Remarks around 50 lines of Matlab: short finite element implementation

A short Matlab implementation for P ₁-x ₁ finite elements on triangles and parallelograms is provided for the numerical solution of elliptic problems with mixed boundary conditions on unstructured grids. According to the shortness of the program and the given documentation, any adaptation from simple model examples to more complex problems can easily be performed. Numerical examples prove the flexibility of the Matlab tool. This revised version was published online in June 2006 with corrections to the Cover Date.     

Pullback attractors for nonautonomous 2D Bénard problem in some unbounded domains

In this paper, we study the 2D Bénard problem, a system with the Navier–Stokes equations for the velocity field coupled with a convection–diffusion equation for the temperature, in an arbitrary domain (bounded or unbounded) satisfying the Poincaré inequality with nonhomogeneous boundary conditions and nonautonomous external force and heat source. The existence of a weak solution to the problem is proved by using the Galerkin method. We then show the existence of a unique minimal finite‐dimensional pullback D_σ‐attractor for the process associated to the problem. Copyright © 2013 John Wiley & Sons, Ltd.     

Discretisation procedures for multi-physics phenomena

Procedures are described for solving the equations governing a multi-physics process. Finite volume techniques are used to discretise, using the same unstructured mesh, the equations of fluid flow, heat transfer with solidification, and solid deformation. These discretised equations ofe then solved in an integrated manner. The computational mechanics environment, PHYSICA, which facilitates the building of multi-physics models, is described. Comparisons between model predictions and experimental data are presented for the casting of metal components.     

Effects of thermal radiation on the boundary layer flow of a Jeffrey fluid over an exponentially stretching surface

In the present investigation we have analyzed the boundary layer flow of a Jeffrey fluid over an exponentially stretching surface. The effects of thermal radiation are carried out for two cases of heat transfer analysis known as (1) Prescribed exponential order surface temperature (PEST) and (2) Prescribed exponential order heat flux (PEHF). The highly nonlinear coupled partial differential equations of Jeffrey fluid flow along with the energy equation are simplified by using similarity transformation techniques based on boundary layer assumptions. The reduced similarity equations are then solved analytically by the homotopy analysis method (HAM). The convergence of the HAM series solution is obtained by plotting (^h/_2p)\hbar-curves for velocity and temperature. The effects of physical parameters on the velocity and temperature profiles are examined by plotting graphs.     

Ritz approximations to two-dimensional boundary value problems

Summary Various Ritz solutions to the plane strain elasticity and the steady state heat flow boundary value problems for a polygonal domain are considered. Historically, two basic approaches have been used in partitioning into finite elements, (i) complete triangulation and (ii) rectangles with boundary triangles. In each case, the Ritz solution is the unique function (or vector of functions) which minimizes an energy functional over a finite dimensional vector spaceS. We consider as choices forS, piecewise linear and cubic functions for complete triangulations; and piecewise bilinear and bicubic functions for the case in which is a union of rectangles and boundary triangles. For the elasticity problem,L ₂ convergence of the components of stress and strain is established for each choice of the spaceS. L ₂ convergence of the displacement vector is also shown for a wide class of boundary conditions. Convergence of the temperature is proven for the heat flow problem also. Numerical comparisons are made of the Ritz solutions based upon each spaceS of trial functions.     

非一致性界面热流固耦合作用的整体求解

提出了非一致性界面热流固耦合作用整体求解的一种方法．热流体求解基于Boussinesq假设和不可压缩的Navier-Stokes方程．流体区域的运动采用任意Lagrange-Euler（ALE）方法．拟固体元方法实现流体区域的变形．使用几何非线性的热弹性动力学描述固体运动．为了保证界面处应力和传热的平衡,采用了基于Gauss积分点的数据交换方法,对热流固耦合最终形成的强非线性方程实现整体求解．数值实例分析表明该方法的健壮性和有效性．     

Numerical study of melting in an enclosure with discrete protruding heat sources

In order to explore the capability of a solid–liquid phase change material (PCM) for cooling electronic or heat storage applications, melting of a PCM in a vertical rectangular enclosure was studied. Three protruding generating heat sources are attached on one of the vertical walls of the enclosure, and generating heat at a constant and uniform volumetric rate. The horizontal walls are adiabatic. The power generated in heat sources is dissipated in PCM (n-eicosane with the melting temperature, T_m = 36 °C) that filled the rectangular enclosure. The advantage of using PCM is that it is able to absorb high amount of heat generated by heat sources due to its relatively high energy density. To investigate the thermal behaviour and thermal performance of the proposed system, a mathematical model based on the mass, momentum and energy conservation equations was developed. The governing equations are next discretised using a control volume approach in a staggered mesh and a pressure correction equation method is employed for the pressure–velocity coupling. The PCM energy equation is solved using the enthalpy method. The solid regions (wall and heat sources) are treated as fluid regions with infinite viscosity and the thermal coupling between solid and fluid regions is taken into account using the harmonic mean of the thermal conductivity method. The dimensionless independent parameters that govern the thermal behaviour of the system were next identified. After validating the proposed mathematical model against experimental data, a numerical investigation was next conducted in order to examine the thermal behaviour of the system by analyzing the flow structure and the heat transfer during the melting process, for a given values of governing parameters.     

2D simulation of fluid-structure interaction using finite element method

This paper deals with pressure-based finite element analysis of fluid–structure systems considering the coupled fluid and structural dynamics. The present method uses two-dimensional fluid elements and structural line elements for the numerical simulation of the problem. The equations of motion of the fluid, considered inviscid and compressible, are expressed in terms of the pressure variable alone. The solution of the coupled system is accomplished by solving the two systems separately with the interaction effects at the fluid–solid interface enforced by an iterative scheme. Non-divergent pressure and displacement are obtained simultaneously through iterations. The Galerkin weighted residual method-based FE formulation and the iterative solution procedure are explained in detail followed by some numerical examples. Numerical results are compared with the existing solutions to validate the code for sloshing with fluid–structure coupling.     

On the existence of weak solutions to the equations of steady flow of heat-conducting fluids with dissipative heating

We study a system of partial differential equations describing the steady flow of a heat conducting incompressible fluid in a bounded three dimensional domain, where the right-hand side of the momentum equation includes the buoyancy force. In the present work we prove the existence of a weak solution under both the smallness and a sign condition on physical parameters α₀ and α₁ which appear on the right hand side.     

Multidisciplinary Simulations with the Coupling Library MpCCI

Extreme high demands on designing accurate prototypes for example in the fields of medical research, aircraft construction, shipbuilding and automotive industry require multidisciplinary simulations. A large number of tools for monodisciplinary simulations are available today. Each of these provides high quality simulation results in a specific physical domain. Now there is also a solution to do multidisciplinary computations: Parallel monodisciplinary codes are coupled with the Mesh based parallel Code Coupling Interface MpCCI to solve multidisciplinary problems with a loose coupled approach. The paper presents applications in the framework of fluid‐structure interaction, which demonstrate the advantages of the parallel coupling library for this kind of problems. The computational fluid dynamics code FLOWer developed at the Institute of Design Aerodynamics/DLR and the structural mechanics code SIMPACK developed at the Institute of Aeroelasticity/DLR are coupled to solve an aeroelastic test problem. The applicability of the coupling library in the field of aeroelasticity is strongly dependent on the integrated interpolations between the involved meshes. In the Institute of Aeroelasticity the aeroelastic analysis tool CAESAR was developed which includes aeroelasticity specific interpolation algorithms. These routines are integrated in MpCCI via a special interface. There are two types of interpolation routines included. The first kind of algorithms is based on the method of finite interpolation elements and the second uses radial basis functions.     

Existence of local strong solutions to fluid–beam and fluid–rod interaction systems

We study an unsteady nonlinear fluid–structure interaction problem. We consider a Newtonian incompressible two-dimensional flow described by the Navier–Stokes equations set in an unknown domain depending on the displacement of a structure, which itself satisfies a linear wave equation or a linear beam equation. The fluid and the structure systems are coupled via interface conditions prescribing the continuity of the velocities at the fluid–structure interface and the action-reaction principle. Considering three different structure models, we prove existence of a unique local-in-time strong solution, for which there is no gap between the regularity of the initial data and the regularity of the solution enabling to obtain a blow up alternative. In the case of a damped beam this is an alternative proof (and a generalization to non zero initial displacement) of the result that can be found in [20]. In the case of the wave equation or a beam equation with inertia of rotation, this is, to our knowledge the first result of existence of strong solutions for which no viscosity is added. The key points consist in studying the coupled system without decoupling the fluid from the structure and to use the fluid dissipation to control, in appropriate function spaces, the structure velocity.     

Uniform stabilization to equilibrium of a nonlinear fluid–structure interaction model

We consider uniform stability to a nontrivial equilibrium of a nonlinear fluid–structure interaction (FSI) defined on a two or three dimensional bounded domain. Stabilization is achieved via boundary and/or interior feedback controls implemented on both the fluid and the structure. The interior damping on the fluid combining with the viscosity effect stabilizes the dynamics of fluid. However, this dissipation propagated from the fluid alone is not sufficient to drive uniformly to equilibrium the entire coupled system. Therefore, additional interior damping on the wave component or boundary porous like damping on the interface is considered. A geometric condition on the interface is needed if only boundary damping on the wave is active. The main technical difficulty is the mismatch of regularity of hyperbolic and parabolic component of the coupled system. This is overcome by considering special multipliers constructed from Stokes solvers. The uniform stabilization result obtained in this article is global for the fully coupled FSI model.     

Calculation of conjugate heat transfer problem with volumetric heat generation using the Galerkin method

A mathematical model of fluid flow across a rod bundle with volumetric heat generation has been built. The rods are heated with volumetric internal heat generation. To construct the model, a volume average technique (VAT) has been applied to momentum and energy transport equations for a fluid and a solid phase to develop a specific form of porous media flow equations. The model equations have been solved with a semi-analytical Galerkin method. The detailed velocity and temperature fields in the fluid flow and the solid structure have been obtained. Using the solution fields, a whole-section drag coefficient C_d and a whole-section Nusselt number Nu have also been calculated. To validate the developed solution procedure, the results have been compared to the results of a finite volume method. The comparison shows an excellent agreement. The present results demonstrate that the selected Galerkin approach is capable of performing calculations of heat transfer in a cross-flow where thermal conductivity and internal heat generation in a solid structure has to be taken into account. Although the Galerkin method has limited applicability in complex geometries, its highly accurate solutions are an important benchmark on which other numerical results can be tested.     

Packing and Covering Triangles in K 4-free Planar Graphs

We show that every K ₄-free planar graph with at most ν edge-disjoint triangles contains a set of at most ${\frac32\nu}$ edges whose removal makes the graph triangle-free. Moreover, equality is attained only when G is the edge-disjoint union of 5-wheels plus possibly some edges that are not in triangles. We also show that the same statement is true if instead of planar graphs we consider the class of graphs in which each edge belongs to at most two triangles. In contrast, it is known that for any c?<?2 there are K ₄-free graphs with at most ν edge-disjoint triangles that need more than cν edges to cover all triangles.     

An efficient linear model and optimisation algorithm for multi-site combined heat and power production

Combined heat and power (CHP) production is an important energy production technology which can help to improve the efficiency of energy production and to reduce the emission of CO₂. Cost-efficient operation of a CHP system can be planned using an optimisation model based on hourly load forecasts. A long-term planning model decomposes into hourly models, which can be formulated as linear programming (LP) problems. Such problems can be solved efficiently using the specialized Power Simplex algorithm. However, Power Simplex can only manage one heat and one power balance. Since heat cannot be transported over long distances, Power Simplex applies only for local CHP planning.In this paper we formulate the hourly multi-site CHP planning problem with multiple heat balances as an LP model with a special structure. We then develop the Extended Power Simplex (EPS) algorithm for solving such models efficiently. Even though the problem can be quite large as the number of demand sites increases, EPS demonstrates very good efficiency. In test runs with realistic models, EPS is from 29 to 85 times faster than an efficient sparse Simplex code using the product form of inverse (PFI). Furthermore, the relative efficiency of EPS improves as the problem size grows.     

Adaptive simplification of huge sets of terrain grid data for geosciences applications

We propose and discuss a new Lepp-surface method able to produce a small triangular approximation of huge sets of terrain grid data by using a two-goal strategy that assures both small approximation error and well-shaped 3D triangles. This is a refinement method which starts with a coarse initial triangulation of the input data, and incrementally selects and adds data points into the mesh as follows: for the edge e having the highest error in the mesh, one or two points close to (one or two) terminal edges associated with e are inserted in the mesh. The edge error is computed by adding the triangle approximation errors of the two triangles that share e, while each L₂-norm triangle error is computed by using a curvature tensor (a good approximation of the surface) at a representative point associated with both triangles. The method produces triangular approximations that capture well the relevant features of the terrain surface by naturally producing well-shaped triangles. We compare our method with a pure L₂-norm optimization method.     

Numerical Hopf bifurcation analysis in nonlinear ordinary and partial differential systems from chemical reactor theory

A code has been developed which will automatically locate and analyze points of Hopf bifurcation in autonomous ordinary differential systems. The code first locates critical value(s) v_c of a user-specified parameter v (the bifurcation parameter) such that a stationary (equilibrium) solution x_*(v) loses linear stability by virtue of a complex conjugate pair of eigenvalues. The code computes x_*(v) during the location of v_c. Then the code computes the various coefficients in a local approximation to the family of periodic solutions which arise, a process which involves computation of second and third partial derivatives by numerical differencing of the user-supplied Jacobian matrix. The current version of the code, called BIFOR2, is fully described in Hassard, Kazarinoff, and Wan, Theory and Applications of Hopf Bifurcation, Cambridge U.P., 1981. In this paper we demonstrate the code in applications to two systems drawn from chemical reactor theory. The first application is to a 4th order ordinary differential system modeling a coupled tank reactor. The second application is to a partial differential system modeling a catalyst particle system. These represent the first applications of the code to chemically reacting systems other than the Brusselator. The second application demonstrates how collocation methods may be used in conjunction with BIFOR2 to perform Hopf bifurcation analysis of partial differential systems.     

A numerical analysis of solid–liquid phase change heat transfer around a horizontal cylinder

A numerical study is conducted to analyze the melting process around a horizontal circular cylinder in the presence of the natural convection in the melt phase. Two boundary conditions are investigated one of constant wall temperature over the surface of the cylinder and the other of constant heat flux. A numerical code is developed using an unstructured finite-volume method and an enthalpy porosity technique to solve for natural convection coupled to solid–liquid phase change. The validity of the numerical code used is ascertained by comparing our results with previously published results.     

Convergence analysis of an approximation to miscible fluid flows in porous media by combining mixed finite element and finite volume methods

This article deals with development and analysis of a numerical method for a coupled system describing miscible displacement of one incompressible fluid by another through heterogeneous porous media. A mixed finite element (MFE) method is employed to discretize the Darcy flow equation combined with a conservative finite volume (FV) method on unstructured grids for the concentration equation. It is shown that the FV scheme satisfies a discrete maximum principle. We derive L^∞ and BV estimates under an appropriate CFL condition. Then we prove convergence of the approximate solutions to a weak solution of the coupled system. Numerical results are presented to see the performance of the method in two space dimensions. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

A multiscale Darcy–Brinkman model for fluid flow in fractured porous media

The aim of this work is to present a reduced mathematical model for describing fluid flow in porous media featuring open channels or fractures. The Darcy’s law is assumed in the porous domain while the Stokes–Brinkman equations are considered in the fractures. We address the case of fractures whose thickness is very small compared to the characteristic diameter of the computational domain, and describe the fracture as if it were an interface between porous regions. We derive the corresponding interface model governing the fluid flow in the fracture and in the porous media, and establish the well-posedness of the coupled problem. Further, we introduce a finite element scheme for the approximation of the coupled problem, and discuss solution strategies. We conclude by showing the numerical results related to several test cases and compare the accuracy of the reduced model compared with the non-reduced one.