Simultaneous surface and tetrahedron mesh adaptation using mesh‐free techniques

We present a method to adapt a tetrahedron mesh together with a surface mesh with respect to a size criterion. The originality of our work lies in the fact that both surface and tetrahedron mesh adaptation are carried out simultaneously and that no CAD is required to adapt the surface mesh. The adaptation procedure consists of splitting or removing interior and surface edges which violate a given size criterion. The enrichment process is based on a bisection technique. In order to guarantee mesh conformity during the refinement process, all possible remeshing configurations of tetrahedra have been examined. Once the tetrahedron mesh has been adapted, surface nodes are projected on a geometrical model. The building of a surface model from discrete data has already been presented in this journal. The method is based on a mesh‐free technique called Hermite Diffuse Interpolation. Surface and volume mesh optimization procedures are carried out during the adaptation and at the end of the process to enhance the mesh. Copyright © 2003 John Wiley & Sons, Ltd.     

A multi‐level adaptive mesh refinement method for level set simulations of multiphase flow on unstructured meshes

An adaptive mesh refinement (AMR) technique is proposed for level set simulations of incompressible multiphase flows. The present AMR technique is implemented for two‐dimensional/three‐dimensional unstructured meshes and extended to multi‐level refinement. Smooth variation of the element size is guaranteed near the interface region with the use of multi‐level refinement. A Courant–Friedrich–Lewy condition for zone adaption frequency is newly introduced to obtain a mass‐conservative solution of incompressible multiphase flows. Finite elements around the interface are dynamically refined using the classical element subdivision method. Accordingly, finite element method is employed to solve the problems governed by the incompressible Navier–Stokes equations, using the level set method for dynamically updated meshes. The accuracy of the adaptive solutions is found to be comparable with that of non‐adaptive solutions only if a similar mesh resolution near the interface is provided. Because of the substantial reduction in the total number of nodes, the adaptive simulations with two‐level refinement used to solve the incompressible Navier–Stokes equations with a free surface are about four times faster than the non‐adaptive ones. Further, the overhead of the present AMR procedure is found to be very small, as compared with the total CPU time for an adaptive simulation. Copyright © 2016 John Wiley & Sons, Ltd.     

An adaptive simplex cut‐cell method for high‐order discontinuous Galerkin discretizations of conjugate heat transfer problems

In this paper, we present a solution framework for high‐order discretizations of conjugate heat transfer problems on non‐body‐conforming meshes. The framework consists of and leverages recent developments in discontinuous Galerkin discretization, simplex cut‐cell techniques, and anisotropic output‐based adaptation. With the cut‐cell technique, the mesh generation process is completely decoupled from the interface definitions. In addition, the adaptive scheme combined with the discontinuous Galerkin discretization automatically adjusts the mesh in each sub‐domain and achieves high‐order accuracy in outputs of interest. We demonstrate the solution framework through several multi‐domained conjugate heat transfer problems consisting of laminar and turbulent flows, curved geometry, and highly coupled heat transfer regions. The combination of these attributes yield nonintuitive coupled interactions between fluid and solid domains, which can be difficult to capture with user‐generated meshes. Copyright © 2016 John Wiley & Sons, Ltd.     

Parallel computing of high‐speed compressible flows using a node‐based finite‐element method

An efficient parallel computing method for high‐speed compressible flows is presented. The numerical analysis of flows with shocks requires very fine computational grids and grid generation requires a great deal of time. In the proposed method, all computational procedures, from the mesh generation to the solution of a system of equations, can be performed seamlessly in parallel in terms of nodes. Local finite‐element mesh is generated robustly around each node, even for severe boundary shapes such as cracks. The algorithm and the data structure of finite‐element calculation are based on nodes, and parallel computing is realized by dividing a system of equations by the row of the global coefficient matrix. The inter‐processor communication is minimized by renumbering the nodal identification number using ParMETIS. The numerical scheme for high‐speed compressible flows is based on the two‐step Taylor–Galerkin method. The proposed method is implemented on distributed memory systems, such as an Alpha PC cluster, and a parallel supercomputer, Hitachi SR8000. The performance of the method is illustrated by the computation of supersonic flows over a forward facing step. The numerical examples show that crisp shocks are effectively computed on multiprocessors at high efficiency. Copyright © 2003 John Wiley & Sons, Ltd.     

A Cartesian ghost‐cell multigrid Poisson solver for incompressible flows

In this paper, a simple Cartesian ghost‐cell multigrid Poisson solver is proposed for simulating incompressible fluid flows. The flow field is discretized efficiently on a rectangular mesh, in which solid bodies are immersed. A small number of ghost mesh cells and their symmetric image cells are distributed in the vicinity of the solid boundary. With the aid of the ghost and image cells, the Dirichlet and Neumann boundary conditions can be implemented effectively. Chorin's fractional‐step projection method is adopted for the coupling of velocity and pressure for the solution of the Navier–Stokes equations. Point‐wise Gauss–Seidel iteration is used to solve the pressure Poisson equation. To speed up the convergence of the solution to the corresponding linear system, sub‐level coarse meshes embedded with ghost and image cells are also introduced and operated in a sequential V‐cycle. Several test cases including the classical ideal incompressible flow around a cylinder, a lid‐driven cavity flow and viscous flow past a fixed/rotating cylinder are presented to demonstrate the accuracy and efficiency of the current approach. Copyright © 2010 John Wiley & Sons, Ltd.     

An adaptive level set method based on two‐level uniform meshes and its application to dislocation dynamics

In this paper, we present an adaptive level set method for motion of high codimensional objects (e.g., curves in three dimensions). This method uses only two (or a few fixed) levels of meshes. A uniform coarse mesh is defined over the whole computational domain. Any coarse mesh cell that contains the moving object is further divided into a uniform fine mesh. The coarse‐to‐fine ratios in the mesh refinement can be adjusted to achieve optimal efficiency. Refinement and coarsening (removing the fine mesh within a coarse grid cell) are performed dynamically during the evolution. In this adaptive method, the computation is localized mostly near the moving objects; thus, the computational cost is significantly reduced compared with the uniform mesh over the whole domain with the same resolution. In this method, the level set equations can be solved on these uniform meshes of different levels directly using standard high‐order numerical methods. This method is examined by numerical examples of moving curves and applications to dislocation dynamics simulations. This two‐level adaptive method also provides a basis for using locally varying time stepping to further reduce the computational cost. Copyright © 2013 John Wiley & Sons, Ltd.     

Finite analytic method for 2D steady fluid flows in heterogeneous porous media with unstructured grids

The finite analytic method (FAM) is developed to solve the 2D steady fluid flows in heterogeneous porous media with full tensor permeability on unstructured grids. The proposed FAM is constructed based upon the power‐law analytic nodal solution in the angular domain with arbitrary shape. When approaching the grid node joining the subdomains, 3 different flow patterns may exist: power‐law flow, linear flow, or the stagnant flow. Based on the nodal analytic solution, the triangle‐based FAM is proposed. Numerical examples show that the proposed numerical scheme makes the convergences much quickly than the traditional methods, typically the weighted harmonic mean method under the cell refinement. In practical applications, the grid refinement parameter n = 2 or n = 3 is recommended, and the relative error of the calculated equivalent permeability will below 4% independent of the heterogeneity. In contrast, when using the traditional numerical scheme the refinement ratio for the grid cell needs to increase dramatically to get an accurate result, especially for strong heterogeneous porous medium.     

Application of the higher‐order Cauchy–Born rule in mesh‐free continuum and multiscale simulation of carbon nanotubes

This paper investigates the application of a recently proposed higher‐order Cauchy–Born rule in the continuum simulation and multiscale analysis of carbon nanotubes (CNTs). A mesh‐free computational framework is developed to implement the numerical computation of the hyper‐elastic constitutive model that is derived from the higher‐order Cauchy–Born rule. The numerical computation reveals that the buckling pattern of a single‐walled carbon nanotube (SWCNT) can be accurately displayed by taking into consideration the second‐order deformation gradient, and fewer mesh‐free nodes can provide a good simulation of homogeneous deformation. The bridging domain method is employed to couple the developed mesh‐free method and the atomistic simulation. The coupling method is used to simulate the bending buckling of an SWCNT and the tensile failure of an SWCNT with a single‐atom vacancy defect, and good computational results are obtained. Copyright © 2008 John Wiley & Sons, Ltd.     

The tetrahedral finite cell method: Higher‐order immersogeometric analysis on adaptive non‐boundary‐fitted meshes

The finite cell method (FCM) is an immersed domain finite element method that combines higher‐order non‐boundary‐fitted meshes, weak enforcement of Dirichlet boundary conditions, and adaptive quadrature based on recursive subdivision. Because of its ability to improve the geometric resolution of intersected elements, it can be characterized as an immersogeometric method. In this paper, we extend the FCM, so far only used with Cartesian hexahedral elements, to higher‐order non‐boundary‐fitted tetrahedral meshes, based on a reformulation of the octree‐based subdivision algorithm for tetrahedral elements. We show that the resulting TetFCM scheme is fully accurate in an immersogeometric sense, that is, the solution fields achieve optimal and exponential rates of convergence for h‐refinement and p‐refinement, if the immersed geometry is resolved with sufficient accuracy. TetFCM can leverage the natural ability of tetrahedral elements for local mesh refinement in three dimensions. Its suitability for problems with sharp gradients and highly localized features is illustrated by the immersogeometric phase‐field fracture analysis of a human femur bone. Copyright © 2016 John Wiley & Sons, Ltd.     

A front remeshing technique for a Lagrangian description of moving interfaces in two‐fluid flows

The numerical analysis of two‐fluid flows involves the treatment of a discontinuity that appears at the separating interface. Classical Lagrangian schemes applied to update the front position between two immiscible incompressible fluids have been long recognized to provide a sharp representation of the interface. However, the main drawback of these approaches is the progressive distortion in the distribution of the markers used to identify the material front. To avoid this problem, an interface remeshing algorithm based on the diffuse approximation of the interface curvature is proposed in this work. In addition, the remeshed front is enforced to preserve the global volume. These new aspects are incorporated in an existing fluid dynamics formulation for the analysis of two‐fluid flows problems. The resulting formulation is called in this work as the moving Lagrangian interface remeshing technique (MLIRT). Copyright © 2005 John Wiley & Sons, Ltd.     

Simulations of flow through fluid/porous layers by a characteristic‐based method on unstructured grids

An upwind characteristic‐based finite volume method on unstructured grids is employed for numerical simulation of incompressible laminar flow and forced convection heat transfer in 2D channels containing simultaneously fluid layers and fluid‐saturated porous layers. Hydrodynamic and heat transfer results are reported for two configurations: the first one is a backward‐facing step channel with a porous block inserted behind the step, and the second one is a partially porous channel with discrete heat sources on the bottom wall. The effects of Darcy numbers on heat transfer augmentation and pressure loss were investigated for low Reynolds laminar flows. The results demonstrate the accuracy and robustness of the numerical scheme proposed, and suggest that partially porous insertion in a channel can significantly improve heat transfer performance with affordable pressure loss. Copyright © 2001 John Wiley & Sons, Ltd.     

A BEM‐level set domain‐decomposition strategy for non‐linear and fragmented interfacial flows

A domain‐decomposition algorithm has been developed to handle two‐phase flows with large deformation, breaking and fragmentation of the interface. The strategy couples a boundary element method with a Navier–Stokes solver combined with a level‐set technique for the tracking of the interface. The former is used in the fluid region where the interface can be modelled as a smooth surface. In the rest of the domain the field solver is applied. This results in an efficient and accurate method. In this paper, the features of the used strategy are described and the challenges connected with the coupling are deeply discussed. The numerical investigation highlighted the importance of a proper rational study when CFD methods are considered. In the present case, a crucial aspect is represented by the domain‐composition step, that is when the information from one solver to the other have to be properly reconstructed and made consistent with the receiver sub‐domain. Copyright © 2006 John Wiley & Sons, Ltd.     

A consistent grayscale‐free topology optimization method using the level‐set method and zero‐level boundary tracking mesh

This paper proposes a level‐set based topology optimization method incorporating a boundary tracking mesh generating method and nonlinear programming. Because the boundary tracking mesh is always conformed to the structural boundary, good approximation to the boundary is maintained during optimization; therefore, structural design problems are solved completely without grayscale material. Previously, we introduced the boundary tracking mesh generating method into level‐set based topology optimization and updated the design variables by solving the level‐set equation. In order to adapt our previous method to general structural optimization frameworks, the incorporation of the method with nonlinear programming is investigated in this paper. To successfully incorporate nonlinear programming, the optimization problem is regularized using a double‐well potential. Furthermore, the sensitivities with respect to the design variables are strictly derived to maintain consistency in mathematical programming. We expect the investigation to open up a new class of grayscale‐free topology optimization. The usefulness of the proposed method is demonstrated using several numerical examples targeting two‐dimensional compliant mechanism and metallic waveguide design problems. Copyright © 2014 John Wiley & Sons, Ltd.     

An adaptive non‐conforming finite‐element method for Reissner–Mindlin plates

Adaptive algorithms are important tools for efficient finite‐element mesh design. In this paper, an error controlled adaptive mesh‐refining algorithm is proposed for a non‐conforming low‐order finite‐element method for the Reissner–Mindlin plate model. The algorithm is controlled by a reliable and efficient residual‐based a posteriori error estimate, which is robust with respect to the plate's thickness. Numerical evidence for this and the efficiency of the new algorithm is provided in the sense that non‐optimal convergence rates are optimally improved in our numerical experiments. Copyright © 2003 John Wiley & Sons, Ltd.     

An adaptive fixed mesh method for modelling and simulating of unsteady two‐phase flows with sharp physical interfaces

We present in this paper a new computational method for simulation of two‐phase flow problems with moving boundaries and sharp physical interfaces. An adaptive interface‐capturing technique (ICT) of the Eulerian type is developed for capturing the motion of the interfaces (free surfaces) in an unsteady flow state. The adaptive method is mainly based on the relative boundary conditions of the zero pressure head, at which the interface is corresponding to a free surface boundary. The definition of the free surface boundary condition is used as a marker for identifying the position of the interface (free surface) in the two‐phase flow problems. An initial‐value‐problem (IVP) partial differential equation (PDE) is derived from the dynamic conditions of the interface, and it is designed to govern the motion of the interface in time. In this adaptive technique, the Navier–Stokes equations written for two incompressible fluids together with the IVP are solved numerically over the flow domain. An adaptive mass conservation algorithm is constructed to govern the continuum of the fluid. The finite element method (FEM) is used for the spatial discretization and a fully coupled implicit time integration method is applied for the advancement in time. FE‐stabilization techniques are added to the standard formulation of the discretization, which possess good stability and accuracy properties for the numerical solution. The adaptive technique is tested in simulation of some numerical examples. With the test problems presented here, we demonstrated that the adaptive technique is a simple tool for modelling and computation of complex motion of sharp physical interfaces in convection–advection‐dominated flow problems. We also demonstrated that the IVP and the evolution of the interface function are coupled explicitly and implicitly to the system of the computed unknowns in the flow domain. Copyright © 2005 John Wiley & Sons, Ltd.     

An anisotropic Zienkiewicz–Zhu‐type error estimator for 3D applications

We extend the anisotropic Zienkiewicz–Zhu a posteriori error estimator of (Proceedings of the ENUMATH‐2009, Uppsala, Sweden, 29 June–3 July 2009) to three dimensions. Like the standard Zienkiewicz–Zhu estimator, the proposed estimator is designed to be independent of the problem at hand, is cheap to compute and easy to implement. In contrast to the standard Zienkiewicz–Zhu estimator, the elementwise counterpart of the proposed estimator explicitly takes into account the geometrical properties of the actual tetrahedron. Thus, in a wide variety of applications, the estimator is able to detect the anisotropic features exhibited by the solution of the governing equations. A metric‐based optimization procedure, rigorously addressed, drives the adaptation of the mesh. It is shown numerically to yield quasi‐optimal triangulations, dictating the accuracy‐vs‐number of elements behaviour. Despite being heuristic to some extent, in practice the overall anisotropic adaptation procedure turns out to be effective. Copyright © 2010 John Wiley & Sons, Ltd.     

An adaptive finite element method for second‐order plate theory

We present a discontinuous finite element method for the Kirchhoff plate model with membrane stresses. The method is based on P₂‐approximations on simplices for the out‐of‐plane deformations, using C⁰‐continuous approximations. We derive a posteriori error estimates for linear functionals of the error and give some numerical examples. Copyright © 2009 John Wiley & Sons, Ltd.     

Multi‐time‐step explicit–implicit method for non‐linear structural dynamics

We present a method with domain decomposition to solve time‐dependent non‐linear problems. This method enables arbitrary numeric schemes of the Newmark family to be coupled with different time steps in each subdomain: this coupling is achieved by prescribing continuity of velocities at the interface. We are more specifically interested in the coupling of implicit/explicit numeric schemes taking into account material and geometric non‐linearities. The interfaces are modelled using a dual Schur formulation where the Lagrange multipliers represent the interfacial forces. Unlike the continuous formulation, the discretized formulation of the dynamic problem is unable to verify simultaneously the continuity of displacements, velocities and accelerations at the interfaces. We show that, within the framework of the Newmark family of numeric schemes, continuity of velocities at the interfaces enables the definition of an algorithm which is stable for all cases envisaged. To prove this stability, we use an energy method, i.e. a global method over the whole time interval, in order to verify the algorithms properties. Then, we propose to extend this to non‐linear situations in the following cases: implicit linear/explicit non‐linear, explicit non‐linear/explicit non‐linear and implicit non‐linear/explicit non‐linear. Finally, we present some examples showing the feasibility of the method. Copyright © 2001 John Wiley & Sons, Ltd.     

Implicit boundary method for finite element analysis using non‐conforming mesh or grid

In the finite element method (FEM), a mesh is used for representing the geometry of the analysis and for representing the test and trial functions by piece‐wise interpolation. Recently, analysis techniques that use structured grids have been developed to avoid the need for a conforming mesh. The boundaries of the analysis domain are represented using implicit equations while a structured grid is used to interpolate functions. Such a method for analysis using structured grids is presented here in which the analysis domain is constructed by Boolean combination of step functions. Implicit equations of the boundary are used in the construction of trial and test functions such that essential boundary conditions are guaranteed to be satisfied. Furthermore, these functions are constructed such that internal elements, through which no boundary passes, have the same stiffness matrix. This approach has been applied to solve linear elastostatic problems and the results are compared with analytical and finite element analysis solutions to show that the method gives solutions that are similar to the FEM in quality but is less computationally expensive for dense mesh/grid and avoids the need for a conforming mesh. Copyright © 2007 John Wiley & Sons, Ltd.