Non-conforming Interface Research Articles

A Reduced Order Model for Domain Decompositions with Non-conforming Interfaces

A Reduced Order Model for Domain Decompositions with Non-conforming Interfaces

A Direct Implicit Discretization Approach for Arbitrary Non-conformal Interface by Using a Virtual Cell

A direct implicit discretization approach is presented for an arbitrary non-conformal interface by using a virtual cell on unstructured meshes. In this study, the virtual cell is supposed to be located at the neighbour of a target cell adjacent to the non-conformal interface. Any computational value on a virtual cell is formulated as a linear combination of values on contribution cells, which are determined by the nearest neighbouring cells or surface intersecting cells. The linear formulation simplifies the process of implementing implicit discretization for the governing equations. The approach is simple and efficient for information transfer through the interface because it avoids mesh stitching or iterations based on explicit discretization for treating topologically inconsistent meshes. Several numerical tests verify that the proposed method is capable and efficient for dealing with the arbitrary non-conformal interface.

Non-conforming Interface Conditions for the Second-Order Wave Equation

Imposition methods of interface conditions for the second-order wave equation with non-conforming grids is considered. The spatial discretization is based on high order finite differences with summation-by-parts properties. Previously presented solution methods for this problem, based on the simultaneous approximation term (SAT) method, have shown to introduce significant stiffness. This can lead to highly inefficient schemes. Here, two new methods of imposing the interface conditions to avoid the stiffness problems are presented: (1) a projection method and (2) a hybrid between the projection method and the SAT method. Numerical experiments are performed using traditional and order-preserving interpolation operators. Both of the novel methods retain the accuracy and convergence behavior of the previously developed SAT method but are significantly less stiff.

An efficient SBFEM-based approach for transient exterior vibro-acoustic analysis of power-law functionally graded shells

An efficient approach for transient exterior vibro-acoustic analysis of power-law functionally graded (FG) shells is developed using the scaled boundary finite element method. In the structural formulation, a shell element is treated as a three-dimensional continuum and its middle surface is represented with a quadrilateral spectral element. Along the thickness, the middle surface is scaled and the displacements are approximated using a quadratic Lagrange interpolation. The assumed natural strain method is applied to treat numerical locking and the integral along the thickness is performed analytically. In the acoustic formulation, velocity potential acts as the basic unknown. An infinite fluid is truncated by a spherical surface. The exterior field is simulated through the improved doubly asymptotic open boundary while the interior region is split into subdomains with their impedance evaluated by the improved continued-fraction method. The structure and fluid are discretized independently. The collocation procedure is utilized to perform the data transfer across the nonconforming interface. A high-order implicit time integration scheme is applied to solve the coupled system of equations. Numerical examples demonstrate that the proposed approach is stable, accurate and highly efficient. The effects of the geometric and material parameters on the vibro-acoustic behaviors of FG shells are systematically studied.

Non-conforming coupling RI-IGABEM for solving multidimensional and multiscale thermoelastic–viscoelastic problems

In this paper, a novel non-conforming coupling radial integration IGABEM (RI-IGABEM) is proposed to solve the multidimensional and multiscale thermoelastic–viscoelastic​ problems. Firstly, the displacement and regularized strain boundary integral equations for the thermoviscoelastic problems are derived. Secondly, the time-dependent shear modulus is expressed in the form of Prony series, and the memory stress is calculated by using the genetic integral, which reduces computing costs. In addition, the radial integration method (RIM) is applied to transform the domain integrals related to body force, temperature and memory stress into equivalent boundary integrals through applied points, which not only allows IGABEM to retain the advantage of discretization only on the boundary, but also reduces the singularity of the integral. Moreover, the adaptive integral method based on hierarchical quadtree decomposition algorithm is proposed to deal with the nearly singular integrals more flexible and efficient at optimal computational cost. The modified surface traction recovery method (TRM) of the thermoviscoelastic mechanics is proposed to solve the strains of the boundary points, which allows us to avoid solving hypersingular integrals. Furthermore, we construct the displacement (or tractions) coupling constraints among the non-conforming interface meshes by a virtual knot insertion technique. The main advantages of this method are simplicity and robustness, as it is problem-independent and only depends on the NURBS meshes on both sides of the interface. Some 2D and 3D examples are applied to prove the accuracy and efficiency of the present method for the multiscale thermoelastic–viscoelastic problems.

A High Order Finite Difference Method for the Elastic Wave Equation in Bounded Domains with Nonconforming Interfaces

A High Order Finite Difference Method for the Elastic Wave Equation in Bounded Domains with Nonconforming Interfaces

Isogeometric FE-BE method with non-conforming coupling interface for solving elasto-thermoviscoelastic problems

The isogeometric finite element (FE) and boundary element (BE) coupling method with non-conforming interface is applied to solve 3D elasto-thermoviscoelastic problems. The thermoviscoelastic constitutive relation is expressed by Stieltjes integral, and the equivalent integral weak form of the problem is derived according to the principle of virtual work, in which the time integral is discretized by the trapezoidal difference approximation formula. The parameters of viscoelastic material are described by Prony series, and the discrete formula can be simplified by recursive relationship to avoid storing all solutions in time steps. The equilibrium and compatibility conditions of FE and BE subdomains on the coupling interface are used to couple two different system matrices, and the coupling of non-conforming non-uniform rational B-splines (NURBS) surfaces in different subdomains is achieved by using virtual knot insertion technique. In addition, the time-temperature equivalent principle is applied to observe the mechanical behavior of viscoelastic materials over a wide temperature range in a short time. Through several numerical examples, the effects of different working conditions and geometrical parameters of combustion chamber on the structure of the solid rocket motor (SRM) are studied, and the correctness and robustness of this method are verified.

Laplace-Fourier transform solution to the electrochemical kinetics of a symmetric lithium cell affected by interface conformity

Lithium (Li)-metal batteries using solid-state electrolytes (SSEs) have attracted extensive attention owing to their high energy density. However, the interface issues arising from the Li stripping/plating cycles pose a major challenge for the applications of these batteries. This paper reports a three-dimensional (3D) time-dependent electrokinetic model, built based on the Nernst–Planck equation with electroneutrality assumption, to quantify the effects of interface conformity, external pressure, electrolyte Young's modulus, and ionic conductivity on the electrochemical behaviors of symmetric solid-state lithium cells. The evolution of interface morphology is related to the surface roughness, elastoplastic deformation, creep, and plating of the Li metal. The Laplace-Fourier domain solution is analytically derived, and the numerical solution is solved by implementing Talbot's Laplace transform method and the continuous convolution-Fourier transform (CC-FT) algorithm. A number of cases are analyzed, and the results reveal that a non-conformal interface with imperfect contact can induce uneven field distributions and that the nonuniformity increases with contact loss, and that the impact of external pressure on the reduction in ionic conductivity of SSE is similar to its effect on the potential drop. The results also suggest that the external pressure of 12.5 MPa can lead to relatively uniform deposition in a ceramic electrolyte system, while only 2 MPa is required in a polymer electrolyte system. In addition, A pressure of at least 2 MPa is needed to maintain the high ionic conductivity level for polymer composite SSEs of 14–16% volume LLZO particles.

A comparison of interpolation techniques for non-conformal high-order discontinuous Galerkin methods

The capability to incorporate moving geometric features within models for complex simulations is a common requirement in many fields. Fluid mechanics within aeronautical applications, for example, routinely feature rotating (e.g. turbines, wheels and fan blades) or sliding components (e.g. in compressor or turbine cascade simulations). With an increasing trend towards the high-fidelity modelling of these cases, in particular combined with the use of high-order discontinuous Galerkin methods, there is therefore a requirement to understand how different numerical treatments of the interfaces between the static mesh and the sliding/rotating part impact on overall solution quality. In this article, we compare two different approaches to handle this non-conformal interface. The first is the so-called mortar approach, where flux integrals along edges are split according to the positioning of the non-conformal grid. The second is a less-documented point-to-point interpolation method, where the interior and exterior quantities for flux evaluations are interpolated from elements lying on the opposing side of the interface. Although the mortar approach has significant advantages in terms of its numerical properties, in that it preserves the local conservation properties of DG methods, in the context of complex 3D meshes it poses notable implementation difficulties which the point-to-point method handles more readily. In this paper we examine the numerical properties of each method, focusing not only on observing convergence orders for smooth solutions, but also how each method performs in under-resolved simulations of linear and nonlinear hyperbolic problems, to inform the use of these methods in implicit large-eddy simulations.

Sliding Non-Conforming Interfaces for Edge Elements in Eddy Current Problems

In this work, a flexible discretization technique for the approximate solution of electromagnetic field problems with rotating parts, based on non-conforming (NC) interfaces of Nitsche-type, is investigated. This approach enables the use of edge elements of the first and second kind for the discretization of the magnetic vector potential without posing severe restrictions on gaps and overlaps of elements on both sides of the rotating interface. It improves the computational efficiency, compared to other NC interface formulations because no additional unknowns are introduced, and edge elements of different polynomial order can be coupled with this approach. The applicability is shown in two numerical examples, involving a cylindrical NC interface between a stationary and a rotating domain.

Non-conforming interface coupling and symmetric iterative solution in isogeometric FE–BE​ analysis

In this paper, a three-dimensional (3D) isogeometric finite element (FE) and boundary element (BE) coupling approach based on non-uniform rational B-splines (NURBS) is developed to simulate the deformed body. Based on the geometric exactness and higher smoothness of isogeometric analysis (IGA), NURBS discretization is applied to the approximation of geometric representation, displacement field and traction field simultaneously. An improved power series expansion method (M-PSEM) is used to evaluate the singular integral and is suitable for high order (≥3) NURBS basis functions, which is the limitation of the traditional method. Adaptive integration scheme based on quadtree subdivision technique is adopted to enable the non-singular integration computation more flexible and efficient at optimal computational cost. The relationship between the tractions and nodal external forces on the interface is established by using the virtual work principle. On the other hand, displacement coupling constraints among the non-conforming (or incomplete coincident) surface meshes are constructed by a virtual knot insertion technique. The main advantages of this method are simplicity and robustness, as it is problem-independent and only depends on the NURBS meshes on both sides of the interface. Introducing an enhanced symmetric matrix related to the BE subdomain, the symmetric iterative coupling method (SICM) is extended to solve the final system coupling equation. The final coupling matrix obtained by this iterative method is symmetric, and its dimension is the same as that of the stiffness matrix of IGA finite element method (IGAFEM). Numerical examples are investigated to assess the accuracy and efficiency of the presented method.

Modeling Nonlinear Steady-State Induction Heating Processes

In this article, an efficient simulation strategy for a fully coupled nonlinear magnetic-thermal problem is presented. The solution-dependent magnetic subproblem is solved with a harmonic balancing scheme, with the main focus on the correct choice of the material model (magnetization curve). To further increase the computational efficiency, a non-conforming interface approach is used, based on the jump operators and penalty terms. This method drastically decreases the meshing time because different mesh sizes can be combined without taking care of element distortions in transition layers between fine and coarse parts of the mesh.

Coupling Nonlinear Element Free Galerkin and Linear Galerkin Finite Volume Solver for 2D Modeling of Local Plasticity in Structural Material

This paper introduces a computational strategy to collaboratively develop the Galerkin Finite Volume Method (GFVM) as one of the most straightforward and efficient explicit numerical methods to solve structural problems encountering material nonlinearity in a small limited area, while the remainder of the domain represents a linear elastic behavior. In this regard, the Element Free Galerkin method (EFG), which is remarkably robust and accurate, but presumably more expensive, has locally been employed as a nonlinear sub-model to cover the shortcomings of the GFVM in the elastoplastic analysis. Since the formulations of these two methods are fundamentally different, the iterative zonal coupling has been accomplished using overlapping Multi-Grid (MG) patches with a non-matching interface and Iterative Global/Local (IGL) approach. The main property of such an algorithm is its non-intrusiveness, which means the complex nonlinear EFG solver is locally utilized over an elastic global GFVM without any geometric modification. This method is verified and investigated with available analytical and numerical solutions which gave quiet promising results showing the robustness and accuracy of the method. The Moving Least-Square approximation (MLS) has widely been applied on transfer level due to the non-conforming interface at the patch edges, and easily allows us to attach complex geometries with different mesh patterns. The new type of Quasi-Newtonian accelerator is adopted on the global material constitutive matrices and its convergence property and accuracy is compared with dynamic Aitken accelerators for two-dimensional problems in MATLAB. Finally, various accelerator types and mapping strategies are also concerned in the examination.

A clipping algorithm on non-conformal interface for heat conduction analysis

In numerical analysis of heat transfer including complicated computational domains or geometries, the domains are divided into several simple domains, and it is possible to generate cells in the simple domains easily. However, for the sake of the isolation of the domains, the methods lead the vertex mismatch between the divided regions, and become non-conformal. Because of the non-conformal meshes, it is difficult to calculate fluxes through the interface. Therefore, a computational treatment for the non-conformal mesh is required to obtain geometrical information between neighboring cells. A clipping method to find the geometrical information are developed, and applied to thermal conduction problems. The clipping method implemented in this study is advantageous in using simple vector operations only to obtain the geometrical information on the non-conformal interface. To test the accuracy of the clipping method, twelve cases with different grid configurations are presented. They show that the method has a high degree of accuracy for calculating the clipped areas. Furthermore, to validate the scheme, the heat conduction problem is performed. And then, the numerical result agrees well with the exact solution. Additionally, we apply the scheme to the fourth case in the ISO 10211 that includes two materials with different aspect ratios. In the vicinity of the interface, the high temperature gradients are presented because of the different thermal conduction coefficients. Therefore, it shows that the present methods can treat the non-conformal meshes with the different neighboring cell size properly.

Order-Preserving Interpolation for Summation-by-Parts Operators at Nonconforming Grid Interfaces

We study non-conforming grid interfaces for summation-by-parts finite difference methods applied to partial differential equations with second derivatives in space. To maintain energy stability, previous efforts have been forced to accept a reduction of the global convergence rate by one order, due to large truncation errors at the non-conforming interface. We avoid the order reduction by generalizing the interface treatment and introducing order preserving interpolation operators. We prove that, given two diagonal-norm summation-by-parts schemes, order preserving interpolation operators with the necessary properties are guaranteed to exist, regardless of the grid-point distributions along the interface. The new methods retain the stability and global accuracy properties of the underlying schemes for conforming interfaces.

A Weak Formulation for Isogeometric Analysis of Vibro-Acoustic Systems with Non-Conforming Interfaces

This paper develops a weak formulation for steady-state dynamic analysis of vibro-acoustic systems in the isogeometric framework. The Non-Uniform Rational B-Splines (NURBS) polynomials are used as basis functions for both geometrical transformation and numerical approximation. Starting from a NURBS-based geometrical projection that transforms a general, complex domain to a regular, spectral domain, an energy-based numerical model is then built, in which NURBS are again used as wholly admissible functions to approximate the structural displacement and the acoustic sound pressure. This process eventually leads to a set of discretized equations, based on which the system’s dynamic properties can be obtained. Besides better accuracy and excellent geometrical adaptivity, the main merit of the proposed method lies in its element-free feature of system modeling, which provides a natural solution for structural-acoustic coupling with non-conforming interfaces. This non-conformity results from either non-matching control points along the boundary or distinct polynomial orders used by adjacent patches, which is inevitable for a vibro-acoustic system given its multi-physics nature. The reliability of the proposed method is validated by a benchmark example. Its accuracy and efficiency are further investigated through a more general, car-like vibro-acoustic system.

Design-order, non-conformal low-Mach fluid algorithms using a hybrid CVFEM/DG approach

A hybrid, design-order sliding mesh algorithm, which uses a control volume finite element method (CVFEM), in conjunction with a discontinuous Galerkin (DG) approach at non-conformal interfaces, is outlined in the context of a low-Mach fluid dynamics equation set. This novel hybrid DG approach is also demonstrated to be compatible with a classic edge-based vertex centered (EBVC) scheme. For the CVFEM, element polynomial, P, promotion is used to extend the low-order P=1 CVFEM method to higher-order, i.e., P=2. An equal-order low-Mach pressure-stabilized methodology, with emphasis on the non-conformal interface boundary condition, is presented. A fully implicit matrix solver approach that accounts for the full stencil connectivity across the non-conformal interface is employed. A complete suite of formal verification studies using the method of manufactured solutions (MMS) is performed to verify the order of accuracy of the underlying methodology. The chosen suite of analytical verification cases range from a simple steady diffusion system to a traveling viscous vortex across mixed-order non-conformal interfaces. Results from all verification studies demonstrate either second- or third-order spatial accuracy and, for transient solutions, second-order temporal accuracy. Significant accuracy gains in manufactured solution error norms are noted even with modest promotion of the underlying polynomial order. The paper also demonstrates the CVFEM/DG methodology on two production-like simulation cases that include an inner block subjected to solid rotation, i.e., each of the simulations include a sliding mesh, non-conformal interface. The first production case presented is a turbulent flow past a high-rate-of-rotation cube (Re, 4000; RPM, 3600) on like and mixed-order polynomial interfaces. The final simulation case is a full-scale Vestas V27 225 kW wind turbine (tower and nacelle omitted) in which a hybrid topology, low-order mesh is used. Both production simulations provide confidence in the underlying capability and demonstrate the viability of this hybrid method for deployment towards high-fidelity wind energy validation and analysis.

비등각격자면을 갖는 복합재질의 전열해석을 위한 클리핑방법

Purpose: As the importance of ecological architecture such as prevention of condensation, thermal analysis with numerical methods became essential in a modern architecture. A grid-based numerical method is widely used, and various grid shapes have been studied. Notably, an unstructured grid has been attracted by means of successful results. Due to the its flexibility of an unstructured grid, it is possible to divide a complex domain into several simple domains in which the grid are generated easily and independently. However, the grid points on the interface between neighbor domains are not matched completely so called non-conformal grid, and thus, the flux between neighbor cells cannot calculate directly. Thus, a technical treatment for the arbitrary interface is required. Method: We employed the clipping method that calculates the connectivity and geometrical information between non-conformal cells on the interface. To validate the clipping method, the heat conduction with a non-conformal interface was performed. Result: The algorithm that we developed represented the interface, and the numerical result agreed with the analytic solution for a heat conduction problem.

Algebraic generation of single domain computational grid for twin screw machines. Part I. Implementation

Special attention is required for generation of computational grids in highly deforming working chambers of twin screw machines for 3D CFD calculations. Two approaches for customised grid generation are practically available. The first is an algebraic grid generation and the second is a differential decomposition method. This paper reports on new developments in the algebraic approach that has the advantages associated with both algebraic and differential methods. Two control functions are introduced for regularisation of the initial algebraic distribution. One is based on an analytical control function in transformed coordinate system while the other uses background blocking structure in order to guide the initial algebraic distribution towards a single computational mesh. This paper presents implementation and grid characteristics of these new functions. Developed grids have been tested and results from flow calculations on a dry air compressor have been validated in part II of the paper [29].It was possible to achieve two distinct characteristics desirable in a twin screw rotor domain mesh. Firstly, it is possible to independently control grid refinement in the interlobe region thereby providing better accuracy in representation of the leakage gaps. Secondly and most importantly, it is possible now to eliminate the non-conformal interface between the two rotor domains thereby producing a single domain structured grid for the rotors, while still maintaining the fully hexahedral cell topology. An improvement in the global orthogonality of the cells was achieved. Despite of a decrement in the Face warp quality, aspect ratio of cells retained similar scale.

Application of isogeometric method to free vibration of Reissner–Mindlin plates with non-conforming multi-patch

The isogeometric method is used to study the free vibration of thick plates based on Mindlin theory. The Non-uniform Rational B-Spline (NURBS) basis functions are employed to build the thick plate’s geometry models and serve as the shape functions for solution field approximation in finite element analysis. The Reissner–Mindlin plates built with multiple NURBS patches are investigated, in which several patches of the model have multi-interface and different patches may share a common point. In order to solve the non-conforming interface problems, Nitsche method is employed to glue different NURBS patches and only refers to the coupling conditions in this work. Various plate shapes, different boundary conditions and several kinds of thickness-span ratios are considered to verify the validity of the presented method. The dimensionless frequencies for different cases are obtained by solving the eigenvalue equation problems and compared with the existing reference solutions or the results calculated by ABAQUS software. Several numerical examples exhibit the effectiveness of the isogeometric approach. It shows that the natural frequencies of the Reissner–Mindlin plate can be successfully predicted by the combination of isogeometric analysis and Nitsche method.