Non-smooth Geometries Research Articles

New aspects of the CISAMR algorithm for meshing domain geometries with sharp edges and corners

Several new algorithmic aspects of the conforming to interface structured adaptive mesh refinement (CISMAR) mesh generation technique are presented for creating high-fidelity finite element models of 3D problems with non-smooth (C0–continuous) geometries. The Stereolithography (STL) file characterizing the domain geometry is first pre-processed to identify its sharp edges and corners, followed by overlaying that on a structured background mesh composed of tetrahedral elements to identify elements cuts by material interfaces/boundaries. Three non-iterative algorithms, namely hierarchical r-adaptivity, element deletion, and Kirigami-inspired sub-tetrahedralization are then introduced to generate a high-quality conforming mesh by relocating background mesh nodes, eliminating degenerate elements, and subdividing non-conforming tetrahedra to match material interfaces and domain boundaries. These algorithms either upgrade or fully replace different phases of the original CISAMR technique, which was limited to modeling problems with smooth interfaces. The example problems provided in this manuscript demonstrate the upgraded CISAMR algorithm to handle domain geometries with complex, non-smooth interfaces.

Bio-inspired evaporation from shaped interfaces: an experimental study

Evaporative interfaces help process heat and substances in a variety of technical realms, from electronic to architectural applications. Because geometry affects the hydraulics, thermal properties and aerodynamics of evaporative devices, their performance can be tuned through design. While non-smooth interfaces are widely exploited to enhance transfer passively, surface area extension in packed volumes is a predominant line of research. This leaves aerodynamic structure-transfer relations and the impact of geometry itself unclear. Meanwhile, protrusions in leaves such as lobes and toothed margins have been associated with enhanced vapor dissipation. This experimental study explores the design space of leaf-inspired structures with evaporating protrusions. Three sets of water-absorbing models with fixed evaporating surface area and unlimited hydraulic supply were tested: (1) paper strips with dimension-equivalent protrusions of varied shape and degree of elongation; (2) cellulose sponges with the same designs as their cross-sectional profile, extruded three-dimensionally; (3) ceramic tiles with grooves of varied cross-section, conceived as building elements for evaporative cooling. Overall, results demonstrate that protrusions affect mass transfer rate and surface temperatures and can be integrated in the design of evaporative exchangers with non-smooth geometries. For the paper models, evaporation rate correlated with protrusion aspect ratio, supporting a functional interpretation of leaf design and its utilization in low-wind plate-fin exchangers. However, the same transfer enhancement was not regained from simply extruding an effective design into three-dimensions. For the ceramic tiles, geometry-driven differences in evaporation depended on the aerodynamic roughness and size of the grooved pattern, and on ventilation. Their outdoor thermal behavior was complex due to a multifaceted interaction with the environment and geometry-related factors such as self-shading and thermal mass. Ultimately, this design effort illustrates the potential of structured interfaces for evaporative exchange and thermoregulating the built environment.

Global existence, uniqueness and stability for nonlinear dissipative bulk-interface interaction systems

We show global well-posedness and exponential stability of equilibria for a general class of nonlinear dissipative bulk-interface systems. They correspond to thermodynamically consistent gradient structure models of bulk-interface interaction. The setting includes nonlinear slow and fast diffusion in the bulk and nonlinear coupled diffusion on the interface. Additional driving mechanisms can be included and non-smooth geometries and coefficients are admissible, to some extent. An important application are volume-surface reaction-diffusion systems with nonlinear coupled diffusion.

Solution of Stokes flow in complex nonsmooth 2D geometries via a linear-scaling high-order adaptive integral equation scheme

We present a fast, high-order accurate and adaptive boundary integral scheme for solving the Stokes equations in complex—possibly nonsmooth—geometries in two dimensions. We apply the panel-based quadratures of Helsing and coworkers to evaluate to high accuracy the weakly-singular, hyper-singular, and super-singular integrals arising in the Nyström discretization, and also the near-singular integrals needed for flow and traction evaluation close to boundaries. The resulting linear system is solved iteratively via calls to a Stokes fast multipole method. We include an automatic algorithm to “panelize” a given geometry, and choose a panel order, which will efficiently approximate the density (and hence solution) to a user-prescribed tolerance. We show that this adaptive panel refinement procedure works well in practice even in the case of complex geometries with large number of corners, or close-to-touching smooth curves. In one example, for instance, a model 2D vascular network with 378 corners required less than 200K discretization points to obtain a 9-digit solution accuracy.

Vertex assigned morphing for parameter free shape optimization of 3-dimensional solid structures

In the past the Vertex Morphing method has mainly been used in the context of optimization of thin walled shell structures or fluid mechanical problems. In this paper an extension of this parameter free shape optimization method to 3-dimensional solid structures based on the free Finite Element code CalculiX is presented. Key element of the whole process is an efficient adjoint sensitivity analysis that can handle many design variables providing gradient information for the response functions of interest, e.g. stresses and mass. Typically, parameter free shape optimal designs result in non-smooth geometries and suffer from mesh dependencies. To avoid these phenomena an intermediate filter operation in combination with several post processing steps of the raw sensitivities is added. In the optimization, the filter size is also used as an additional design parameter controlling the curvature of the optimal shape. In many engineering optimization applications constraints have to be considered. These restrictions are included in the optimization problem by the use of Rosen’s gradient projection method. For the successive shape update a pseudo linear-elastic static Finite Element analysis with multiple point constraints and a varying Young’s modulus is performed in every design iteration. Finally, the capabilities of the extended Vertex Morphing method are demonstrated for an academic example and a geometrically complex part from aerospace industry.

Spinors on Singular Spaces and the Topology of Causal Fermion Systems

Causal fermion systems and Riemannian fermion systems are proposed as a framework for describing non-smooth geometries. In particular, this framework provides a setting for spinors on singular spaces. The underlying topological structures are introduced and analyzed. The connection to the spin condition in differential topology is worked out. The constructions are illustrated by many simple examples like the Euclidean plane, the two-dimensional Minkowski space, a conical singularity, a lattice system as well as the curvature singularity of the Schwarzschild space-time. As further examples, it is shown how complex and Kähler structures can be encoded in Riemannian fermion systems.

A mortar finite element approach for point, line, and surface contact

SummaryAn approach for investigating finite deformation contact problems with frictional effects with a special emphasis on nonsmooth geometries such as sharp corners and edges is proposed in this contribution. The contact conditions are separately enforced for point contact, line contact, and surface contact by employing 3 different sets of Lagrange multipliers and, as far as possible, a variationally consistent discretization approach based on mortar finite element methods. The discrete unknowns due to the Lagrange multiplier approach are eliminated from the system of equations by employing so‐called dual or biorthogonal shape functions. For the combined algorithm, no transition parameters are required, and the decision between point contact, line contact, and surface contact is implicitly made by the variationally consistent framework. The algorithm is supported by a penalty regularization for the special scenario of nonparallel edge‐to‐edge contact. The robustness and applicability of the proposed algorithms are demonstrated with several numerical examples.

Asymptotic expansions and effective boundary conditions: a short review for smooth and nonsmooth geometries with thin layers

Problems involving materials with thin layers arise in various application fields. We present here the use of asymptotic expansions for linear elliptic problems to derive and justify so-called ap-proximate or effective boundary conditions. We first recall the known results of the literature, and then discuss the optimality of the error estimates in the smooth case. For non-smooth geometries, the results of [18, 57] are commented and adapted to a model problem, and two improvements of the approximate model are proposed to increase its numerical performance.

A fourth-order Cartesian grid embedded boundary method for Poisson’s equation

Author(s): Devendran, D; Graves, DT; Johansen, H; Ligocki, T | Abstract: In this paper, we present a fourth-order algorithm to solve Poisson's equation in two and three dimensions. We use a Cartesian grid, embedded boundary method to resolve complex boundaries. We use a weighted least squares algorithm to solve for our stencils. We use convergence tests to demonstrate accuracy and we show the eigenvalues of the operator to demonstrate stability. We compare accuracy and performance with an established second-order algorithm. We also discuss in depth strategies for retaining higher-order accuracy in the presence of nonsmooth geometries.

Fast iterative boundary element methods for high-frequency scattering problems in 3D elastodynamics

The fast multipole method is an efficient technique to accelerate the solution of large scale 3D scattering problems with boundary integral equations. However, the fast multipole accelerated boundary element method (FM-BEM) is intrinsically based on an iterative solver. It has been shown that the number of iterations can significantly hinder the overall efficiency of the FM-BEM. The derivation of robust preconditioners for FM-BEM is now inevitable to increase the size of the problems that can be considered. The main constraint in the context of the FM-BEM is that the complete system is not assembled to reduce computational times and memory requirements. Analytic preconditioners offer a very interesting strategy by improving the spectral properties of the boundary integral equations ahead from the discretization. The main contribution of this paper is to combine an approximate adjoint Dirichlet to Neumann (DtN) map as an analytic preconditioner with a FM-BEM solver to treat Dirichlet exterior scattering problems in 3D elasticity. The approximations of the adjoint DtN map are derived using tools proposed in [40]. The resulting boundary integral equations are preconditioned Combined Field Integral Equations (CFIEs). We provide various numerical illustrations of the efficiency of the method for different smooth and non-smooth geometries. In particular, the number of iterations is shown to be completely independent of the number of degrees of freedom and of the frequency for convex obstacles.

Second kind integral equation formulation for the mode calculation of optical waveguides

We present a second kind integral equation (SKIE) formulation for calculating the electromagnetic modes of optical waveguides, where the unknowns are only on material interfaces. The resulting numerical algorithm can handle optical waveguides with a large number of inclusions of arbitrary irregular cross section. It is capable of finding the bound, leaky, and complex modes for optical fibers and waveguides including photonic crystal fibers (PCF), dielectric fibers and waveguides. Most importantly, the formulation is well conditioned even in the case of nonsmooth geometries. Our method is highly accurate and thus can be used to calculate the propagation loss of the electromagnetic modes accurately, which provides the photonics industry a reliable tool for the design of more compact and efficient photonic devices. We illustrate and validate the performance of our method through extensive numerical studies and by comparison with semi-analytical results and previously published results.

A High-Order Locally Corrected Nyström Scheme for Charge-Current Integral Equations

A high-order Nystrom scheme is developed for electromagnetic (EM) scattering problems of smooth objects. The implementation is based on different well-known charge-current formulations. This study aims to develop a novel quadrature scheme to compute different singular integrals that appear on integral equation methods in EMs. The proposed quadrature scheme is used on a locally corrected Nystrom method, and applied to charge-current integral equations to obtain a fast, accurate, and a low-frequency stable method to solve EM scattering problems. The proposed method allows obtaining accuracy in the range 4–10 digits in the near electric and magnetic scattered field for certain simple geometries and a wide range of frequencies. We also do a preliminary research on nonsmooth geometries obtaining a certain degree of accuracy depending on the particular charge-current integral equation used.

Modelling of a Phase Change Material melting process heated from below using spectral collocation methods

Purpose – Mathematical and numerical models are developed to study the melting of a Phase Change Material (PCM) inside a 2D cavity. The bottom of the cell is heated at constant and uniform temperature or heat flux, assuming that the rest of the cavity is completely adiabatic. The paper used suitable numerical methods to follow the interface temporal evolution with a good accuracy. The purpose of this paper is to show how the evolution of the latent energy absorbed to melt the PCM depends on the temperature imposed on the lower wall of the cavity. Design/methodology/approach – The problem is written with non-homogeneous boundary conditions. Momentum and energy equations are numerically solved in space by a spectral collocation method especially oriented to this situation. A Crank-Nicolson scheme permits the resolution in time. Findings – The results clearly show the evolution of multicellular regime during the process of fusion and the kinetics of phase change depends on the boundary condition imposed on the bottom cell wall. Thus the charge and discharge processes in energy storage cells can be controlled by varying the temperature in the cell PCM. Substantial modifications of the thermal convective heat and mass transfer are highlighted during the transient regime. This model is particularly suitable to follow with a good accuracy the evolution of the solid/liquid interface in the process of storage/release energy. Research limitations/implications – The time-dependent physical properties that induce non-linear coupled unsteady terms in Navier-Stokes and energy equations are not taken into account in the present model. The present model is actually extended to these coupled situations. This problem requires smoother geometries. One can try to palliate this disadvantage by constructing smoother approximations of non-smooth geometries. The augmentation of polynomials developments orders increases strongly the computing time. When the external heat flux or temperature imposed at the PCM is much greater than the temperature of the PCM fusion, one must choose carefully some data to assume the algorithms convergence. Practical implications – Among the areas where this work can be used, are: buildings where the PCM are used in insulation and passive cooling; thermal energy storage, the PCM stores energy by changing phase, solid to liquid (fusion); cooling and transport of foodstuffs or pharmaceutical or medical sensitive products, the PCM is used in the food industry, pharmaceutical and medical, to minimize temperature variations of food, drug or sensitive materials; and the textile industry, PCM materials in the textile industry are used in microcapsules placed inside textile fibres. The PCM intervene to regulate heat transfer between the body and the outside. Originality/value – The paper's originality is reflected in the precision of its results, due to the use of a high-accuracy numerical approximation based on collocation spectral methods, and the choice of Chebyshev polynomials basis in both axial and radial directions.

A New Level-Set Model for the Representation of Non-Smooth Geometries

In Starinshak et al. (J Comput Phys 262(1):1---16, 2014), we proposed a new level-set model for representing multimaterial flows in multiple space dimensions. Rather than associating each level-set function with the boundary of a material, the new model associates each level-set function with a pair of materials and the interface that separates them. In this paper, we extend the model to represent geometries with non-smooth boundaries. The model uses multiple level-set functions to describe the shape boundary, typically with one level-set function per smooth boundary segment. Sign information is collected from all level-set functions and a voting algorithm is used to determine the interior/exterior of the geometric shape. The model is well suited for representing boundaries with singularities; it offers significant improvement over standard level-set approaches, both in shape preservation and area conservation; and it eliminates the need for costly redistancing of the level-set function. Numerical examples illustrate the superior performance of the proposed model.

Missing boundary data reconstruction via an approximate optimal control

An approximate optimal control formulation of the Cauchy problem for elliptic equations is considered. A cost functional adding a fading through the iterations regularizing term borrowed from the domain decomposition communauty is proposed. Convergence of the descretized finite elements solution to the continuous one is proved. Numerical experiments involving smooth, non-smooth geometries as well as anisotropy highlight the capability of the present missing boundary data recovering process.

Method of fundamental solutions with regularization techniques for Cauchy problems of elliptic operators

In this paper we combine the method of fundamental solutions with various regularization techniques to solve Cauchy problems of elliptic differential operators. The main idea is to approximate the unknown solution by a linear combination of fundamental solutions whose singularities are located outside the solution domain. To solve effectively the discrete ill-posed resultant matrix, we use three regularization strategies under three different choices for the regularization parameter. Several examples on problems with smooth and non-smooth geometries in 2D and 3D spaces using under-, equally, and over-specified Cauchy data on an accessible boundary are given. Numerical results indicate that the generalized cross-validation and L -curve choice rulers for Tikhonov regularization and damped singular value decomposition strategy are most effective when using the same numbers of collocation and source points. It has also been observed that the use of more Cauchy data will greatly improve the accuracy of the approximate solution.

Laplacian decomposition and the boundary element method for solving Stokes problems

The solution of the equations governing the steady incompressible slow viscous fluid flow is analysed using a novel technique based on a Laplacian decomposition instead of the more traditional approaches based on the biharmonic streamfunction formulation or the velocity-pressure formulation. This results in the need to solve the Laplace equations for the pressure and other auxiliary harmonic functions which arise from the ideas of Almansi's decomposition. These equations, which become coupled through the boundary conditions, are numerically solved using the boundary element method (BEM). Results both on the boundary and inside the solution domain are presented and discussed for a simple benchmark test example and a few applications in smooth and non-smooth geometries in order to illustrate that the Laplacian decomposition in combination with BEM provides an efficient technique, in terms of accuracy and convergence, to investigate numerically a Stokes flow.

A local error analysis of the boundary-concentrated hp-FEM

The boundary concentrated finite element method is a variant of the hp-version of the FEM that is particularly suited for the numerical treatment of elliptic boundary value problems with smooth coefficients and boundary conditions with low regularity or non-smooth geometries. In this paper we consider the case of the discretization of a Dirichlet problem with exact solution u ∈ H 1+� () and investigate the local error in various norms. We show that for a � > 0 these norms behave as O(N −�−� ), where N denotes the dimension of the underlying finite element space. Furthermore, we present a new Gauss-Lobatto based interpolation operator that is adapted to the case non-uniform polynomial degree distributions.

Rotation-free finite element for the non-linear analysis of beams and axisymmetric shells

In this paper a finite element for the non-linear analysis of two-dimensional beams and axisymmetric shells is presented. The element uses classical thin shell assumptions (no transverse shear strains). The main feature of the element is that it has no rotational degrees of freedom. Curvatures are computed using geometrical information from the patch of three elements formed by the main element and the two neighbor (adjacent) elements. Special attention is devoted to non-smooth geometries and branching shells. An elastic–plastic material law is considered. Large strains are treated using a logarithmic strain measure and a through-the-thickness numerical integration of the constitutive equations. Several examples are presented including linear problems to study convergence properties, and non-linear problems for both elastic and elastic–plastic materials and large strains.

Decomposition contact response (DCR) for explicit finite element dynamics

We propose a new explicit contact algorithm for finite element discretized solids and shells with smooth and non-smooth geometries. The equations of motion are integrated in time with a predictor-corrector-type algorithm. After each predictor step, the impenetrability constraints and the exchange of momenta between the impacting bodies are considered and enforced independently. The geometrically inadmissible penetrations are removed using closest point projections or similar updates. Penetration is measured using the signed volume of intersection described by the contacting surface elements, which is well-defined for both smooth and non-smooth geometries. For computing the instantaneous velocity changes that occur during the impact event, we introduce the decomposition contact response method. This enables the closed-form solution of the jump equations at impact, and applies to non-frictional as well as frictional contact, as exemplified by the Coulomb frictional model. The overall algorithm has excellent momentum and energy conservation characteristics, as several numerical examples demonstrate. Copyright © 2005 John Wiley & Sons, Ltd.