Finite volume scheme based on cell-vertex reconstructions for anisotropic diffusion problems with discontinuous coefficients

We propose a new second-order finite volume scheme for non-homogeneous and anisotropic diffusion problems based on cell to vertex reconstructions involving minimization of functionals to provide the coefficients of the cell to vertex mapping. The method handles complex situations such as large preconditioning number diffusion matrices and very distorted meshes. Numerical examples are provided to show the effectiveness of the method

Monotonic solution of heterogeneous anisotropic diffusion problems

Anisotropic problems arise in various areas of science and engineering, for example groundwater transport and petroleum reservoir simulations. The pure diffusive anisotropic time-dependent transport problem is solved on a finite number of nodes, that are selected inside and on the boundary of the given domain, along with possible internal boundaries connecting some of the nodes. An unstructured triangular mesh, that attains the Generalized Anisotropic Delaunay condition for all the triangle sides, is automatically generated by properly connecting all the nodes, starting from an arbitrary initial one. The control volume of each node is the closed polygon given by the union of the midpoint of each side with the "anisotropic" circumcentre of each final triangle. A structure of the flux across the control volume sides similar to the standard Galerkin Finite Element scheme is derived. A special treatment of the flux computation, mainly based on edge swaps of the initial mesh triangles, is proposed in order to obtain a stiffness M-matrix system that guarantees the monotonicity of the solution. The proposed scheme is tested using several literature tests and the results are compared with analytical solutions, as well as with the results of other algorithms, in terms of convergence order. Computational costs are also investigate

Discontinuous Galerkin approximations in computational mechanics: hybridization, exact geometry and degree adaptivity

Discontinuous Galerkin (DG) discretizations with exact representation of the geometry and local polynomial degree adaptivity are revisited. Hybridization techniques are employed to reduce the computational cost of DG approximations and devise the hybridizable discontinuous Galerkin (HDG) method. Exact geometry described by non-uniform rational B-splines (NURBS) is integrated into HDG using the framework of the NURBS-enhanced finite element method (NEFEM). Moreover, optimal convergence and superconvergence properties of HDG-Voigt formulation in presence of symmetric second-order tensors are exploited to construct inexpensive error indicators and drive degree adaptive procedures. Applications involving the numerical simulation of problems in electrostatics, linear elasticity and incompressible viscous flows are presented. Moreover, this is done for both high-order HDG approximations and the lowest-order framework of face-centered finite volumes (FCFV).Peer ReviewedPostprint (author's final draft

Monotonic solution of flow and transport problems in heterogeneous media using Delaunay unstructured triangular meshes

Transport problems occurring in porous media and including convection, diffusion and chemical reactions, can be well represented by systems of Partial Differential Equations. In this paper, a numerical procedure is proposed for the fast and robust solution of flow and transport problems in 2D heterogeneous saturated media. The governing equations are spatially discretized with unstructured triangular meshes that must satisfy the Delaunay condition. The solution of the flow problem is split from the solution of the transport problem and it is obtained with an approach similar to the Mixed Hybrid Finite Elements method, that always guarantees the M-property of the resulting linear system. The transport problem is solved applying a prediction/correction procedure. The prediction step analytically solves the convective/reactive components in the context of a MAST Finite Volume scheme. The correction step computes the anisotropic diffusive components in the context of a recently proposed Finite Elements scheme. Massa balance is locally and globally satisfied in all the solution steps. Convergence order and computational costs are investigated and model results are compared with literature on

The mixed virtual element discretization for highly-anisotropic problems: the role of the boundary degrees of freedom

In this paper, we discuss the accuracy and the robustness of the mixed Virtual Element Methods when dealing with highly anisotropic diffusion problems. In particular, we analyze the performance of different approaches which are characterized by different sets of both boundary and internal degrees of freedom in the presence of a strong anisotropy of the diffusion tensor with constant or variable coefficients. A new definition of the boundary degrees of freedom is also proposed and tested

Experiments and modeling of the chemo-mechanically coupled behavior of polymeric gels

Polymeric materials consist of mutually entangled or chemically crosslinked long njitmolecular chains which form a polymer network. Due to their molecular structure, the njitpolymeric materials are known to undergo large deformation in response to various njitenvironmental stimuli, such as temperature, chemical potential and light. When a polymer network is exposed to a suitable chemical solvent, the solvent molecules are able to diffuse inside the network, causing it to undergo a large volumetric deformation, known as swelling. In addition to volumetric deformation, this process involves the chemical mixing of the polymer network and solvent molecules, and is typically environmentally responsive. A polymeric material in this mixed and swollen state is known as a polymeric gel. Swollen polymers, or polymeric gels, find their application in the oil industry, soft robotics, drug delivery and microfluidic channels. Moreover, most of the organs inside our body are gel-like in structure, which makes this class of materials important for biomedical applications and tissue engineering. An important distinction between biological tissues and much of the previous literature on the mechanics of polymeric gels is that most biological tissues contain fibers. The existence of these fibers embedded in the material, causes the properties to be significantly different along the fiber direction. Recent years have seen the development of a vast number of large deformation continuum-level constitutive models aimed to capture the coupled diffusion-deformation behavior of polymeric gels. However, there is an insufficient amount of experimental data to complement such theoretical research. Thus, despite numerous potential applications, many aspects of polymeric gel behavior remain elusive. In addition, the diffusion-deformation behavior is known to be affected by the external stimuli. In the current state of the art there is a lack of theoretical models and robust simulation capabilities to account for the influence of such stimuli, hindering further advances in technologies involving polymeric gels. The purpose of this research is to bridge the gap between the experimental and theoretical studies, and provide reliable finite element simulation capabilities for polymeric gels. More specifically, the aim is to (i) experimentally characterize the behavior of polymeric gels, (ii) develop new experimentally motivated constitutive models and (iii) implement the models numerically for use in a finite element software. The final result of this research is a robust finite element method (FEM) code that can be used for simulations in the commercial software package Abaqus. Towards the goal, an experimental procedure is designed to thoroughly investigate the behavior of polymeric gels, and provide a direction for the development of novel constitutive models. The procedure involves mechanical testing of dry polymeric material, free swelling with suitable solvents, and mechanical testing when fully swollen. The experimental observations provide transformational insights in the mechanical behavior of polymeric gels, and are utilized to develop a continuum-level constitutive model. Further, the presence of embedded fibers in a swellable polymer matrix leads to anisotropy in the overall behavior. In order to capture this response, a constitutive model for fiber-reinforced polymeric gels is developed, that explicitly takes into account anisotropy in both the mechanical and diffusive behavior. The constitutive model is implemented as user element subroutine (UEL) in the commercial finite element software package Abaqus/Standard. Numerical simulations are performed to show the behavior of the model, and qualitative comparisons are made to experiments of a soft robotic gripper. In addition, many polymeric gels are known to respond or activate when exposed to a light stimulus. This light-driven alteration of the behavior is known to be caused by the photochemical reactions occurring inside the polymer network. Thus, the overall response of light-activated polymeric gels is affected by the mechanical stress, solvent content, and the extent of photochemical reaction caused by light irradiation. To account for such response of a polymeric gel, a continuum level constitutive model is developed and numerically implemented in Abaqus/Standard as a user element (UEL) subroutine