An unfitted interface penalty method for the numerical approximation of contrast problems

We aim to approximate contrast problems by means of a numerical scheme which does not require that the computational mesh conforms with the discontinuity between coefficients. We focus on the approximation of diffusion-reaction equations in the framework of finite elements. In order to improve the unsatisfactory behavior of Lagrangian elements for this particular problem, we resort to an enriched approximation space, which involves elements cut by the interface. Firstly, we analyze the H¹-stability of the finite element space with respect to the position of the interface. This analysis, applied to the conditioning of the discrete system of equations, shows that the scheme may be ill posed for some configurations of the interface. Secondly, we propose a stabilization strategy, based on a scaling technique, which restores the standard properties of a Lagrangian finite element space and results to be very easily implemented. We also address the behavior of the scheme with respect to large contrast problems ending up with a choice of Nitsche?s penalty terms such that the extended finite element scheme with penalty is robust for the worst case among small sub-elements and large contrast problems. The theoretical results are finally illustrated by means of numerical experiments.     

无网格Galerkin法与有限元耦合新算法

通过构造新的斜坡函数,把无网格Galerkin法与有限元耦合算法应用到全域范围,并使其能适应不同连接域内单元结点构成,既满足了本质边界条件实现的需要,又能方便灵活的布置无网格点和有限元法中的单元,满足复杂计算要求．计算结果与理论解比较表明所提出的方法是可行和有效的．     

A multiscale finite element method for the solution of transport equations

A multiscale Galerkin finite element scheme based on the residual free bubble function method is proposed to generate stable and accurate solutions for the transport equations namely diffusion-reaction (DR), convection-diffusion (CD) and convection-diffusion-reaction (CDR) equations. These equations show multiscale behavior in reaction or convection dominated situations. The idea is based on the approximation of the definite integral of the interpolation function within the element, instead of the function approximation. The numerical experiments are performed using the bilinear Lagrangian elements. To validate the approach, the numerical results obtained for a benchmark problem are compared with the analytical solution in a wide range of Peclet and Damköhler numbers. The results show that the developed method is capable of generating stable and accurate solutions.     

Numerical approaches for computation of fronts

Moving fronts and pulses appear in many engineering applications like flame propagation and a falling liquid film. Standard computation methods are inappropriate since the problem is defined over an infinite domain and a steady-state solution exists only for a certain front velocity. This work presents a transformation that converts the original problem into a boundary-value problem within a finite domain, in a way that preserves the behavior at the boundaries. Good low-order approximations can be obtained as demonstrated by two examples. In another approach, a central element of adjustable length is incorporated into a three-element structure where the edge-elements obey known asymptotic solutions. That yields multiplicity of travelling fronts in an infinite domain but it successfully approximates standing wave solutions in a finite domain. The approximate solutions are shown to obey the qualitative features known for the exact solutions, like asymptotic solutions or the bifurcation set–the boundary where a new solution emerges or disappears.     

The application of an implicit finite element algorithm with a frontal solution technique to the analysis of time dependent viscous flows

A finite element algorithm is described which implements the Galerkin approximation to the Navier-Stokes equations and incorporates five predominate features. Although none of these five features is unique to this algorithm, their orchestration, as described in this paper, results in an algorithm which is not only easy to implement but also stable, accurate, and robust, as well as computationally efficient. The zero stress natural boundary condition is implemented which permits calculation of the outflow velocity distribution. A nine-node, Lagrangian, isoparametric, quadrilateral elementis used to represent the velocity while the pressure uses a four-node, Lagrangian, superparametric element coincident with the velocity element. The easily implemented, computationally efficient frontal solution technique is used to assemble the element coefficient matrices, impose the boundary conditions, and solve the resulting linear system of equations. An implicit backward Euler time integration rule provides a very stable solution method for time dependent problems. A Picard scheme with a relatively large radius of convergence is used for iteration of the non-linear equations at each time step. Results are given from the calculation of two dimensional, steady-state and time-dependent convection dominated flows of viscous incompressible fluids.     

Electromagnetic finite elements based on a four-potential variational principle

We derive electromagnetic finite elements based on a variational principle that uses the electromagnetic four-potential as primary variable. This choice is used to construct elements suitable for downstream coupling with mechanical and thermal finite elements for the analysis of electromagnetic/mechanical systems that involve superconductors. The main advantages of the four-potential as a basis for finite element formulation are: the number of degrees of freedom per node remains modest as the problem dimensionality increases, jump discontinuities on interfaces are naturally accomodated, and statics as well as dynamics may be treated without any a priori approximations. The new elements are tested on an axisymmetric problem under steady-state forcing conditions. The results are in excellent agreement with analytical solutions.     

Solution-precipitation creep – micromechanical modelling and numerical results

Our aim is to present a continuum mechanical model for solution-precipitation creep as well as to compare the numerical results based on that model with experimental observations. The formulation of the problem is based on the minimization of a Lagrangian consisting of elastic power and dissipation. Elastic energy is chosen to be in a standard form but dissipation is strongly adapted to the solution-precipitation process by introducing two new quantities: the velocity of material transport within the crystallite-interfaces and the normal velocity of precipitation or solution respectively. The model enables one to give an analytical solution for the case of a single crystal and numerical solution based on a finite element method for more complex, polycrystalline materials. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

湍流边界层中固体小颗粒湍流运动的Lagrangian模型

给出了固体小颗粒在边界层中的Lagrangian运动方程,方程中包括受壁面影响的粘性阻力,Saffman升力及Magus升力等.使用频谱法,得到了颗粒响应流体的Lagrangian能谱的表达式,使用这些结果研究了各种响应特性.本文的结果清楚地表明了固体个颗粒在湍流扩散过程中,其湍流扩散是可能大于流体的.     

Pure Lagrangian and semi-Lagrangian finite element methods for the numerical solution of Navier–Stokes equations

In this paper we propose a unified formulation to introduce Lagrangian and semi-Lagrangian velocity and displacement methods for solving the Navier–Stokes equations. This formulation allows us to state classical and new numerical methods. Several examples are given. We combine them with finite element methods for spatial discretization. In particular, we propose two new second-order characteristics methods in terms of the displacement, one semi-Lagrangian and the other one pure Lagrangian. The pure Lagrangian displacement methods are useful for solving free surface problems and fluid-structure interaction problems because the computational domain is independent of the time and fluid–solid coupling at the interphase is straightforward. However, for moderate to high-Reynolds number flows, they can lead to high distortion in the mesh elements. When this happens it is necessary to remesh and reinitialize the transformation to the identity. In order to assess the performance of the obtained numerical methods, we solve different problems in two space dimensions. In particular, numerical results for a sloshing problem in a rectangular tank and the flow in a driven cavity are presented.     

A cellular automaton technique for modelling of a binary dendritic growth with convection

A two-dimensional model for the simulation of a binary dendritic growth with convection has been developed in order to investigate the effects of convection on dendritic morphologies. The model is based on a cellular automaton (CA) technique for the calculation of the evolution of solid/liquid (s/l) interface. The dynamics of the interface controlled by temperature, solute diffusion and Gibbs–Thomson effects, is coupled with the continuum model for energy, solute and momentum transfer with liquid convection. The solid fraction is calculated by a governing equation, instead of some approximate methods such as lever rule method [A. Jacot, M. Rappaz, Acta Mater. 50 (2002) 1909–1926.] or interface velocity method [L. Nastac, Acta Mater. 47 (1999) 4253; L. Beltran-Sanchez, D.M. Stefanescu, Mat. and Mat. Trans. A 26 (2003) 367.]. For the dendritic growth without convection, mesh independency of simulation results is achieved. The simulated steady-state tip velocity are compared with the predicted values of LGK theory [Lipton, M.E. Glicksmanm, W. Kurz, Metall. Trans. 18(A) (1987) 341.] as a function of melt undercooling, which shows good agreement. The growth of dendrite arms in a forced convection has been investigated. It was found that the dendritic growth in the upstream direction was amplified, due to larger solute gradient in the liquid ahead of the s/l interface caused by melt convection. In the isothermal environment, the calculated results under very fine mesh are in good agreement with the Oseen–Ivanstov solution for the concentration-driven growth in a forced flow.     

Unstructured finite volume method for matrix free explicit solution of stress-strain fields in two dimensional problems with curved boundaries in equilibrium condition

In this paper, a plane stress structural solver which uses a matrix free unstructured finite volume method based on Galerkin approach is introduced for solution of weak form of two dimensional Cauchy equations on linear triangular element meshes. The developed shape function free Galerkin finite volume structural solver explicitly computes stresses and displacements in cartesian coordinate directions for the two dimensional solid mechanic problems in equilibrium condition. The accuracy of the introduced algorithm is assessed by comparison of computed results of two plane-stress cases with curved boundaries under uniformly distributed loads with available analytical solutions. The results of the introduced method are presented in terms of stress and strain contours and its effective parameters on convergence behaviour to equilibrium condition are assessed.     

A new coupled model for alloy solidification

A new coupled model in the binary alloy solidification has been developed. The model is based on the cellular automaton (CA) technique to calculate the evolution of the interface governed by temperature, solute diffusion and Gibbs-Thomson effect. The diffusion equation of temperature with the release of latent heat on the solid/liquid (S/L) interface is valid in the entire domain. The temperature diffusion without the release of latent heat and solute diffusion are solved in the entire domain. In the interface cells, the     

A new coupled model for alloy solidification

A new coupled model in the binary alloy solidification has been developed. The model is based on the cellular automaton (CA) technique to calculate the evolution of the interface governed by temperature, solute diffusion and Gibbs-Thomson effect. The diffusion equation of temperature with the release of latent heat on the solid/liquid (S/L) interface is valid in the entire domain. The temperature diffusion without the release of latent heat and solute diffusion are solved in the entire domain. In the interface cells, the energy and solute conservation, thermodynamic and chemical potential equilibrium are adopted to calculate the temperature, solid concentration, liquid concentration and the increment of solid fraction. Compared with other models where the release of latent heat is solved in implicit or explicit form according to the solid/liquid (S/L) interface velocity, the energy diffusion and the release of latent heat in this model are solved at different scales, i.e. the macro-scale and micro-scale. The variation of solid fraction in this model is solved using several algebraic relations coming from the chemical potential equilibrium and thermodynamic equilibrium which can be cheaply solved instead of the calculation of S/L interface velocity. With the assumption of the solute conservation and energy conservation, the solid fraction can be directly obtained according to the thermodynamic data. This model is natural to be applied to multiple (< 2) spatial dimension case and multiple (< 2) component alloy. The morphologies of equiaxed dendrite are obtained in numerical experiments.     

Vibration analysis of rectangular plates coupled with fluid

The approach developed in this paper applies to vibration analysis of rectangular plates coupled with fluid. This case is representative of certain key components of complex structures used in industries such as aerospace, nuclear and naval. The plates can be totally submerged in fluid or floating on its free surface. The mathematical model for the structure is developed using a combination of the finite element method and Sanders’ shell theory. The in-plane and out-of-plane displacement components are modelled using bilinear polynomials and exponential functions, respectively. The mass and stiffness matrices are then determined by exact analytical integration. The velocity potential and Bernoulli’s equation are adopted to express the fluid pressure acting on the structure. The product of the pressure expression and the developed structural shape function is integrated over the structure-fluid interface to assess the virtual added mass due to the fluid. Variation of fluid level is considered in the calculation of the natural frequencies. The results are in close agreement with both experimental results and theoretical results using other analytical approaches.     

Continuous source in tidal flow: A numerical study of the transport equation

A detailed comparative numerical study of a continuous source discharging in a tidal flow has been performed. The advective term in the one-dimensional transport equation consists of a constant freshwater velocity and an isotropic oscillating component. Various finite element solutions are investigated. Compared to the analytical solution of the dynamic steady state concentration, the numerical results for typical estuarine conditions clearly indicate the superiority of the collocation method with Hermite basis functions. An extended Fourier series analysis that accounts for the source condition is developed to explain the numerical behaviour of the different schemes.     

Numerical modelling of 3D sloshing experiments in rectangular tanks

This work encompasses numerical and experimental studies of three-dimensional (3D) sloshing problems. The two-fluid viscous flow, which is solved within a stabilized finite element context, involves liquid and gaseous phases. The free surface is captured with a level set (LS) method, including the bounded renormalization with continuous penalization technique, to avoid the well-known spreading of the marker function. Specifically, this technique is improved with a volume-preserving algorithm for long-term analyses. To verify the numerical model, the responses of free-sloshing cases are compared with analytical solutions and other results computed using a Lagrangian technique. These simulations assess the influence of considering two-dimensional (2D) and 3D analyses, as well as the effects of depth and viscosity. This work presents data obtained from a forced sloshing experiment that is specifically devoted to 3D free surface behaviour. Free surface evolution measurements are used to validate the numerical method. Moreover, the effect of the initial conditions used to promote 3D behaviour in the modelling is evaluated.     

An Eulerian‐Lagrangian Concept for the Numerical Simulation of Flat Steady‐State Hot Rolling Processes

For the finite element analysis of stationary flat hot rolling processes, a new and efficient mathematical model was developed. The method is based on an intermediary Eulerian‐Lagrangian concept, where an Eulerian coordinate is employed in the rolling direction, while Lagrangian coordinates are used in the direction of the thickness and width of the strip. This approach yields an efficient algorithm, where the time is eliminated as an independent variable in the steady‐state case. Further, the vector of independent field variables consists of a velocity component in Eulerian and of displacement components in Lagrangian directions. Due to this concept, the free surface deformations can be accounted for directly and the problems encountered with pure Eulerian or Lagrangian models now appear with reduced complexity and can thus be tackled more easily. The general formalism was applied to different practical hot rolling situations, ranging from thick slabs to ultra‐thin hot strips. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations

Summary The Lagrange-Galerkin method is a numerical technique for solving convection — dominated diffusion problems, based on combining a special discretisation of the Lagrangian material derivative along particle trajectories with a Galerkin finite element method. We present optimal error estimates for the Lagrange-Galerkin mixed finite element approximation of the Navier-Stokes equations in a velocity/pressure formulation. The method is shown to be nonlinearly stable.     

Steady state numerical simulation of the particle collection efficiency of a new urban sustainable gravity settler using design of experiments by FVM

The aim of this work is to analyze the efficiency of a new sustainable urban gravity settler to avoid the solid particle transport, to improve the water waste quality and to prevent pollution problems due to rain water harvesting in areas with no drainage pavement. In order to get this objective, it is necessary to solve particle transport equations along with the turbulent fluid flow equations since there are two phases: solid phase (sand particles) and fluid phase (water). In the first place, the turbulent flow is modelled by solving the Reynolds-averaged Navier-Stokes (RANS) equations for incompressible viscous flows through the finite volume method (FVM) and then, once the flow velocity field has been determined, representative particles are tracked using the Lagrangian approach. Within the particle transport models, a particle transport model termed as Lagrangian particle tracking model is used, where particulates are tracked through the flow in a Lagrangian way. The full particulate phase is modelled by just a sample of about 2,000 individual particles. The tracking is carried out by forming a set of ordinary differential equations in time for each particle, consisting of equations for position and velocity. These equations are then integrated using a simple integration method to calculate the behaviour of the particles as they traverse the flow domain. The entire FVM model is built and the design of experiments (DOE) method was used to limit the number of simulations required, saving on the computational time significantly needed to arrive at the optimum configuration of the settler. Finally, conclusions of this work are exposed.     

求解流固耦合问题的一种四步分裂有限元算法

基于arbitrary Lagrangian Eulerian (ALE) 有限元方法,发展了一种求解流固耦合问题的弱耦合算法．将半隐式四步分裂有限元格式推广至求解ALE描述下的Navier-Stokes（N-S）方程,并在动量方程中引入迎风流线（streamline upwind/Petrov-Galerkin, SUPG）稳定项以消除对流引发的速度场数值振荡;采用Newmark-β法对结构方程进行时间离散;运用经典的Galerkin有限元法求解修正的Laplace方程以实现网格更新,每个计算步施加网格总变形量防止结构长时间、大位移运动时的网格质量恶化．运用上述算法对弹性支撑刚性圆柱体的流致振动问题进行了数值模拟,计算结果与已有结果相吻合,初步验证了该算法的正确性和有效性．