with pressure and shear dependent viscosity together with the necessary background of the generalized Navier-Stokes equations. In general the viscosity of a material can be constant, e.g. water and this kind of fluid is called as Newtonian fluid. However the flow can be complicated for quasi-Newtonian fluid, where the viscosity can depend on some physical quantity. For example, the viscosity of Bingham fluid is a function of the shear rate. Moreover even further complications can arise when the dependencies of both shear rate and pressure occur for the viscosity as in the case of the granular materials, e.g. Poliquen model. The Navier-Stokes equations in primitive variables (velocity-pressure) are regarded as the privilege answer to incorporate these phenomena. The modification of the viscous stresses leads to generalized Navier-Stokes equations extending the range of their validity to such flow. The resulting equations are mathematically more complex than the Navier-Stokes equations and several problems arise from the numerical point of view. Firstly, the difficulty of approximating incompressible velocity fields and secondly, poor conditioning and possible lack of differentiability of the involved nonlinear functions due to the material laws. The difficulty related to the approximation of incompressible velocity fields is treated by applying the conforming Stokes element Q2/P1 and the lack of differentiability is taken care of by regularization. Then the continuous Newton method as linearization technique is applied and the method consists of working directly on the variational integrals. Next the corresponding continuous Jacobian operators are derived and consequently a convergence rate of the nonlinear iterations independent of the mesh refinement is achieved. This continuous approach is advantageous: Firstly the explicit accessibility of the Jacobian allows a robust method with respect to the starting guess and secondly it avoids the delicate task of choosing the step-length which is required for divided differences approaches. We denote the full Jacobian matrix on the discrete level by A and separate it into two parts: A1 and A2 corresponding to Fixed point and Newton method respectively. A fundamental issue for the continuous Newton method arises when the problem is not ready for it at the initial state due to the poor condition of the 'bad-part' A2 of the Jacobian. Although the Newton method is popular for its local quadratic convergence behavior, however the solver may show unpredictable and undesirable divergent behavior if A2 is poor conditioned. This particular difficulty is handled by our Adaptive Newton method, where we introduce a charateristic function f(Qn), which depends solely on the relative residual change Qn and controls the weighing parameter δn for the 'bad-part' A2 resulting in the swinging back and forth of the solver between Fixed point and Newton state. Finally the new Adaptive Newton method is validated for the Bingham fluid for the benchmark geometry Flow around cylinder and a test case of 2D Couette flow for (modified) Poliquen model having the scope of real world applications is studied to fulfill the objective need of performance

High order ADER-DG schemes for the simulation of linear seismic waves induced by nonlinear dispersive free-surface water waves

In this paper, we propose a unified and high order accurate fully-discrete one-step ADER Discontinuous Galerkin method for the simulation of linear seismic waves in the sea bottom that are generated by the propagation of free surface water waves. A hyperbolic reformulation of the Serre-Green-Naghdi model for nonlinear dispersive free surface flows is coupled with a first order velocity-stress formulation for linear elastic wave propagation in the sea bottom. Cartesian non-conforming meshes are defined and the coupling is achieved by an appropriate time-dependent pressure boundary condition in the three-dimensional domain for the elastic wave propagation, where the pressure is a combination of hydrostatic and non-hydrostatic pressure in the water column above the sea bottom. The use of a first order hyperbolic reformulation of the nonlinear dispersive free surface flow model leads to a straightforward coupling with the linear seismic wave equations, which are also written in first order hyperbolic form. It furthermore allows the use of explicit time integrators with a rather generous CFL-type time step restriction associated with the dispersive water waves, compared to numerical schemes applied to classical dispersive models. Since the two systems employed are written in the same form of a first order hyperbolic system they can also be efficiently solved in a unique numerical framework. We choose the family of arbitrary high order accurate discontinuous Galerkin finite element schemes. The developed methodology is carefully assessed by first considering several benchmarks for each system separately showing a good agreement with exact and numerical reference solutions. Finally, also coupled test cases are addressed. Throughout this paper we assume the elastic deformations in the solid to be sufficiently small so that their influence on the free surface water waves can be neglected