Locally adaptive phase-field models and transition to fracture

This thesis proposes a new computational model for the efficient simulation of crack propagation, through the combination of a phase-field model in small subdomains around crack tips and a discontinuous model in the rest of the domain. The combined model inherits the advantages of both approaches. The phase-field model determines crack propagation at crack tips, and the discontinuous model explicitly describes the crack elsewhere, enabling to use a coarser discretization and thus reducing the computational cost. In crack-tip subdomains, the discretization is refined to capture the phase-field solution, while in the discontinuous part, sharp cracks are incorporated into the coarse background discretization by the eXtended Finite Element Method (XFEM). As crack-tip subdomains move with crack growth, the discretization is automatically updated and phase-field bands are replaced by sharp cracks in the wake of cracks. The first step is the development of an adaptive refinement strategy for phase-field models. To this end, two alternatives are proposed. Both of them consider two types of elements, standard and refined, which are mapped into a fixed background mesh. In refined elements, the space of approximation is uniformly $h$-refined. Continuity between elements of different type is imposed in weak form to handle the non-conformal approximations in a natural way, without spreading of refinement nor having to deal with hanging nodes, leading to a very local refinement along cracks. The first adaptive strategy relies on a Hybridizable Discontinuous Galerkin (HDG) formulation of the problem, in which continuity between elements is imposed in weak form. The second one is based on a more efficient Continuous Galerkin (CG) formulation; a continuous FEM approximation is used in the standard and refined regions and, then, continuity on the interface between regions is imposed in weak form by Nitsche's method. The proposed strategies robustly refine the discretization as cracks propagate and can be easily incorporated into a working code for phase-field models. However, the computational cost can be further reduced by transitioning to the discontinuous in the combined model. In the wake of crack tips, the phase-field diffuse cracks are replaced by XFEM discontinuous cracks and elements are derefined. The combined model is studied within the adaptive CG formulation. Numerical experiments include branching and coalescence of cracks, and a fully 3D test.En aquesta tesi es proposa un nou model computacional per a simular la propagació de fractures de manera eficient, a partir de la combinació d’un model de camp de fase en petits subdominis al voltant dels extrems de les fissures, i d’un model discontinu a la resta del domini. El model combinat manté els avantatges de tots dos tipus de model. El model continu determina la propagació de la fissura, i el model discontinu descriu explícitament la fissura en gairebé tot del domini, amb una discretització més grollera i el conseqüent estalvi en cost computacional. Als subdominis de camp de fase, la discretització es refina per tal d’aproximar bé la solució, mentre que a la part discontínua, les fissures s’incorporen a la discretització grollera a partir de l’eXtended Finite Element Method (XFEM). A mesura que les fissures es propaguen pel domini, la discretització s’actualitza automàticament i, lluny dels extrems, la representació suavitzada de les fissures a partir del camp de fase es reemplaça per una representació discontínua. El primer pas és definir una estratègia de refinament adaptatiu pels models continus de camp de fase. En aquesta tesi es proposen dues alternatives diferents. Totes dues consideren dos tipus d’elements, estàndards i refinats, que es mapen a la malla inicial. Als elements refinats, l’espai d’aproximació es refina uniformement. La continuïtat entre elements de tipus diferent s’imposa en forma feble per facilitar el tractament de les aproximacions no conformes, sense que s’escampi el refinament ni haver d’imposar restriccions als nodes de la interfície, donant lloc a un refinament molt localitzat. La primera estratègia adaptativa es basa en una formulació Hybridizable Discontinuous Galerkin (HDG) del problema, que imposa continuïtat entre elements en forma feble. La segona es basa en una formulació contínua més eficient; es fa servir una aproximació contínua del Mètode dels Elements Finits a les regions estàndards i refinades i, aleshores, a la interfície entre les dues regions s’imposa la continuïtat en forma feble amb el mètode de Nitsche. Les estratègies adaptatives refinen la discretització a mesura que les fissures es propaguen, i es poden afegir a un codi per a models de camp de fase de manera senzilla. No obstant, el cost computacional es pot reduir encara més fent servir el model combinat. Lluny dels extrems de les fissures, la representació suavitzada del camp de fase es substitueix per discontinuïtats en una discretització de XFEM, i els elements es desrefinen. El model combinat es formula a partir de l’estratègia adaptativa contínua. Els exemples numèrics inclouen bifurcació i coalescència de fissures, i un exemple en 3D

The diffuse Nitsche method: Dirichlet constraints on phase-field boundaries

We explore diffuse formulations of Nitsche's method for consistently imposing Dirichlet boundary conditions on phase-field approximations of sharp domains. Leveraging the properties of the phase-field gradient, we derive the variational formulation of the diffuse Nitsche method by transferring all integrals associated with the Dirichlet boundary from a geometrically sharp surface format in the standard Nitsche method to a geometrically diffuse volumetric format. We also derive conditions for the stability of the discrete system and formulate a diffuse local eigenvalue problem, from which the stabilization parameter can be estimated automatically in each element. We advertise metastable phase-field solutions of the Allen-Cahn problem for transferring complex imaging data into diffuse geometric models. In particular, we discuss the use of mixed meshes, that is, an adaptively refined mesh for the phase-field in the diffuse boundary region and a uniform mesh for the representation of the physics-based solution fields. We illustrate accuracy and convergence properties of the diffuse Nitsche method and demonstrate its advantages over diffuse penalty-type methods. In the context of imaging based analysis, we show that the diffuse Nitsche method achieves the same accuracy as the standard Nitsche method with sharp surfaces, if the inherent length scales, i.e., the interface width of the phase-field, the voxel spacing and the mesh size, are properly related. We demonstrate the flexibility of the new method by analyzing stresses in a human vertebral body

Meshless methods for shear-deformable beams and plates based on mixed weak forms

Thin structural theories such as the shear-deformable Timoshenko beam and Reissner-Mindlin plate theories have seen wide use throughout engineering practice to simulate the response of structures with planar dimensions far larger than their thickness dimension. Meshless methods have been applied to construct numerical methods to solve the shear deformable theories. Similarly to the finite element method, meshless methods must be carefully designed to overcome the well-known shear-locking problem. Many successful treatments of shear-locking in the finite element literature are constructed through the application of a mixed weak form. In the mixed weak form the shear stresses are treated as an independent variational quantity in addition to the usual displacement variables. We introduce a novel hybrid meshless-finite element formulation for the Timoshenko beam problem that converges to the stable first-order/zero-order finite element method in the local limit when using maximum entropy meshless basis functions. The resulting formulation is free from the effects shear-locking. We then consider the Reissner-Mindlin plate problem. The shear stresses can be identified as a vector field belonging to the Sobelov space with square integrable rotation, suggesting the use of rotated Raviart-Thomas-Nedelec elements of lowest-order for discretising the shear stress field. This novel formulation is again free from the effects of shear-locking. Finally we consider the construction of a generalised displacement method where the shear stresses are eliminated prior to the solution of the final linear system of equations. We implement an existing technique in the literature for the Stokes problem called the nodal volume averaging technique. To ensure stability we split the shear energy between a part calculated using the displacement variables and the mixed variables resulting in a stabilised weak form. The method then satisfies the stability conditions resulting in a formulation that is free from the effects of shear-locking.Open Acces

Direct immersogeometric fluid flow analysis using B-rep CAD models

We present a new method for immersogeometric fluid flow analysis that directly uses the CAD boundary representation (B-rep) of a complex object and immerses it into a locally refined, non-boundary-fitted discretization of the fluid domain. The motivating applications include analyzing the flow over complex geometries, such as moving vehicles, where the detailed geometric features usually require time-consuming, labor-intensive geometry cleanup or mesh manipulation for generating the surrounding boundary-fitted fluid mesh. The proposed method avoids the challenges associated with such procedures. A new method to perform point membership classification of the background mesh quadrature points is also proposed. To faithfully capture the geometry in intersected elements, we implement an adaptive quadrature rule based on the recursive splitting of elements. Dirichlet boundary conditions in intersected elements are enforced weakly in the sense of Nitsche\u27s method. To assess the accuracy of the proposed method, we perform computations of the benchmark problem of flow over a sphere represented using B-rep. Quantities of interest such as drag coefficient are in good agreement with reference values reported in the literature. The results show that the density and distribution of the surface quadrature points are crucial for the weak enforcement of Dirichlet boundary conditions and for obtaining accurate flow solutions. Also, with sufficient levels of surface quadrature element refinement, the quadrature error near the trim curves becomes insignificant. Finally, we demonstrate the effectiveness of our immersogeometric method for high-fidelity industrial scale simulations by performing an aerodynamic analysis of an agricultural tractor directly represented using B-rep

An X-FEM and Level Set computational approach for image-based modeling. Application to homogenization.

International audienceThe advances in material characterization by means of imaging techniques require powerful computational methods for numerical analysis. The present contribution focuses on highlighting the advantages of coupling the Extended Finite Elements Method (X-FEM) and the level sets method, applied to solve microstructures with complex geometries. The process of obtaining the level set data starting from a digital image of a material structure and its input into an extended finite element framework is presented. The coupled method is validated using reference examples and applied to obtain homogenized properties for heterogeneous structures. Although the computational applications presented here are mainly two dimensional, the method is equally applicable for three dimensional problems

Development of discontinuous Galerkin method for nonlocal linear elasticity

Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2007.Includes bibliographical references (p. 75-81).A number of constitutive theories have arisen describing materials which, by nature, exhibit a non-local response. The formulation of boundary value problems, in this case, leads to a system of equations involving higher-order derivatives which, in turn, results in requirements of continuity of the solution of higher order. Discontinuous Galerkin methods are particularly attractive toward this end, as they provide a means to naturally enforce higher interelement continuity in a weak manner without the need of modifying the finite element interpolation. In this work, a discontinuous Galerkin formulation for boundary value problems in small strain, non-local linear elasticity is proposed. The underlying theory corresponds to the phenomenological strain-gradient theory developed by Fleck and Hutchinson within the Toupin-Mindlin framework. The single-field displacement method obtained enables the discretization of the boundary value problem with a conventional continuous interpolation inside each finite element, whereas the higher-order interelement continuity is enforced in a weak manner. The proposed method is shown to be consistent and stable both theoretically and with suitable numerical examples.by Ram Bala Chandran.S.M

A Hierarchical Multiscale Method for Nonlocal Fine-scale Models via Merging Weak Galerkin and VMS Frameworks

Many problems in natural sciences and engineering involve phenomena that possess a wide spectrum of material, spatial and temporal scales. Most often these scales are nonlocal and therefore pose great challenge to current computational techniques. Of specific interest to the objectives of this proposal are advection dominated viscous flows leading to anisotropic turbulence where all the scales are nonlocal and they concurrently interact all across. Another class of problems from mathematical physics that we address involves propagating steep fronts wherein interacting discontinuities in the underlying physical fields challenge the stability of the numerical methods. Modeling such problems raises open mathematical questions: How should the information obtained from a model at one level be incorporated into a model at a different level? And secondly, how should these scales be made to communicate seamlessly if both coarse and fine scales are inherently non-local? Focus of the proposed research is a unifying mathematical framework for the development of robust numerical methods in two areas of Computational Fluid Dynamics (CFD): (i) Methods for anisotropic turbulence that are derived consistently from the Navier-Stokes equations via facilitating two-way coupling between global and local scales, and (i) Methods that have sound variational structures for the modeling of problems with steep gradients and propagating discontinuities. Emphasis is placed throughout on variationally consistent interscale coupling with rigorous treatment of the continuity conditions that are critical for the mathematical and algorithmic stability.NSF - DMS-16-20231, 2016-17Ope

Rotation-Based Mixed Formulations for an Elasticity-Poroelasticity Interface Problem

In this paper we introduce a new formulation for the stationary poroelasticity equations written using the rotation vector and the total fluid-solid pressure as additional unknowns, and we also write an extension to the elasticity-poroelasticity problem. The transmission conditions are imposed naturally in the weak formulation, and the analysis of the effective governing equations is conducted by an application of Fredholm's alternative. We also propose a monolithically coupled mixed finite element method for the numerical solution of the problem. Its convergence properties are rigorously derived and subsequently confirmed by a set of computational tests that include applications to subsurface flow in reservoirs as well as to dentistry-oriented problems.Fondo Nacional de Desarrollo Científico y Tecnológico/[11160706]/FONDECYT/ChilePrograma Concurso Apoyo a Centros Científicos y Tecnológicos de Excelencia/[AFB170001]/PIA/ChileUCR::Sedes Regionales::Sede de OccidenteUCR::Vicerrectoría de Docencia::Ciencias Básicas::Facultad de Ciencias::Escuela de Matemátic

Adaptive meshing methodology based on topometric data for dambreak inundation assessments

Flood maps are the final products of dam failure studies that are required by dam safety regulations. A flood limit, which represents the maximum envelope reached by flood waves, is generally the result of a dam-break scenario simulated by a hydraulic numerical model. However, the numerical model uses only a limited portion of the available bathymetry data to build the terrain model (2D mesh plus topometric elevation at nodes). This is particularly so in the cases where the topo-metric data recorded by LIDAR was estimated in several million points. But the hydraulic numerical models rarely exceed hundreds of thousands of nodes, in particular because of the computer constraints and time associated with the operation of these models. The production of the final flood map requires consistency between projected levels and elevations for all points on the map. This verification may be tedious for a large area with several small secondary valleys of tributary streams that have not been represented by the original hydraulic numerical model. The aim of this work is to propose an automatic remeshing strategy that uses the envelope of the maximum dimensions reached by the original model coupled with the available LIDAR data to produce an improved mesh that can accurately capture the wet/dry fronts and the overflows of the secondary valleys. This model helps us to consider the maximum slope inside each element on the basis of the real data, instead of controlling the slope for not having negative depth or controlling the velocity. The algorithm is based on a few basic steps: (i) find the elements cut by the envelope of the wet/dry interfaces; (ii) project the topometric points onto the cut elements; (iii) if these points are very close to the interface, if they are found in a valley, or if they are more elevated than the corresponding cut elements, then these points will be added to the previous nodes and included in a subsequent triangulation step; and (iv) re-run the simulation on the new mesh. This algorithm has been implemented and validated in the study of a dambreak flow with a complex river topography on the Eastmain River and the Romaine-Puyjalon River