A p-multigrid method enhanced with an ILUT smoother and its comparison to h-multigrid methods within Isogeometric Analysis

Over the years, Isogeometric Analysis has shown to be a successful alternative to the Finite Element Method (FEM). However, solving the resulting linear systems of equations efficiently remains a challenging task. In this paper, we consider a p-multigrid method, in which coarsening is applied in the approximation order p instead of the mesh width h. Since the use of classical smoothers (e.g. Gauss-Seidel) results in a p-multigrid method with deteriorating performance for higher values of p, the use of an ILUT smoother is investigated. Numerical results and a spectral analysis indicate that the resulting p-multigrid method exhibits convergence rates independent of h and p. In particular, we compare both coarsening strategies (e.g. coarsening in h or p) adopting both smoothers for a variety of two and threedimensional benchmarks

Core-annular flow through a horizontal pipe: hydrodynamic counterbalancing of buoyancy force on core

A theoretical investigation has been made of core-annular flow: the flow of a high-viscosity liquid core surrounded by a low-viscosity liquid annular layer through a horizontal pipe. Special attention is paid to the question of how the buoyancy force on the core, caused by a density difference between the core and the annular layer, is counterbalanced. From earlier studies it is known that at the interface between the annular layer and the core waves are present that move with respect to the pipe wall. In the present study the core is assumed to consist of a solid center surrounded by a high-viscosity liquid layer. Using hydrodynamic lubrication theory taking into account the flow in the low-viscosity liquid annular layer and in the high-viscosity liquid core layer the development of the interfacial waves is calculated. They generate pressure variations in the core layer and annular layer that can cause a net force on the core. Steady eccentric core-annular flow is found to be possible

Smoothness-Increasing Accuracy-Conserving (SIAC) filters for derivative approximations of discontinuous Galerkin (DG) solutions over nonuniform meshes and near boundaries

Accurate approximations for the derivatives are usually required in many application areas such as biomechanics, chemistry and visualization applications. With the help of Smoothness-Increasing Accuracy-Conserving (SIAC) filtering, one can enhance the derivatives of a discontinuous Galerkin solution. However, current investigations of derivative filtering are limited to uniform meshes and periodic boundary conditions, which do not meet practical requirements. The purpose of this paper is twofold: to extend derivative filtering to nonuniform meshes and propose position-dependent derivative filters to handle filtering near the boundaries. Through analyzing the error estimates for SIAC filtering, we extend derivative filtering to nonuniform meshes by changing the scaling of the filter. For filtering near boundaries, we discuss the advantages and disadvantages of two existing position-dependent filters and then extend them to position-dependent derivative filters, respectively. Further, we prove that with the position-dependent derivative filters, the filtered solutions can obtain a better accuracy rate compared to the original discontinuous Galerkin approximation with arbitrary derivative orders over nonuniform meshes. One- and two-dimensional numerical results are provided to support the theoretical results and demonstrate that the position-dependent derivative filters, in general, enhance the accuracy of the solution for both uniform and nonuniform meshes

A General Algorithm for Reusing Krylov Subspace Information. I. Unsteady Navier-Stokes

A general algorithm is developed that reuses available information to accelerate the iterative convergence of linear systems with multiple right-hand sides A x = b (sup i), which are commonly encountered in steady or unsteady simulations of nonlinear equations. The algorithm is based on the classical GMRES algorithm with eigenvector enrichment but also includes a Galerkin projection preprocessing step and several novel Krylov subspace reuse strategies. The new approach is applied to a set of test problems, including an unsteady turbulent airfoil, and is shown in some cases to provide significant improvement in computational efficiency relative to baseline approaches

A comparison of Krylov methods for Shifted Skew-Symmetric Systems

It is well known that for general linear systems, only optimal Krylov methods with long recurrences exist. For special classes of linear systems it is possible to find optimal Krylov methods with short recurrences. In this paper we consider the important class of linear systems with a shifted skew-symmetric coefficient matrix. We present the MRS3 solver, a minimal residual method that solves these problems using short vector recurrences. We give an overview of existing Krylov solvers that can be used to solve these problems, and compare them with the MRS3 method, both theoretically and by numerical experiments. From this comparison we argue that the MRS3 solver is the fastest and most robust of these Krylov method for systems with a shifted skew-symmetric coefficient matrix.Comment: 23 pages, 3 figure

A biomechanical mathematical model for the collagen bundle distribution-dependent contraction and subsequent retraction of healing dermal wounds

A continuum hypothesis-based, biomechanical model is presented for the simulation of the collagen bundle distribution-dependent contraction and subsequent retraction of healing dermal wounds that cover a large surface area. Since wound contraction mainly takes place in the dermal layer of the skin, solely a portion of this layer is included explicitly into the model. This portion of dermal layer is modeled as a heterogeneous, orthotropic continuous solid with bulk mechanical properties that are locally dependent on both the local concentration and the local geometrical arrangement of the collagen bundles. With respect to the dynamic regulation of the geometrical arrangement of the collagen bundles, it is assumed that a portion of the collagen molecules are deposited and reoriented in the direction of movement of (myo)fibroblasts. The remainder of the newly secreted collagen molecules are deposited by ratio in the direction of the present collagen bundles. Simulation results show that the distribution of the collagen bundles influences the evolution over time of both the shape of the wounded area and the degree of overall contraction of the wounded area. Interestingly, these effects are solely a consequence of alterations in the initial overall distribution of the collagen bundles, and not a consequence of alterations in the evolution over time of the different cell densities and concentrations of the modeled constituents. In accordance with experimental observations, simulation results show furthermore that ultimately the majority of the collagen molecules ends up permanently oriented toward the center of the wound and in the plane that runs parallel to the surface of the skin

A vector valued Stefan problem from aluminium industry

Dissolution of stoichiometric multi-component particles in ternary alloys is an important process occurring during the heat treatment of as-cast aluminium alloys prior to hot-extrusion. A mathematical model is proposed to describe such a process. In this model an equation is given to determine the position of the particle interface in time, using two diffusion equations which are coupled by nonlinear boundary conditions at the interface. Moreover the well-posedness of the moving boundary problem is investigated using the maximum principle for the parabolic partial differential equation. Furthermore, for an unbounded domain and planar co-ordinates an analytical asymptotic approximation based on self-similarity is derived. This asymptotic approximation gives insight into the well-posedness of the problem