Grid refinement tests of a least-squares finite element method for advective transport of reactive species

A time splitting least-squares finite element method (LSFEM) and the ‘stiff ODEs’ solver LSODE are used to simulate the advective transport of reactive species. Specifically, the rotating cone problem with chemical reactions serves as a model to test the algorithm. A non-linear filter is used to suppress spurious oscillations at each advective time step. All simulations are carried out by using linear and quadratic elements on two mesh systems. Results from the coarse mesh system suggest that the use of robust numerical methods alone will not be able to provide accurate results and point to the need of grid refinement. The grid refinement tests are evaluated by pollutant peak concentrations, pollutant concentration distributions, average relative errors, species mass conservation and distribution ratios and CPU times. Results from the fine mesh system are satisfactory and imply that accurate simulations of the transport of reactive species require adequate grid resolution and robust numerical methods for each individual advective and reactive step.     

Numerical simulation of incompressible flows using adaptive unstructured meshes and the pseudo-compressibility hypothesis

The numerical simulation of incompressible viscous flows, using finite elements with automatic adaptive unstructured meshes and the pseudo-compressibility hypothesis, is presented in this work. Special emphasis is given to the automatic adaptive process of unstructured meshes with linear tetrahedral elements in order to get more accurate solutions at relatively low computational costs. The behaviour of the numerical solution is analyzed using error indicators to detect regions where some important physical phenomena occur. An adaptive scheme, consisting in a mesh refinement process followed by a nodal re-allocation technique, is applied to the regions in order to improve the quality of the numerical solution. The error indicators, the refinement and nodal re-allocation processes as well as the corresponding data structure (to manage the connectivity among the different entities of a mesh, such as elements, faces, edges and nodes) are described. Then, the formulation and application of a mesh adaptation strategy, which includes a refinement scheme, a mesh smoothing technique, very simple error indicators and an adaptation criterion based in statistical theory, integrated with an algorithm to simulate complex two and three dimensional incompressible viscous flows, are the main contributions of this work. Two numerical examples are presented and their results are compared with those obtained by other authors.     

Stabilized low-order finite elements for frictional contact with the extended finite element method

Contact problem suffers from a numerical instability similar to that encountered in incompressible elasticity, in which the normal contact pressure exhibits spurious oscillation. This oscillation does not go away with mesh refinement, and in some cases it even gets worse as the mesh is refined. Using a Lagrange multipliers formulation we trace this problem to non-satisfaction of the LBB condition associated with equal-order interpolation of slip and normal component of traction. In this paper, we employ a stabilized finite element formulation based on the polynomial pressure projection (PPP) technique, which was used successfully for Stokes equation and for coupled solid-deformation–fluid-diffusion using low-order mixed finite elements. For the frictional contact problem the polynomial pressure projection approach is applied to the normal contact pressure in the framework of the extended finite element method. We use low-order linear triangular elements (tetrahedral elements for 3D) for both slip and normal pressure degrees of freedom, and show the efficacy of the stabilized formulation on a variety of plane strain, plane stress, and three-dimensional problems.     

Spurious reflection of elastic waves in nonuniform finite element grids

Elastic wave propagation in a one-dimensional grid of finite elements whose size if uniform from element to element except at one node is analyzed, using complex variables. It is found that spurious wave reflection, along with an increase of amplitude of the diffracted wave, takes place when a wave passes between two finite elements of different sizes. Spurious reflection is significant only for relatively small wavelengths (less than ten times the size of the larger elements) and is important even for very small differences in element size (10%). If the wave arrives from finite elements of smaller size, the transmitted wave has a larger amplitude than the incident wave although the mean energy flux is less. The consistent mass matrix is found to give much smaller spurious reflections than the lumped mass matrix and to enable resolution of smaller wavelengths. This contrasts with the fact that for numerical stability (and suppression of spurious grid oscillations) the lumped mass matrix is better, and for suppression of wave dispersion a combination of lumped and consistent mass matrices is best. The study is restricted to explicit time-step algorithm, second-order (central) difference formulas, and finite elements with linear spatial expansions. In this case it is found that the time step has negligible effects.     

Discontinuous upwind residual distribution: A route to unconditional positivity and high order accuracy

This paper proposes an approach to the approximation of time-dependent hyperbolic conservation laws which is both second order accurate in space and time (for any sufficiently smooth solution profile, even one containing turning points) and free of spurious oscillations for any time-step. The numerical algorithm is based on the concept of fluctuation distribution, applied on a space-time mesh of triangular prisms, for which second order accurate schemes already exist which are oscillation-free if the time-step satisfies a CFL-type constraint. This restriction is lifted here by combining the concept of a two-layer scheme with a representation of the solution which is allowed to be discontinuous-in-time. Numerical results are presented in two space dimensions, using unstructured meshes of space-time triangular prisms, for the scalar advection equation, Burgers’ equation and the Euler equations of gas dynamics.     

High Accuracy Schemes for DNS and Acoustics

High-accuracy schemes have been proposed here to solve computational acoustics and DNS problems. This is made possible for spatial discretization by optimizing explicit and compact differencing procedures that minimize numerical error in the spectral plane. While zero-diffusion nine point explicit scheme has been proposed for the interior, additional high accuracy one-sided stencils have also been developed for ghost cells near the boundary. A new compact scheme has also been proposed for non-periodic problems—obtained by using multivariate optimization technique. Unlike DNS, the magnitude of acoustic solutions are similar to numerical noise and that rules out dissipation that is otherwise introduced via spatial and temporal discretizations. Acoustics problems are wave propagation problems and hence require Dispersion Relation Preservation (DRP) schemes that simultaneously meet high accuracy requirements and keeping numerical and physical dispersion relation identical. Emphasis is on high accuracy than high order for both DNS and acoustics. While higher order implies higher accuracy for spatial discretization, it is shown here not to be the same for time discretization. Specifically it is shown that the 2nd order accurate Adams-Bashforth (AB)—scheme produces unphysical results compared to first order accurate Euler scheme. This occurs, as the AB-scheme introduces a spurious computational mode in addition to the physical mode that apportions to itself a significant part of the initial condition that is subsequently heavily damped. Additionally, AB-scheme has poor DRP property making it a poor method for DNS and acoustics. These issues are highlighted here with the help of a solution for (a) Navier–Stokes equation for the temporal instability problem of flow past a rotating cylinder and (b) the inviscid response of a fluid dynamical system excited by simultaneous application of acoustic, vortical and entropic pulses in an uniform flow. The last problem admits analytic solution for small amplitude pulses and can be used to calibrate different methods for the treatment of non-reflecting boundary conditions as well.     

An adaptive finite element scheme to solve fluid-structure vibration problems on non-matching grids

This paper deals with the computation of the vibration modes of a system consisting of a linear elastic solid interacting with an acoustic fluid. A finite element method based on meshes for each medium not matching on the fluid-solid interface is analyzed. Optimal order of convergence is proved for the approximation of the eigenfunctions, as well as a double order for the eigenvalues. Numerical tests confirming the theoretical results and showing the advantage of using non-matching grids are reported. Finally, an a posteriori error estimator for this method is introduced and combined with a mesh refinement strategy. The efficiency of this adaptive technique is tested with further numerical experiments. Received: 30 January 2001 / Accepted: 30 May 2001     

A priori and a posteriori error analysis for a large-scale ocean circulation finite element model

We consider the finite element solution of the stream function–vorticity formulation for a large-scale ocean circulation model. First, we study existence and uniqueness of solution for the continuous and discrete problems. Under appropriate regularity assumptions we prove that the stream function can be computed with an error of order h in H¹-seminorm. Second, we introduce and analyze an h-adaptive mesh refinement strategy to reduce the spurious oscillations and poor resolution which arise when convective terms are dominant. We propose an a posteriori anisotropic error indicator based on the recovery of the Hessian from the finite element solution, which allows us to obtain well adapted meshes. The numerical experiments show an optimal order of convergence of the adaptive scheme. Furthermore, this strategy is efficient to eliminate the oscillations around the boundary layer.     

Finite element MAC scheme in general curvilinear coordinates

A combined study of F.D. and F.E. methods for 2-D incompressible Navier-Stokes flows is undertaken. In primitive variable formulation, major difficulties are connected to spurious numerical oscillations which may arise from the enforcement of the incompressibility constraint. With regard to this problem, various F.D. schemes differ essentially according to the variable location on the mesh points, while F.E. schemes are analogously differentiated by the interpolation functions adopted for the different variables. In the present paper, we propose an F.E. analog of MAC scheme, which can be accomplished by different interpolation functions for the two velocity components. This new F.E. scheme-although based on low order approximations-eliminates all spurious oscillations. The extension to curvilinear quadrilateral elements—which is needed in order to achieve geometrical versatility—requires the problem to be formulated in general curvilinear coordinates and the contravariant velocity components to be assumed as variables. Some numerical results are presented and discussed, in order to assess the capabilities of the proposed model.     

Online Mesh Refinement for Parallel Atmospheric Models

Forecast precisions of climatological models are limited by computing power and time available for the executions. As more and faster processors are used in the computation, the resolution of the mesh adopted to represent the Earth’s atmosphere can be increased, and consequently the numerical forecast is more accurate. However, a finer mesh resolution, able to include local phenomena in a global atmosphere integration, is still not possible due to the large number of data elements to compute in this case. To overcome this situation, different mesh refinement levels can be used at the same time for different areas of the domain. Thus, our paper evaluates how mesh refinement at run time (online) can improve performance for climatological models.The online mesh refinement (OMR) increases dynamically mesh resolution in parts of a domain,when special atmosphere conditions are registered during the execution. Experimental results show that the execution of a model improved by OMR provides better resolution for the meshes, without any significant increase of execution time. The parallel performance of the simulations is also increased through the creation of threads in order to explore different levels of parallelism.     

Modeling fully nonlinear shallow-water waves and their interactions with cylindrical structures

Introduction of a time-accurate stabilized finite-element approximation for the numerical investigation of fully nonlinear shallow-water waves is presented in this paper. To make the time approximation match the order of accuracy of the spatial representation of the triangular elements by the Galerkin method, the fourth-order time integration of implicit multistage Padé method is used for the development of the numerical scheme. The Streamline-Upwind Petrov-Galerkin (SUPG) method with cross-wind diffusion is employed to stabilize the scheme and suppress the spurious oscillations, usually common in the numerical computation of convection dominated flow problems. The performance of numerical stabilization and accuracy is addressed. Numerical results obtained for cases representing propagation of solitary waves, collisions of two solitary waves, and wave-structure interactions show fairly good agreement with experimental measurements and other published numerical solutions. The comparisons between results from present fully nonlinear wave model and weekly nonlinear wave model are presented and discussed.     

Assessment of implicit operators for the upwind point Gauss-Seidel method on unstructured meshes

The effect of the numerical dissipation level of implicit operators on the stability and convergence characteristics of the upwind point Gauss-Seidel (GS) method for solving the Euler equations was studied through the von Neumann stability analysis and numerical experiments. The stability analysis for linear model equations showed that the point GS method is unstable even for very small CFL numbers when the numerical dissipation level of the implicit operator is equivalent to that of the explicit operator. The stability restriction is rapidly alleviated as the dissipation level of the implicit operator increases. The instability predicted by the linear stability analysis was further amplified as the flow problems became stiffer due to the presence of the shock wave or the refinement of the mesh. It was found that for the efficiency and the robustness of the upwind point GS method, the numerical flux of the implicit operator needs to be more dissipative than that of the explicit operator.     

An iterative procedure for collapse analysis of reinforced concrete plates

An iterative procedure is proposed for evaluating the ultimate load of a laterally loaded plate discretized by finite elements. The procedure regards reinforced concrete plates, but it can be extended to metallic plates without any conceptual change. The stress and displacement fields are approximated by means of a finite element model with constant stress and linear displacement fields. Consequently, any load distribution is represented by the equivalent system of nodal forces for a given mesh. In the set of mechanisms compatible with the assumed discretization the best upper bound to the collapse multiplier of the actual load is obtained via linear programming. By dualization a sequence of linear programming problems is obtained which allows an evaluation of a lower bound of the collapse multiplier for the equivalent load system. When the mesh gets finer and finer, the value obtained does not change substantially anymore. This value can be regarded as an estimate of the collapse multiplier for the original load system. Some numerical examples of plates subjected to uniform pressure confirm the reliability of this approximate multiplier.     

The Modeling of Diffuse Boundaries in the 2-D Digital Waveguide Mesh

The digital waveguide mesh can be used to simulate the propagation of sound waves in an acoustic system. The accurate simulation of the acoustic characteristics of boundaries within such a system is an important part of the model. One significant property of an acoustic boundary is its diffusivity. Previous approaches to simulating diffuse boundaries in a digital waveguide mesh are effective but exhibit limitations and have not been analyzed in detail. An improved technique is presented here that simulates diffusion at boundaries and offers a high degree of control and consistency. This technique works by rotating wavefronts as they pass through a special diffusing layer adjacent to the boundary. The waves are rotated randomly according to a chosen probability function and the model is lossless. This diffusion model is analyzed in detail, and its diffusivity is quantified in the form of frequency dependent diffusion coefficients. The approach used to measuring boundary diffusion is described here in detail for the 2-D digital waveguide mesh and can readily be extended for the 3-D case.     

Application of mortar elements to diffuse-interface methods

An adaptive 2D mesh refinement technique based on mortar spectral elements applied to diffuse-interface methods is presented. The refinement algorithm tracks the movement of the 2D diffuse-interface and subsequently refines the mesh locally at that interface, while coarsening the mesh in the rest of the computational domain, based on error estimators. Convergence of the method is validated using a Gaussian distribution problem and results are presented for a Cahn–Hilliard diffuse-interface model applied to capture the transient dynamics of polymer blends.     

Mesh generation and mesh refinement procedures for the analysis of concrete shells

In this paper, a mesh generation and mesh refinement procedure for adaptive finite element (FE) analyses of real-life surface structures are proposed. For mesh generation, the advancing front method is employed. FE meshes of curved structures are generated in the respective 2D parametric space of the structure. Thereafter, the 2D mesh is mapped onto the middle surface of the structure. For mesh refinement, two different modes, namely uniform and adaptive mesh refinement, are considered. Remeshing in the context of adaptive mesh refinement is controlled by the spatial distribution of the estimated error of the FE results. Depending on this distribution, remeshing may result in a partial increase and decrease, respectively, of the element size. In contrast to adaptive mesh refinement, uniform mesh refinement is characterized by a reduction of the element size in the entire domain. The different refinement strategies are applied to ultimate load analysis of a retrofitted cooling tower. The influence of the underlying FE discretization on the numerical results is investigated.     

A posteriori error estimation and adaptive mesh refinement for combined thermal-stress finite element analysis

Local and global error estimators and an associated h-based adaptive mesh refinement schemes are proposed for coupled thermal-stress problems. The error estimators are based on the “flux smoothing” technique of Zienkiewicz and Zhu with important modifications to improve convergence performance and computational efficiency. Adaptive mesh refinement is based on the concept of adaptive accuracy criteria, previously presented by the authors for stress-based problems and extended here for coupled thermal-stress problems. Three methods of mesh refinement are presented and numerical results indicate that the proposed method is the most efficient in terms of number of adaptive mesh refinements required for convergence in both the thermal and stress solutions. Also, the proposed method required a smaller number of active degrees of freedom to obtain an accurate solution.     

Transient fluid–structure interaction with non-matching spatial and temporal discretizations

This paper presents a review of spatial and temporal discretization schemes for unsteady flow interacting with structure. Two types of spatial coupling schemes are analyzed: (i) point-to-element projection and (ii) common-refinement based projection. It is shown that the point-to-element projection schemes may yield inaccurate load transfer from the source fluid mesh to the target solid mesh, leading to a weak instability in the form of spurious oscillations and overshoots in the interface solution. The common-refinement scheme resolves this problem by providing an accurate transfer of discrete interface conditions across non-matching meshes. With respect to the temporal discretization, three coupling techniques are assessed: (i) conventional sequential staggered (CSS); (ii) generalized serial staggered (GSS) and (iii) combined interface boundary condition (CIBC). Traditionally, continuity of velocity and traction along interfaces are satisfied through algebraic interface conditions applied in a sequential fashion, which is often referred to as staggered computation. In existing partitioned staggered procedures, the interface conditions may undermine stability and accuracy of coupled fluid–structure simulations. By utilizing the CIBC technique on the velocity and traction boundary conditions, a staggered coupling procedure can be constructed with similar order of accuracy and stability as standalone computations. The effectiveness of spatial and temporal coupling schemes is investigated with the aid of simple 1D examples and new 2D subsonic flow-shell aeroelastic simulations.     

On the simulation of wave propagation with a higher-order finite volume scheme based on Reproducing Kernel Methods

In this work we study the dispersion and dissipation characteristics of a higher-order finite volume method based on Moving Least Squares approximations (FV-MLS), and we analyze the influence of the kernel parameters on the properties of the scheme. Several numerical examples are included. The results clearly show a significant improvement of dispersion and dissipation properties of the numerical method if the third-order FV-MLS scheme is used compared with the second-order one. Moreover, with the explicit fourth-order Runge–Kutta scheme the dispersion error is lower than with the third-order Runge–Kutta scheme, whereas the dissipation error is similar for both time-integration schemes. It is also shown than a CFL number lower than 0.8 is required to avoid an unacceptable dispersion error.     

Considerations on higher-order finite elements for multilayered plates based on a Unified Formulation

In this work, a numerical assessment of a series of multilayered finite plate elements is proposed. The finite elements are derived within the “Unified Formulation”, a technique developed by Carrera for an accurate modeling of laminates. The Unified Formulation affords an implementation-friendly possibility to derive a large number of two-dimensional, axiomatic models for plates and shells. An accurate model for multilayered components naturally involves transverse shear and normal stresses, as well as higher-order displacement assumptions in thickness direction. The aim of this work is to give a first insight of the numerical properties of finite elements relying on these formulations. Some considerations concerning the numerical difficulties associated to thickness locking phenomena are presented. A detailed numerical analysis is performed to study the shear locking phenomenon. For the selected case study, once the latter spurious stiffening effect is suppressed with classical or advanced numerical techniques, the resulting elements are shown to behave robustly and accurately.