Topology optimization of flow domains using the lattice Boltzmann method

We consider the optimal design of two- (2D) and three-dimensional (3D) flow domains using the lattice Boltzmann method (LBM) as an approximation of Navier-Stokes (NS) flows. The problem is solved by a topology optimization approach varying the effective porosity of a fictitious material. The boundaries of the flow domain are represented by potentially discontinuous material distributions. NS flows are traditionally approximated by finite element and finite volume methods. These schemes, while well established as high-fidelity simulation tools using body-fitted meshes, are effected in their accuracy and robustness when regular meshes with zero-velocity constraints along the surface and in the interior of obstacles are used, as is common in topology optimization. Therefore, we study the potential of the LBM for approximating low Mach number incompressible viscous flows for topology optimization. In the LBM the geometry of flow domains is defined in a discontinuous manner, similar to the approach used in material-based topology optimization. In addition, this non-traditional discretization method features parallel scalability and allows for high-resolution, regular fluid meshes. In this paper, we show how the variation of the porosity can be used in conjunction with the LBM for the optimal design of fluid domains, making the LBM an interesting alternative to NS solvers for topology optimization problems. The potential of our topology optimization approach will be illustrated by 2D and 3D numerical examples.     

An Algebraic Multigrid Method for Higher-order Finite Element Discretizations

In this paper, we will design and analyze a class of new algebraic multigrid methods for algebraic systems arising from the discretization of second order elliptic boundary value problems by high-order finite element methods. For a given sparse stiffness matrix from a quadratic or cubic Lagrangian finite element discretization, an algebraic approach is carefully designed to recover the stiffness matrix associated with the linear finite element disretization on the same underlying (but nevertheless unknown to the user) finite element grid. With any given classical algebraic multigrid solver for linear finite element stiffness matrix, a corresponding algebraic multigrid method can then be designed for the quadratic or higher order finite element stiffness matrix by combining with a standard smoother for the original system. This method is designed under the assumption that the sparse matrix to be solved is associated with a specific higher order, quadratic for example, finite element discretization on a finite element grid but the geometric data for the underlying grid is unknown. The resulting new algebraic multigrid method is shown, by numerical experiments, to be much more efficient than the classical algebraic multigrid method which is directly applied to the high-order finite element matrix. Some theoretical analysis is also provided for the convergence of the new method.     

Volumetric mesh generation from T-spline surface representations

A new approach to obtain a volumetric discretization from a T-spline surface representation is presented. A T-spline boundary zone is created beneath the surface, while the core of the model is discretized with Lagrangian elements. T-spline enriched elements are used as an interface between isogeometric and Lagrangian finite elements. The thickness of the T-spline zone and thereby the isogeometric volume fraction can be chosen arbitrarily large such that pure Lagrangian and pure isogeometric discretizations are included. The presented approach combines the advantages of isogeometric elements (accuracy and smoothness) and classical finite elements (simplicity and efficiency).Different heat transfer problems are solved with the finite element method using the presented discretization approach with different isogeometric volume fractions. For suitable applications, the approach leads to a substantial accuracy gain.     

A Parallel Subgrid Stabilized Finite Element Method Based on Two-Grid Discretization for Simulation of 2D/3D Steady Incompressible Flows

Based on domain decomposition and two-grid discretization, a parallel subgrid stabilized finite element method for simulation of 2D/3D steady convection dominated incompressible flows is proposed and analyzed. In this method, a subgrid stabilized nonlinear Navier–Stokes problem is first solved on a coarse grid where the stabilization term is based on an elliptic projection defined on the same coarse grid, and then corrections are calculated in overlapped fine grid subdomains by solving a linearized problem. By the technical tool of local a priori estimate for finite element solution, error bounds of the approximate solution are estimated. Algorithmic parameter scalings of the method are derived. Numerical results are also given to demonstrate the effectiveness of the method.     

Structural optimization using global stress-deviation objective function via the level-set method

The paper deals with minimum stress design using a novel stress-related objective function based on the global stress-deviation measure. The shape derivative, representing the shape sensitivity analysis of the structure domain, is determined for the generalized form of the global stress-related objective function. The optimization procedure is based on the domain boundary evolution via the level-set method. The elasticity equations are, instead of using the usual ersatz material approach, solved by the extended finite element method. The Hamilton-Jacobi equation is solved using the streamline diffusion finite element method. The use of finite element based methods allows a unified numerical approach with only one numerical framework for the mechanical problem as also for the boundary evolution stage. The numerical examples for the L-beam benchmark and the notched beam are given. The results of the structural optimization problem, in terms of maximum von Mises stress corresponding to the obtained optimal shapes, are compared for the commonly used global stress measure and the novel global stress-deviation measure, used as the stress-related objective functions.     

Parallel multigrid finite volume computation of three-dimensional thermal convection

A parallel implementation of the finite volume method for three-dimensional, time-dependent, thermal convective flows is presented. The algebraic equations resulting from the finite volume discretization, including a pressure equation which consumes most of the computation time, are solved by a parallel multigrid method. A flexible parallel code has been implemented on the Intel Paragon, the Cray T3D, and the IBM SP2 by using domain decomposition techniques and the MPI communication software. The code can use 1D, 2D, or 3D partitions as required by different geometries, and is easily ported to other parallel systems. Numerical solutions for air (Prandtl number Pr = 0.733) with various Rayleigh numbers up to 10⁷ are discussed.     

Fictitious domain formulations of unilateral problems: analysis and algorithms

The present article deals with fictitious domain methods for numerical realization of scalar variational inequalities with the Signorini type conditions on the boundary. Two variants are introduced and analyzed. A discretization is done by finite elements. It leads to a system of non-smooth, piecewise linear equations. This system is solved by the semismooth Newton method. Numerical experiments confirm the efficiency of this approach.     

Numerical solutions of the one-phase classical Stefan problem using an enthalpy green element formulation

The enthalpy method is exploited in tackling a heat transfer problem involving a change of state. The resulting governing equation is then solved with a hybrid finite element - boundary element technique known as the Green element method (GEM). Two methods of approximation are employed to handle the time derivative contained in the discrete element equation. The first involves a finite difference method, while the second utilizes a Galerkin finite element approach. The performance of both methods are assessed with a known closed form solution. The finite element based time discretization, despite its greater challenge, yields less reliable numerical results. In addition a numerical stability test of both methods based on a Fourier series analysis explain the dispersive characters of both techniques, and confirms that replication of correct results is largely attributed to their ability to handle the harmonics of small wavelengths which are usually dominant in the vicinity of a front.     

A Finite Element Method on Convex Polyhedra

We present a method for animating deformable objects using a novel finite element discretization on convex polyhedra. Our finite element approach draws upon recently introduced 3D mean value coordinates to define smooth interpolants within the elements. The mathematical properties of our basis functions guarantee convergence. Our method is a natural extension to linear interpolants on tetrahedra: for tetrahedral elements, the methods are identical. For fast and robust computations, we use an elasticity model based on Cauchy strain and stiffness warping. This more flexible discretization is particularly useful for simulations that involve topological changes, such as cutting or fracture. Since splitting convex elements along a plane produces convex elements, remeshing or subdivision schemes used in simulations based on tetrahedra are not necessary, leading to less elements after such operations. We propose various operators for cutting the polyhedral discretization. Our method can handle arbitrary cut trajectories, and there is no limit on how often elements can be split.     

Finite element analysis of viscous,incompressible fluid flow: Part 1 : Basic methodology

A numerical procedure is developed for the analysis of general two-dimensional flows of viscous, incompressible fluids using the finite element method. The partial differential equations describing the continuum motion of the fluid are discretized by using an integral energy balance approach in conjunction with the finite element approximation. The nonlinear algebraic equations resulting from the discretization process are solved using a Picard iteration technique.A number of computational procedures are developed that allow significant reductions to be made in the computational effort required for the analysis of many flow problems. These techniques include a coarse-to-fine-mesh rezone procedure for the detailed study of regions of particular interest in a flow field and a special finite element to model far-field regions in external flow problems.     

离散化微分模型方程的反应器网络综合

构造基于全混流和交叉流反应器的反应器网络超结构,并建立优化模型,优化模型由复杂的高度非线性的微分／代数方程所构成。本文采用有限元正交配置法离散化微分方程的策略来简化超结构数学模型,将离散化所得的代数方程组与其它约柬条件一起,作为反应器网络超结构的数学模型,然后运用数学软件优化求解。实例研究表明,优化结果与文献相一致或优于文献,表明离散化法求解含有微分模型方程的反应器网络综合问题是有效的。     

A stabilized finite element method for stochastic incompressible Navier–Stokes equations

The major emphasis of this work is the development of a stabilized finite element method for solving incompressible Navier–Stokes equations with stochastic input data. The polynomial chaos expansion is used to represent stochastic processes in the variational problem, resulting in a set of deterministic variational problems to be solved for each Wiener polynomial chaos. To obtain the chaos coefficients in the corresponding deterministic incompressible Navier–Stokes equations, we combine the modified method of characteristics with the finite element discretization. The obtained Stokes problem is solved using a robust conjugate-gradient algorithm. This algorithm avoids projection procedures and any special correction for the pressure. These numerical techniques associate the geometrical flexibility of the finite element method with the ability offered by the modified method of characteristics to solve convection-dominated problems using time steps larger than its Eulerian counterpart. Numerical results are shown for the benchmark problems of driven cavity flow and backward-facing step flow. We also present numerical results for a problem of stochastic natural convection. It is found that the proposed stabilized finite element method offers a robust and accurate approach for solving the stochastic incompressible Navier–Stokes equations, even when high Reynolds and Rayleigh numbers are used in the simulations.     

Effective approach for the identification of boundary conditions in distributed-parameter systems

This paper is concerned with the identification of boundary conditions in parabolic-type distributed systems with boundaries of irregular shape. In the present approach, finite element discretization in the spatial domain and orthogonal functions expansion in the time domain are adopted to reduce the partial differential equation to a set of algebraic equations. The boundary conditions are then estimated by the method of least squares using state observations taken at a few interior points. The present approach is very straightforward and, at least as shown in illustrative examples, the results are in excellent agreement with exact results.     

A finite element based level set method for two-phase incompressible flows

We present a method that has been developed for the efficient numerical simulation of two-phase incompressible flows. For capturing the interface between the phases the level set technique is applied. The continuous model consists of the incompressible Navier–Stokes equations coupled with an advection equation for the level set function. The effect of surface tension is modeled by a localized force term at the interface (so-called continuum surface force approach). For spatial discretization of velocity, pressure and the level set function conforming finite elements on a hierarchy of nested tetrahedral grids are used. In the finite element setting we can apply a special technique to the localized force term, which is based on a partial integration rule for the Laplace–Beltrami operator. Due to this approach the second order derivatives coming from the curvature can be eliminated. For the time discretization we apply a variant of the fractional step θ-scheme. The discrete saddle point problems that occur in each time step are solved using an inexact Uzawa method combined with multigrid techniques. For reparametrization of the level set function a new variant of the fast marching method is introduced. A special feature of the solver is that it combines the level set method with finite element discretization, Laplace–Beltrami partial integration, multilevel local refinement and multigrid solution techniques. All these components of the solver are described. Results of numerical experiments are presented.     

A 3D Crouzeix-Raviart mortar finite element

A mortar finite element discretization of the second order elliptic problem in three dimensions, on non-matching grids, using the 3D Crouzeix-Raviart (CR) finite element in each subdomain, is proposed in this paper. The overall discretization is based on using only the nodal values on the mortar side of a subdomain interface for the calculation of the mortar projection, as opposed to applying the conventional approach where some nodal values in the interior of a subdomain are also required. Since the interior degrees of freedom disappear completely from the computation of the mortar projection, the proposed algorithm becomes less intricate and more flexible as compared to the conventional approach. An error estimate is given, and some numerical experiments are presented.     

Adaptive finite element analysis of structures subjected to transient dynamic loads using time marching scheme

Presents a method of obtaining accurate response using the finite element analysis for linear elastodynamic problems under transient dynamic loading. The accuracy obtained would depend on the discretization of both space and time. An integrated method is presented where both space and time discretization errors are evaluated and iteratively converges to a solution of desired accuracy through adaptive refinement. In this paper, it is proposed to solve the response using Newmark-β method. A numbers of examples are solved to demonstrate the effectiveness of the procedure.     

On the numerical solution of the stokes equations using the finite element method

A numerical solution of the stationary Stokes equations is considered based on the work of Crouzeix and Raviart [1]. The finite element method is used to discretize the partial differential equations, and a direct discretization of the velocity field and pressure is given which is applicable in both two and three dimensions. It is shown that not every arbitrary element can be used, and a condition is given to check whether or not an element is admissible. The system of linear equations is solved using the method of Powell and Hestenes for constrained optimization (see [2]).     

A non-reflecting boundary in fluid-structure interaction

A very efficient non-reflecting boundary condition is derived for the seismic response analysis of a submerged structure, such as a dam or an offshore structure, interacting with a compressible fluid domain of unbounded extent. The fluid-structure system is assumed to be two-dimensional and the analysis is conducted in the frequency domain. In the finite element discretization, pressure and displacements are considered to be the basic nodal unknowns for the fluid domain and the structure, respectively. The implementation of the proposed boundary condition in any existing finite element code, based on such a formulation, is extremely simple. Some fluid-structure systems are analysed to demonstrate the effectiveness and efficiency of the proposed method.