On pressure robustness and independent determination of displacement and pressure in incompressible linear elasticity

We investigate the possibility to determine the divergence-free displacement $\mathbf{u}$ \emph{independently} from the pressure reaction $p$ for a class of boundary value problems in incompressible linear elasticity. If not possible, we investigate if it is possible to determine it \emph{pressure robustly}, i.e. pollution free from the pressure reaction. For convex domains there is but one variational boundary value problem among the investigated that allows the independent determination. It is the one with essential no-penetration conditions combined with homogeneous tangential traction conditions. Further, in most but not all investigated cases, the weakly divergence-free displacement can be computed pressure robustly provided the total body force is decomposed into its direct sum of divergence- and rotation-free components using a Helmholtz decomposition. The elasticity problem is solved using these components as separate right-hand sides. The total solution is obtained using the superposition principle. We employ a $(\mathbf{u},p)$ higher-order finite element formulation with discontinuous pressure elements. It is \emph{inf-sup} stable for polynomial degree $p\ge 2$ but not pressure robust by itself. We propose a three step procedure to solve the elasticity problem preceded by the Helmholtz decomposition of the total body force. The extra cost for the three-step procedure is essentially the cost for the Helmholtz decomposition of the assembled total body force, and the small cost of solving the elasticity problem with one extra right-hand side. The results are corroborated by theoretical derivations as well as numerical results.Comment: 31 pages, 31 references and 48 figure

An anisotropic mesh adaptation method for the finite element solution of heterogeneous anisotropic diffusion problems

Heterogeneous anisotropic diffusion problems arise in the various areas of science and engineering including plasma physics, petroleum engineering, and image processing. Standard numerical methods can produce spurious oscillations when they are used to solve those problems. A common approach to avoid this difficulty is to design a proper numerical scheme and/or a proper mesh so that the numerical solution validates the discrete counterpart (DMP) of the maximum principle satisfied by the continuous solution. A well known mesh condition for the DMP satisfaction by the linear finite element solution of isotropic diffusion problems is the non-obtuse angle condition that requires the dihedral angles of mesh elements to be non-obtuse. In this paper, a generalization of the condition, the so-called anisotropic non-obtuse angle condition, is developed for the finite element solution of heterogeneous anisotropic diffusion problems. The new condition is essentially the same as the existing one except that the dihedral angles are now measured in a metric depending on the diffusion matrix of the underlying problem. Several variants of the new condition are obtained. Based on one of them, two metric tensors for use in anisotropic mesh generation are developed to account for DMP satisfaction and the combination of DMP satisfaction and mesh adaptivity. Numerical examples are given to demonstrate the features of the linear finite element method for anisotropic meshes generated with the metric tensors.Comment: 34 page

A Two-stage Method for Inverse Medium Scattering

We present a novel numerical method to the time-harmonic inverse medium scattering problem of recovering the refractive index from near-field scattered data. The approach consists of two stages, one pruning step of detecting the scatterer support, and one resolution enhancing step with mixed regularization. The first step is strictly direct and of sampling type, and faithfully detects the scatterer support. The second step is an innovative application of nonsmooth mixed regularization, and it accurately resolves the scatterer sizes as well as intensities. The model is efficiently solved by a semi-smooth Newton-type method. Numerical results for two- and three-dimensional examples indicate that the approach is accurate, computationally efficient, and robust with respect to data noise.Comment: 18 pages, 5 figure

Modeling of bone conduction of sound in the human head using hp-finite elements: Code design and verification

We focus on the development of a reliable numerical model for investigating the bone-conduction of sound in the human head. The main challenge of the problem is the lack of fundamental knowledge regarding the transmission of acoustic energy through non-airborne pathways to the cochlea. A fully coupled model based on the acoustic/elastic interaction problem with a detailed resolution of the cochlea region and its interface with the skull and the air pathways, should provide an insight into this fundamental, long standing research problem. To this aim we have developed a 3D hp-finite element code that supports elements of all shapes (tetrahedra, prisms and pyramids) to better capture the geometrical features of the head. We have tested the code on a multilayered sphere and employed it to solve an idealized model of head. In the future we hope to attack a model with a more realistic geometry

Resolving the sign conflict problem for hp–hexahedral Nédélec elements with application to eddy current problems

The eddy current approximation of Maxwell’s equations is relevant for Magnetic Induction Tomography (MIT), which is a practical system for the detection of conducting inclusions from measurements of mutual inductance with both industrial and clinical applications. An MIT system produces a conductivity image from the measured fields by solving an inverse problem computationally. This is typically an iterative process, which requires the forward solution of a Maxwell’s equations for the electromagnetic fields in and around conducting bodies at each iteration. As the (conductivity) images are typically described by voxels, a hexahedral finite element grid is preferable for the forward solver. Low order Nédélec (edge element) discretisations are generally applied, but these require dense meshes to ensure that skin effects are properly captured. On the other hand, hp–Nédélec finite elements can ensure the skin effects in conducting components are accurately captured, without the need for dense meshes and, therefore, offer possible advantages for MIT. Unfortunately, the hierarchic nature of hp–Nédélec basis functions introduces edge and face parameterisations leading to sign conflict issues when enforcing tangential continuity between elements. This work describes a procedure for addressing this issue on general conforming hexahedral meshes and an implementation of a hierarchic hp–Nédélec finite element basis within the deal.II finite element library. The resulting software is used to simulate Maxwell forward problems, including those set on multiply connected domains, to demonstrate its potential as an MIT forward solver

Efficient Large Scale Electromagnetics Simulations Using Dynamically Adapted Meshes with the Discontinuous Galerkin Method

A framework for performing dynamic mesh adaptation with the discontinuous Galerkin method (DGM) is presented. Adaptations include modifications of the local mesh step size (h-adaptation) and the local degree of the approximating polynomials (p-adaptation) as well as their combination. The computation of the approximation within locally adapted elements is based on projections between finite element spaces (FES), which are shown to preserve an upper limit of the electromagnetic energy. The formulation supports high level hanging nodes and applies precomputation of surface integrals for increasing computational efficiency. Error and smoothness estimates based on interface jumps are presented and applied to the fully hp-adaptive simulation of two examples in one-dimensional space. A full wave simulation of electromagnetic scattering form a radar reflector demonstrates the applicability to large scale problems in three-dimensional space.Comment: 33 pages, 8 figures, submitted to Journal of Computational and Applied Mathematic

Adaptive energy minimisation for hp-finite element methods

This article is concerned with the numerical solution of convex variational problems. More precisely, we develop an iterative minimisation technique which allows for the successive enrichment of an underlying discrete approximation space in an adaptive manner. Specifically, we outline a new approach in the context of hp-adaptive finite element methods employed for the efficient numerical solution of linear and nonlinear second-order boundary value problems. Numerical experiments are presented which highlight the practical performance of this new hp-refinement technique for both one- and two-dimensional problems

A new methodology for anisotropic mesh refinement based upon error gradients

We introduce a new strategy for controlling the use of anisotropic mesh refinement based upon the gradients of an a posteriori approximation of the error in a computed finite element solution. The efficiency of this strategy is demonstrated using a simple anisotropic mesh adaption algorithm and the quality of a number of potential a posteriori error estimates is considered

Seasonal Pattern of Batrachochytrium dendrobatidis Infection and Mortality in Lithobates areolatus: Affirmation of Vredenburg's “10,000 Zoospore Rule”

To fully comprehend chytridiomycosis, the amphibian disease caused by the chytrid fungus Batrachochytrium dendrobatidis (Bd), it is essential to understand how Bd affects amphibians throughout their remarkable range of life histories. Crawfish Frogs (Lithobates areolatus) are a typical North American pond-breeding species that forms explosive spring breeding aggregations in seasonal and semipermanent wetlands. But unlike most species, when not breeding Crawfish Frogs usually live singly—in nearly total isolation from conspecifics—and obligately in burrows dug by crayfish. Crayfish burrows penetrate the water table, and therefore offer Crawfish Frogs a second, permanent aquatic habitat when not breeding. Over the course of two years we sampled for the presence of Bd in Crawfish Frog adults. Sampling was conducted seasonally, as animals moved from post-winter emergence through breeding migrations, then back into upland burrow habitats. During our study, 53% of Crawfish Frog breeding adults tested positive for Bd in at least one sample; 27% entered breeding wetlands Bd positive; 46% exited wetlands Bd positive. Five emigrating Crawfish Frogs (12%) developed chytridiomycosis and died. In contrast, all 25 adult frogs sampled while occupying upland crayfish burrows during the summer tested Bd negative. One percent of postmetamorphic juveniles sampled were Bd positive. Zoospore equivalents/swab ranged from 0.8 to 24,436; five out of eight frogs with zoospore equivalents near or >10,000 are known to have died. In summary, Bd infection rates in Crawfish Frog populations ratchet up from near zero during the summer to over 25% following overwintering; rates then nearly double again during and just after breeding—when mortality occurs—before the infection wanes during the summer. Bd-negative postmetamorphic juveniles may not be exposed again to this pathogen until they take up residence in crayfish burrows, or until their first breeding, some years later