A finite element immersed boundary method for fluid flow around moving objects

This paper presents the extension of a recently proposed immersed boundary method to the solution of the flow around moving objects. Solving the flow around objects with complex shapes may involve extensive meshing work that has to be repeated each time a change in the geometry is needed. Mesh generation and solution interpolation between successive grids may be costly and introduce errors if the geometry changes significantly during the course of the computation. These drawbacks are avoided when the solution algorithm can tackle grids that do not fit the shape of immersed objects. This work presents an extension of our recently developed finite element Immersed Boundary (IB) method to transient applications involving the movement of immersed fluid/solid interfaces. As for the fixed solid boundary case, the method produces solutions of the flow satisfying accurately boundary conditions imposed on the surface of immersed bodies. The proposed algorithm enriches the finite element discretization of interface elements with additional degrees of freedom, the latter being eliminated at element level. The boundary of immersed objects is defined using a time dependent level-set function. Solutions are shown for various flow problems and the accuracy of the present approach is measured with respect to solutions on body-conforming meshes.     

A finite element method for flow separation

The finite element model has been developed in order to solve separation pattern of the flow past an obstruction in a two-dimensional flow field. The Helmholtz-Poisson form of the Reynolds equations are solved alternately until a stable flow separation in the neighbourhood of the obstruction is obtained. In order to check the results of the finite element model, an experimental separation pattern using Pitot-tube measurements has been conducted. The computed and the experimental flow separation patterns show a good agreement.     

A finite element study of hull-deckhouse interaction

A medium sized three dimensional elastic finite element analysis which investigates the force interaction and the stress distribution in the hull and deckhouse of a ship is presented. SESAM 69 was used to carry out the analysis and the 5000 degree of freedom model is described in detail. The results show that the deckhouse contributes significantly to the effective section modulus at midships but increases the longitudinal and shear stresses in the vicinity of the deckhouse ends and the vertical stresses in the hull bulkheads. Comparisons are made with stress levels evaluated by simple beam theory and by Caldwell's approximate theory.     

A stabilized mixed finite element method for Darcy flow

We develop new stabilized mixed finite element methods for Darcy flow. Stability and an a priori error estimate in the “stability norm” are established. A wide variety of convergent finite elements present themselves, unlike the classical Galerkin formulation which requires highly specialized elements. An interesting feature of the formulation is that there are no mesh-dependent parameters. Numerical tests confirm the theoretical results.     

Accuracy and convergence of a finite element algorithm for laminar boundary layer flow

The Galerkin-weighted residuals formulation is employed to derive an implicit finite element solution algorithm for a generally non-linear initial-boundary value problem. Solution accuracy and convergence with discretization refinement are quantized in several error norms, for the non-linear parabolic partial differential equation system governing laminar boundary layer flow, using linear, quadratic and cubic functions. Richardson extrapolation is used to isolate integration truncation error in all norms, and Newton iteration is employed for all equation solutions performed in double-precision. The mathematical theory supporting accuracy and convergence concepts for linear elliptic equation appears extensible to the non-linear equations characteristic of laminar boundary layer flow.     

Computational and experimental investigation into aerodynamic interference between slender bodies in supersonic flow

Aerodynamic interference can occur between high-speed bodies when in close proximity. A complex flowfield develops where shock and expansion waves from a generator body impinge upon the adjacent receiver body and modify its aerodynamic characteristics. The aims of this paper are to validate a computational prediction method, to use the predicted solutions to interpret the measured results and to provide a deeper understanding of the associated flow physics.The interference aerodynamics for two slender bodies were investigated through a parametric wind tunnel study where the effect of axial stagger was investigated for different receiver body incidence angles. Measurements included forces and moments, surface pressures and shadowgraph visualisations. Supporting computational predictions provided a deeper understanding of the underlying aerodynamics and flow mechanisms. Good agreement was found between the measured and predicted interference loads and surface pressures for all configurations.The interference loads are strongly dependent upon the axial impingement location of the primary shockwave. These induced interference loads change polarity as the impingement location moves aft over the receiver. Distinct interference characteristics are observed when the receiver is placed at high positive incidence, where the impinging shock has a strong effect on the crossflow separation location. Overall, the observed interference effects are expected to modify the subsequent body trajectories and may increase the likelihood of a collision.     

A fast algorithm for point-location in a finite element mesh

An algorithm for the point-location problem in 2D finite element meshes as a special case of plane straight-line graphs (PSLG) is presented. The element containing a given point P is determined combining a quadtree data structure to generate a quaternary search tree and a local search wave using adjacency information. The preprocessing construction of the search tree has a complexity ofO(n·log(n)) and requires only pointer swap operations. The query time to locate a start element for local search isO(log(n)) and the final point search by ‘point-in-polygon’ tests is independent of the total number of elements in the mesh and thus determined in constant time. Although the theoretical efficiency estimates are only given for quasi-uniform meshes, it is shown in numerical examples, that the algorithm performs equally well for meshes with extreme local refenement.     

A finite element based level-set method for multiphase flow applications

A numerical method for simulating incompressible two-dimensional multiphase flow is presented. The method is based on a level-set formulation discretized by a finite-element technique. The treatment of the specific features of this problem, such as surface tension forces acting at the interfaces separating two immiscible fluids, as well as the density and viscosity jumps that in general occur across such interfaces, have been integrated into the finite-element framework. Using a method based on the weak formulation of the Navier-Stokes equations has its advantages. In this formulation, the singular surface tension forces are included through line integrals along the interfaces, which are easily approximated quantities. In addition, differentiation of the discontinuous viscosity is avoided. The discontinuous density and viscosity are included in the finite element integrals. A strategy for the evaluation of integrals with discontinuous integrands has been developed based on a rigorous analysis of the errors associated with the evaluation of such integrals. Numerical tests have been performed. For the case of a rising buoyant bubble the results are in good agreement with results from a front-tracking method. The run presented here is a run including topology changes, where initially separated areas of one fluid merge in different stages due to buoyancy effects. Received: 1 March 1999 / Accepted: 17 June 1999     

Three dimensional finite element modeling of thermomechanical frictional contact between finite deformation bodies using R-minimum strategy

An algorithm for analyzing the transient thermal coupling with the frictional contact between the multiple elastic–plastic bodies in finite deformation is presented using the R-minimum strategy. An arbitrarily shaped contact element strategy, named as node-to-point contact element strategy, is proposed to handle the thermomechanical frictional contact between finite deformation bodies. Assuming the material properties to be temperature dependent, the constitutive equations for both the thermomechanical frictional contact and the thermal-elastic–plastic materials are deduced respectively and applied in our finite element code. Finally, two examples are presented to show the efficiency and usefulness of this algorithm.     

A higher order finite element algorithm for the unsteady Navier-Stokes equations

A higher order element, the Tocher 10 or C₀ Cubic on triangles, is the base for formulation of a finite element algorithm for numerical calculation of fluid flows governed by the unsteady Navier-Stokes equations. Results from the calculation of supersonic free shear layer flow are numerically accurate and in excellent agreement with finite difference solutions. Diverse characteristics for these two classes of methods emerge when the requirements of core storage and computer time are considered.     

A simple algorithm for three-dimensional finite element analysis of contact problems

This paper addresses the numerical solution of three-dimensional frictionless contact problems by a finite element method. The two-body contact problem is considered in the context of fully non-linear kinematics. The impenetrability constraint is satisfied via a classical penalty formulation. The contacting surfaces are discretized by means of projections of the interacting element faces onto suitably chosen flat surfaces. Attention is focused on the efficiency of the overall algorithm. Numerical simulations are conducted for a series of test problems to assess the performance of the proposed methodology.     

A solution algorithm for linear constraint equations in finite element analysis

A general solution algorithm is presented for the incorporation of a general set of linear constraint equations into a linear algebraic system; such situations arise in the application of the finite element method to a variety of physical problems. Implementation of the algorithm, without need for pre-arranging the equations, into an equation solver using Gauss elimination is developed. The method is most attractive as compared to other approaches for constrained systems.     

A new method for the coupling of finite element and boundary element discretized subdomains of elastic bodies

A new and efficient approach for the coupling of subregions of elastic solids discretized by means of finite elements (FE) and boundary elements (BE), respectively, is presented. The method is characterized by so-called ‘bi-condensation’ of nodal degrees of freedom followed by the transformation of the resulting BEM-related traction-displacement equations for the interface(s) of the BE subregion(s) and the FE subdomain(s) to ‘FEM-like’ force-displacement relations which are assembled with the FEM-related force-displacement equations for the interface(s). The presented ‘local FE coupling approach’ is computationally more economic than a global coupling approach since it only requires the inversion of BEM-related coefficient matrices referred to the interfaces of BE subregions and FE subdomains. Depending on whether the principle of virtual displacements or the principle of minimum of potential energy is used for the generation of force-displacement equations for the coupling interface(s), unsymmetric or symmetric coefficient matrices are obtained. Since the two principles are mechanically equivalent, identical results would be achieved in the limit of finite discretizations.The numerical investigation has shown that, depending on the problem and the discretization, the results obtained on the basis of symmetric coefficient matrices may be poor. This applies to ‘edge problems’ characterized by discontinuous tractions along the edges. On the basis of unsymmetric coefficient matrices, however, satisfactory results are obtained even for relatively coarse discretizations.     

An algorithm for partitioning of finite element meshes

An efficient algorithm is developed for partitioning unstructured finite element meshes. A new graph model is presented and employed for transforming the connectivity properties of meshes. This method leads to a load balance partitioning in which the number of interface nodes are confined to the smallest possible, and aspect ratios of subdomains are desired values. Examples are included to illustrate the performance and efficiency of the presented method.     

A finite element approach for multiphase fluid flow in porous media

The main scope of this work is to carry out a mathematical framework and its corresponding finite element (FE) discretization for the partially saturated soil consolidation modelling in presence of an immiscible pollutant. A multiphase system with the interstitial voids in the grain matrix filled with water (liquid phase), water vapour and dry air (gas phase) and with pollutant substances, is assumed. The mathematical model addressed in this work was developed in the framework of mixture theory considering the pollutant saturation-suction coupling effects. The ensuing mathematical model involves equations of momentum balance, energy balance and mass balance of the whole multiphase system. Encouraging outcomes were achieved in several different examples.     

A variational multiscale higher-order finite element formulation for turbomachinery flow computations

The Variational MultiScale approach for finite elements addresses the inclusion of the effect of fine scales of the solution in the coarse problem. In this framework advective–diffusive–reactive equations modelling turbomachinery flows feature both advection and reaction induced instabilities to be tackled. To this end, this work deals with a new method called V-SGS (Variable-SubGrid Scale), designed for quadratic elements, with a variable ‘intrinsic time’ parameter. Two-dimensional tests have been considered to compare V-SGS against SUPG and other stabilization devices, including the flow around a NACA 4412 airfoil to assess its reliability in the handling of advanced turbulence closures on cruder meshes.     

稀疏有限元线性系统的并行算法实现

在对称多处理机系统上,提出了一种求解稀疏对称有限元线性系统的正规化精确并行逆算法。该算法以一种避免数据依赖的反对角运动方法为基础,使用OpenMP编译指导来实现。诸如加速比和效率等数值实验结果的推出,说明在一个对称多处理机系统上,所提出的算法求解方法能更好地提高性能,获得更大的加速。     

Application of the moving finite element method to moving boundary Stefan problems

The moving finite element method is modified to be an effective solution for a version of the classical ice-water Stefan problem. Also, the cylindrical Stefan problem is numerically solved and compared with an existing perturbation solution.     

A computational algorithm for system modelling based on bi-dimensional finite element techniques

This paper develops a modelling methodology for studying relations defined a priori in two dimensions, z = u(x, y), for which experimental data are known. An innovation is that the methodology is based on bi-dimensional finite element techniques, in which the model’s value consists of a finite number of points, making it possible to obtain its value at any point. Its application permits obtaining representational models of the relation. A computational algorithm is presented; its program has been called Finit Trap 2D, which generates families of models. It has defined criteria for model selection based on information parameters.This scientific research technique complements those existing in scientific literature for generating mathematical models based on experimental data [Cortés M, Villacampa Y, Mateu J, Usó, JL. A new methodology for modelling highly structured systems. Environmental modelling and software 2000;15(5):461–70, S-PLUS 2000. Guide to statistics, vosl. 1–2. Mathsoft Inc.; 1999, SPSS 12.0. Guide to statistics, Mathsoft Inc.; 2003, Verdu F. Un algoritmo para la construcción múltiple de modelos matemáticos no lineales y el estudio de su estabilidad, Thesis Doctoral, Universidad de Alicante; 2001, Verdu F, Villacampa Y. A computational algorithm for the multiple generation of nonlinear mathematical models and stability study. Advances in Engineering Software 2008;39(5):430–7, Villacampa Y, Cortés M, Vives F, Usó JL, Castro MA. A new computational algorithm to construct mathematical models. In: Ecosystems and sustainable development II. Advances in ecological sciences, vol. 2, WIT Press: Southampton, Boston; 1999], and will be useful in the effort to simplify the model.It should be emphasised that representational models have been generated from bi-dimensional finite element models. This will naturally lead to their future application in processes described by partial differential equation whose coefficients are functions, A(x, y) for which only experimental data are known.