Independent Meshes Research Articles

Transient Simulation of Multiscale Structures Using the Nonconformal Domain Decomposition Laguerre-FDTD Method

In this paper, a full-wave transient domain decomposition method is proposed for solving multiscale electromagnetic problems. The proposed scheme is based on the unconditionally stable Laguerre finite-difference time-domain (Laguerre-FDTD) method. The entire computational domain is partitioned into several subdomains with independent meshing according to the physical sizes of each subdomain. Standard Laguerre-FDTD method is used to form the interior field equations of each subdomain, whereas interface field equations are formed by applying the equivalency between finite-element method and FDTD method. In addition, field continuity is enforced through the use of two sets of Lagrange multipliers at the domain interface. Schur complement is implemented for extracting the interface problem and each domain can be evaluated independently. Numerical results for multiscale structures are presented to demonstrate the capability, accuracy, and efficiency of the proposed method.

Finite Element Discretization Error Analysis of a General Interfacial Stress Functional

The finite element discretization of a variable interfacial stress tensor ${\sigma}^{\Gamma}(x)$ is considered; variable interfacial tension is included as a special case. Interfacial stress, for example, occurs in two-phase flow problems. In the weak formulation, the interfacial stress gives rise to a functional which is supported on the interface $\Gamma$. A finite element discretization of this functional is presented and analyzed. The discretization admits almost independent meshes for the approximation of the interface and the approximation of the flow variables. The main result is an $\mathcal{O}(h^{k+1/2})$-error-bound for the interfacial stress functional in a natural norm if the discrete interface is an $\mathcal{O}(h^{k+1})$-approximation of $\Gamma$. The bound is shown to be sharp in numerical experiments.

Numerical modeling of gas–liquid–solid interactions: Gas–liquid free surfaces interacting with deformable solids

A numerical framework to model and simulate gas–liquid–solid three-phase flows is proposed and implemented in this work. The overall algorithm adopts a non-boundary-fitted approach that avoids frequent mesh-updating procedures by defining independent meshes and explicit interfacial points to represent each phase. Here, we couple our existing solvers, the immersed finite element method (IFEM) and the connectivity-free front tracking (CFFT) method that model fluid–solid and gas–liquid interactions, respectively, for the three-phase models. A modified IFEM algorithm is derived for modeling fluid–solid interactions that accounts for the dynamics of the solid, while the CFFT is used here to simulate gas–liquid multi-fluid flows that uses explicit interfacial points to represent the gas–liquid interface and for its easy handling of interface topology changes. Instead of defining different levels simultaneously as used in level sets, an indicator function naturally couples the two methods together to represent and track each of the three phases. Several 2-D and 3-D testing cases are performed to show the robustness and capability of the coupled numerical framework in dealing with complex three-phase problems, in particular free surfaces interacting with deformable solids.

A gluing method for non-matching meshes

This paper presents a gluing method for composite meshes. Different meshes are generated independently and are glued together using some new elements to connect them, referred to as extension elements. The resulting global mesh is non-conforming and consists of connected overlapping meshes. The method is inherently implicit, parallel and versatile, in the sense that it is PDE independent. The most cited gluing method is probably the Chimera method, used for overset grids, where patch meshes are superimposed onto a background mesh. The method employed here was originally devised for such situations and is now applied to disjoint or overlapping meshes. One of the advantages of the method is that the meshes do not have to coincide and can present a gap between them. The method is illustrated through some simple examples to demonstrate the mesh convergence. Finally, we consider the solution of the airflow in the complete respiratory system, by joining independent meshes for the large and small airways, and the simulation of the flow passing through a bypass in a stenosed artery.

A Chimera method for the incompressible Navier–Stokes equations

SUMMARYThe Chimera method was developed three decades ago as a meshing simplification tool. Different components are meshed independently and then glued together using a domain decomposition technique to couple the equations solved on each component. This coupling is achieved via transmission conditions (in the finite element context) or by imposing the continuity of fluxes (in the finite volume context). Historically, the method has then been used extensively to treat moving objects, as the independent meshes are free to move with respect to the others. At each time step, the main task consists in recomputing the interpolation of the transmission conditions or fluxes. This paper presents a Chimera method applied to the Navier–Stokes equations. After an introduction on the Chimera method, we describe in two different sections the two independent steps of the method: the hole cutting to create the interfaces of the subdomains and the coupling of the subdomains. Then, we present the Navier–Stokes solver considered in this work. Implementation aspects are then detailed in order to apply efficiently the method to this specific parallel Navier–Stokes solver. We conclude with some examples to demonstrate the reliability and application of the proposed method. Copyright © 2014 John Wiley & Sons, Ltd.

Efficient Reluctance Network Formulation for Electrical Machine Design Using Optimization

The aim of this paper is to discuss electrical machines building and computation while taking into account the change of reluctance network (RN) topology required in an optimization process. Indeed, as electrical machine RN changes during an electrical period, this model has to consider several rotor positions to analyze harmonic influence on machine performance (e.g., its torque and losses). Such a requirement can also be important in applications where machine geometry changes shortly during sizing process by optimization. These two cases are addressed in this paper. Using established independent meshes of the initial RN, this paper proposes an algorithm to deal with all topologies necessary for an optimization process. This aims to reduce the computation time, increase the model, and improve the solving stability. An output selectivity approach is also applied on the RN modeling to improve the computation time of the optimization process. The simulation results are compared to finite-element methods and the optimization performance is presented.

Use of independent rotation field in the large displacement analysis of beams

This paper examines the effect of using independent finite rotation field in the large displacement analysis of flexible bodies. This finite rotation description is at the core of the large rotation vector formulation (LRVF), which has been used in the dynamic analysis of bodies experiencing large rotation and deformation. The LRVF employs two independently interpolated meshes for describing the flexible body dynamics: the position mesh and the rotation mesh. The use of these two geometrically independent meshes can lead to coordinate and geometric invariant redundancy that can be the source of fundamental problems in the analysis of large deformations. It is demonstrated in this paper that the two geometry meshes can define different space curves, which can differ by arbitrary rigid-body displacements. The material points of the two meshes occupy different positions in the deformed configuration, and as a consequence, the geometries of the two meshes can differ significantly. The paper also discusses other issues including the inextensibility of the rotation mesh. Simple examples are presented in order to shed light on these fundamental issues.

System-Level Thermal Modeling Using Nonconformal Domain Decomposition and Model-Order Reduction

To keep up with increasing modeling complexity arising from 3-D integration, novel thermal modeling methods are required to tackle large-scale 3-D systems. In this paper, we propose a system-level thermal modeling method using nonconformal domain decomposition and Krylov space-based model-order reduction (MOR) for both steady-state and transient analysis. To efficiently model a 3-D system, the system is divided into separate domains with independent meshing grids using nonconformal domain decomposition. As a result, reduced-order models can be created for individual domains with MOR ports, which can reduce the computational challenge of performing MOR for the entire system with a large number of unknowns and ports. The connectivity between domains is captured using coupling matrices via Lagrange multipliers and Schur complements. In addition, the proposed method can efficiently handle varying design parameters, such as air convection coefficient and thermal conductivity without performing parameterized MOR. The modeling results show that the proposed method can efficiently simulate 3-D systems with hundreds of ports with a speed-up of 20×, compared with just thermal modeling using domain decomposition.

Dynamic limit equilibrium analysis of sliding block for rock slope based on nonlinear FEM

Traditional rigid body limit equilibrium method (RBLEM) was adopted for the stability evaluation and analysis of rock slope under earthquake scenario. It is not able to provide the real stress distribution of the structure, while the strength reduction method relies on the arbitrary decision on the failure criteria. The dynamic limit equilibrium solution was proposed for the stability analysis of sliding block based on 3-D multi-grid method, by incorporating implicit stepping integration FEM. There are two independent meshes created in the analysis: One original 3-D FEM mesh is for the simulation of target structure and provides the stress time-history, while the other surface grid is for the simulation of sliding surface and could be selected and designed freely. As long as the stress time-history of the geotechnical structure under earthquake scenario is obtained based on 3-D nonlinear dynamic FEM analysis, the time-history of the force on sliding surface could be derived by projecting the stress time-history from 3-D FEM mesh to surface grid. After that, the safety factor time-history of the sliding block will be determined through applying limit equilibrium method. With those information in place, the structure’s aseismatic stability can be further studied. The above theory and method were also applied to the aseismatic stability analysis of Dagangshan arch dam’s right bank high slope and compared with the the result generated by Quasi-static method. The comparative analysis reveals that the method not only raises the FEM’s capability in accurate simulation of complicated geologic structure, but also increases the flexibility and comprehensiveness of limit equilibrium method. This method is reliable and recommended for further application in other real geotechnical engineering.

On Simulations of Discrete Fracture Network Flows with an Optimization-Based Extended Finite Element Method

Following the approach introduced in [SIAM J. Sci. Comput., 35 (2013), pp. B487--B510], we consider the formulation of the problem of fluid flow in a system of fractures as a PDE constrained optimization problem, with discretization performed using suitable extended finite elements; the method allows independent meshes on each fracture, thus completely circumventing meshing problems usually related to the discrete fracture network (DFN) approach. The application of the method to DFNs of medium complexity is fully analyzed here, accounting for several issues related to viable and reliable implementations of the method in complex problems.

A Gluing Method for Non-matching Meshes

This paper presents a gluing method for composite meshes. Different meshes are generated independently and are glued together using some extension elements to connect them. The resulting global mesh is non-conforming and consists of connected overlapping meshes. The method is inherently implicit, parallel and versatile, in the sense that it is PDE independent. The most cited gluing method is probably the Chimera method, used for overset grids, where patch meshes are superimposed onto a background mesh. The method employed here was originally devised for such situations and is now applied to disjoint or overlapping meshes. One of the advantages of the method is that the meshes do not have to coincide and can present a gap between them. The method is illustrated through some simple examples to demonstrate the mesh convergence and finally applied to the solution of the airflow in the complete respiratory system, by joining independent meshes for the large and small airways.

A PDE-Constrained Optimization Formulation for Discrete Fracture Network Flows

We investigate a new numerical approach for the computation of the three-dimensional flow in a discrete fracture network that does not require a conforming discretization of partial differential equations on complex three-dimensional systems of planar fractures. The discretization within each fracture is performed independently of the discretization of the other fractures and of their intersections. An independent meshing process within each fracture is a very important issue for practical large-scale simulations, making mesh generation easier. Some numerical simulations are given to show the viability of the method. The resulting approach can be naturally parallelized for dealing with systems with a huge number of fractures.

Multimesh anisotropic adaptivity for the Boltzmann transport equation

This article presents a new adaptive finite element based method for the solution of the spatial dimensions of the Boltzmann transport equation. The method applies a curvature based error metric to locate the under and over resolved regions of a solution and this, in turn, is used to guide the refinement and coarsening of the spatial mesh. The error metrics and re-meshing procedures are designed such that they enable anisotropic resolution to form in the mesh should it be appropriate to do so. The adaptive mesh enables the appropriate resolution to be applied throughout the whole domain of a problem and so increase the efficiency of the solution procedure. Another new approach is also described that allows independent adaptive meshes to form for each of the energy group fluxes. The use of independent meshes can significantly improve computational efficiency when solving problems where the different group fluxes require high resolution over different regions. The mesh to mesh interpolation is made possible through the use of a ‘supermeshing’ procedure that ensures the conservation of particles when calculating the group to group scattering sources. Finally it is shown how these methods can be incorporated within a solver to resolve both fixed source and eigenvalue problems. A selection of both fixed source and eigenvalue problems are solved in order to demonstrate the capabilities of these methods.

The effect of interfacial shear strength on damping behavior of carbon nanotube reinforced composites

The effect of interfacial shear strength (ISS) on the mechanical and damping properties of carbon nanotube reinforced composites (CNT-RCs) is investigated in the present study using a multiscale simulation. The atomic lattice of CNTs is modeled with the modified molecular structural mechanics (MMSM) approach and reduced to an equivalent beam element (EBE) which is used as the basic building block for the construction of full length CNTs embedded in the polymer. Linear material properties are assigned to the EBEs, while a Maxwell–Wiechert material model is used for modeling the viscoelastic behavior of the polymer. The interfacial load transfer mechanism between the lateral surface of the carbon nanotube and the surrounding matrix is taken into account with a nonlinear bond-slip friction-type model. Finite element models of representative volume elements (RVEs) are constructed comprised of two independent meshes: a structured with solid elements for the matrix and a series of embedded EBEs for the full length CNTs inside the matrix. Straight as well as wavy CNTs are considered. In the case of wavy CNTs, random CNT geometries are generated using the spectral representation method with evolutionary power spectra (EPS) which are derived from processing scanning electron microscope (SEM) images. Stochastic average properties were derived with Monte Carlo simulation. The mechanical and damping properties of the CNT-RCs are assessed on the basis of sensitivity analyses with respect to various weight fractions and interfacial shear strength (ISS) values. Numerical results are presented, showing the significant effect of the ISS as well as the influence of CNT waviness on the damping behavior of CNT-RCs.

Structural‐acoustic coupling on non‐conforming meshes with quadratic shape functions

SUMMARYFully coupled finite element/boundary element models are a popular choice when modelling structures that are submerged in heavy fluids. To achieve coupling of subdomains with non‐conforming discretizations at their common interface, the coupling conditions are usually formulated in a weak sense. The coupling matrices are evaluated by integrating products of piecewise polynomials on independent meshes. The case of interfacing elements with linear shape functions on unrelated meshes has been well covered in the literature. This paper presents a solution to the problem of evaluating the coupling matrix for interfacing elements with quadratic shape functions on unrelated meshes. The isoparametric finite elements have eight nodes (Serendipity) and the discontinuous boundary elements have nine nodes (Lagrange). Results using linear and quadratic shape functions on conforming and non‐conforming meshes are compared for an example of a fluid‐loaded point‐excited sphere. It is shown that the coupling error decreases when quadratic shape functions are used. Copyright © 2012 John Wiley & Sons, Ltd.

A mixed finite element method for Darcy flow in fractured porous media with non-matching grids

We consider an incompressible flow problem in a N -dimensional fractured porous domain (Darcy’s problem). The fracture is represented by a (N − 1)-dimensional interface, exchanging fluid with the surrounding media. In this paper we consider the lowest-order (ℝ T0 , ℙ0 ) Raviart-Thomas mixed finite element method for the approximation of the coupled Darcy’s flows in the porous media and within the fracture, with independent meshes for the respective domains. This is achieved thanks to an enrichment with discontinuous basis functions on triangles crossed by the fracture and a weak imposition of interface conditions. First, we study the stability and convergence properties of the resulting numerical scheme in the uncoupled case, when the known solution of the fracture problem provides an immersed boundary condition. We detail the implementation issues and discuss the algebraic properties of the associated linear system. Next, we focus on the coupled problem and propose an iterative porous domain/fracture domain iterative method to solve for fluid flow in both the porous media and the fracture and compare the results with those of a traditional monolithic approach. Numerical results are provided confirming convergence rates and algebraic properties predicted by the theory. In particular, we discuss preconditioning and equilibration techniques to make the condition number of the discrete problem independent of the position of the immersed interface. Finally, two and three dimensional simulations of Darcy’s flow in different configurations (highly and poorly permeable fracture) are analyzed and discussed.

Positive electrical circuits and their reachability

Positive electrical circuits and their reachabilityConditions for the positivity of linear electrical circuits composed of resistances, coils, capacitors and voltage (current) sources are established. It is shown that: 1) the electrical circuit composed of resistors, coils and voltage source is positive for any values of their resistances, inductances and source voltages if and only if the number of coils is less or equal to the number of its linearly independent meshes, 2) the electrical circuit is not positive for any values of its resistances, capacitances and source voltages if each its branch contains resistor, capacitor and voltage source, 3) the positiven-meshes electrical circuit with only one inductance in each linearly independent mesh is reachable if all resistances of branches belonging to two linearly independent meshes are zero.

A mixed hybrid Mortar method for solving flow in discrete fracture networks

We consider flow in discrete fracture networks made of 2D domains in intersection and solved with a mixed hybrid finite element method (MHFEM). The discretization within each fracture is performed in two steps: first, borders and intersections are discretized, second, based on these discretizations, a 2D mesh is built. Independent meshing process within each subdomain is of interest for practical use since it makes it easier to refine the chosen subdomains and to perform parallel computation. This article shows how MHFEM is well adapted for integrating a Mortar method to enforce the continuity of the fluxes and heads at the non-matching grids. Some numerical simulations are given to show the efficiency of the method in the case of a preferential orientation of the fractures where a comparison with the 2D solution is possible.

Inductively Heated Incompressible Flow of Electrically Conductive Liquid in Pipe

A novel computer model of inductively heated incompressible flow of electrically conductive liquid in a pipe is presented. The numerical solution of this multiply coupled problem is realized by a higher-order finite element method using the code Hermes2D developed and written by the authors. The algorithm contains a number of original features such as hanging nodes of any order or in time adaptively changing mutually independent meshes for the computation of particular time-dependent field quantities. The methodology is illustrated by a typical example whose results are discussed.

A comparison of two SGBEM solutions of an unsteady heat conduction interface problem

A comparison of two approaches for solving time dependence in an unsteady heat conduction problem with a material or other interface, based on a boundary element technique is presented. A quadrature algorithm is used in the first approach for calculating convolutional integrals, which leads directly to the time-domain solution of the problem; the other approach uses the Laplace transform and a numerical inversion of the Laplace domain solution. The new numerical techniques lead together with the Symmetric Galerkin Boundary Element Method applied for space variables and a weak formulation of interface conditions to an interface problem with generally curved interfaces and to independent meshing of each side of the interface. The discussion on numerical results is focused on the non-matching discretization of the interface. The obtained data are also compared with known analytical solutions, if available, and discussed in the cases of different material properties pertinent to substructures on both sides of the interface.