Bivariate splines for fluid flows

We discuss numerical approximations of the 2D steady-state Navier-Stokes equations in stream function formulation using bivariate splines of arbitrary degree d and arbitrary smoothness r with r<d. We derive the discrete Navier-Stokes equations in terms of B-coefficients of bivariate splines over a triangulation, with curved boundary edges, of any given domain. Smoothness conditions and boundary conditions are enforced through Lagrange multipliers. The pressure is computed by solving a Poisson equation with Neumann boundary conditions. We have implemented this approach in MATLAB and our numerical experiments show that our method is effective. Numerical simulations of several fluid flows will be included to demonstrate the effectiveness of the bivariate spline method.     

The optimal design of structures using ACO and EFG

This paper presents an ant colony optimization (ACO) algorithm incorporating the element free Galerkin (EFG) method for topology optimization of continuum structures. The EFG method is used to derive shape functions using the moving least squares approximation. The essential boundary conditions are enforced by the Lagrange multiplier method. Several numerical examples are presented to show the validity and feasibly of the proposed method. The common numerical instabilities of the ACO algorithm do not exist in the results.     

Development of a meshless Galerkin boundary node method for viscous fluid flows

In this paper, a meshless Galerkin boundary node method is developed for boundary-only analysis of the interior and exterior incompressible viscous fluid flows, governed by the Stokes equations, in biharmonic stream function formulation. This method combines scattered points and boundary integral equations. Some of the novel features of this meshless scheme are boundary conditions can be enforced directly and easily despite the meshless shape functions lack the delta function property, and system matrices are symmetric and positive definite. The error analysis and convergence study of both velocity and pressure are presented in Sobolev spaces. The performance of this approach is illustrated and assessed through some numerical examples.     

The boundary element-free method for 2D interior and exterior Helmholtz problems

The boundary element-free method (BEFM) is developed in this paper for numerical solutions of 2D interior and exterior Helmholtz problems with mixed boundary conditions of Dirichlet and Neumann types. A unified boundary integral equation is established for both interior and exterior problems. By using the improved interpolating moving least squares method to form meshless shape functions, mixed boundary conditions in the BEFM can be satisfied directly and easily. Detailed computational formulas are derived to compute weakly and strongly singular integrals over linear and higher order integration cells. Three numerical integration procedures are developed for the computation of strongly singular integrals. Numerical examples involving acoustic scattering and radiation problems are presented to show the accuracy and efficiency of the meshless method.     

A Roadmap to Well Posed and Stable Problems in Computational Physics

All numerical calculations will fail to provide a reliable answer unless the continuous problem under consideration is well posed. Well-posedness depends in most cases only on the choice of boundary conditions. In this paper we will highlight this fact, and exemplify by discussing well-posedness of a prototype problem: the time-dependent compressible Navier–Stokes equations. We do not deal with discontinuous problems, smooth solutions with smooth and compatible data are considered. In particular, we will discuss how many boundary conditions are required, where to impose them and which form they should have in order to obtain a well posed problem. Once the boundary conditions are known, one issue remains; they can be imposed weakly or strongly. It is shown that the weak and strong boundary procedures produce similar continuous energy estimates. We conclude by relating the well-posedness results to energy-stability of a numerical approximation on summation-by-parts form. It is shown that the results obtained for weak boundary conditions in the well-posedness analysis lead directly to corresponding stability results for the discrete problem, if schemes on summation-by-parts form and weak boundary conditions are used. The analysis in this paper is general and can without difficulty be extended to any coupled system of partial differential equations posed as an initial boundary value problem coupled with a numerical method on summation-by parts form with weak boundary conditions. Our ambition in this paper is to give a general roadmap for how to construct a well posed continuous problem and a stable numerical approximation, not to give exact answers to specific problems.     

Element free method for static and free vibration analysis of spatial thin shell structures

The implementation of the element free Galerkin method (EFG) for spatial thin shell structures is presented in this paper. Both static deformation and free vibration analyses are considered. The formulation of the discrete system equations starts from the governing equations of stress resultant geometrically exact theory of shear flexible shells. Moving least squares approximation is used in both the construction of shape functions based on arbitrarily distributed nodes as well as in the surface approximation of general spatial shell geometry. Discrete system equations are obtained by incorporating these interpolations into the Galerkin weak form. The formulation is verified through numerical examples of static stress analysis and frequency analysis of spatial thin shell structures. For static load analysis, essential boundary conditions are enforced through penalty method and Lagrange multipliers while boundary conditions for frequency analysis are imposed through a weak form using orthogonal transformation techniques. The EFG results compare favorably with closed-form solutions and that of finite element analyses.     

A semi analytical method for the free vibration of doubly-curved shells of revolution

In this paper, a semi analytical method is used to investigate the free vibration of doubly-curved shells of revolution with arbitrary boundary conditions. The doubly-curved shells of revolution are divided into their segments in the meridional direction, and the theoretical model for vibration analysis is formulated by applying Flügge’s thin shell theory. Regardless of the boundary conditions, the displacement functions of shell segments are composed by the Jacobi polynomials along the revolution axis direction and the standard Fourier series along the circumferential direction. The boundary conditions at the ends of the doubly-curved shells of revolution and the continuous conditions at two adjacent segments were enforced by the penalty method. Then, the natural frequencies of the doubly-curved shells are obtained by using the Rayleigh–Ritz method. For arbitrary boundary conditions, this method does not require any changes to the mathematical model or the displacement functions, and it is very effective in the analysis of free vibration for doubly-curved shells of revolution. The credibility and exactness of proposed method are compared with the results of finite element method (FEM), and some numerical results are reported for free vibration of the doubly-curved shells of revolution under classical and elastic boundary conditions. Results of this paper can provide reference data for future studies in related field.     

δ-SPH model for simulating violent impact flows

A smoothed particle hydrodynamics model with numerical diffusive terms, hereinafter referred to as δ-SPH [1] is used to analyze violent water flows. The boundary conditions on solid surfaces of arbitrary shape are enforced with a new technique based on fixed ghost particles. The violent impacts studied result from dam-break water flows striking obstacles of different shapes. The numerical results are validated against experimental data from the literature and solutions from a Navier–Stokes Level-Set solver. Predicted impact pressures are also compared with analytical solutions. The proposed scheme thus proves to be accurate and robust for the prediction of global and local loads of impact flows on structures.     

双螺杆挤出机流场数值模拟中流道进出口边界条件的探讨

在对双螺杆挤出机流场的数值模拟中,流道进出口边界条件的设置一直是一个颇具争议的问题。由于事先无法获得计算域进出口平面上的真实边界条件,研究人员在进行双螺杆挤出机的流场分析时,大都采用放松边界条件。为了考察放松边界条件对双螺杆挤出机流场数值模拟结果的影响,本文采用聚合物流动分析软件POLYFLOW,在流量恒定的前提下对双螺杆挤出机流道进出口给定三种不同分布形式的速度边界条件,对其流场进行了数值模拟。数值计算结果表明,在体积流量恒定的条件下,流道进出口不同分布形式的速度边界条件对流场的影响主要集中在进出口附近区域,但对离进出口边界较远的流场影响很小。一般而言,当计算域所对应的螺杆较长时,可以忽略流道进出口的放松边界条件所引起的误差;当计算域较短时,不宜直接采用放松边界条件,而应根据螺杆的实际构型．在计算域的进出口增加适当长度的发展段。     

A hybrid approach to optimal control problems with bounded state

This paper considers the numerical solution of optimal control problems involving a functional I subject to differential constraints, a state inequality constraint, and terminal constraints. The problem is to find the state x(t), the control u(t), and the parameter π so that the functional is minimized, while the constraints are satisfied to a predetermined accuracy.The approach taken is a sequence of two-phase processes or cycles, each composed of a gradient phase and a restoration phase. The gradient phase involves a single iteration and is designed to decrease the functional, while the constraints are satisfied to first order. The restoration phase involves one or several iterations and is designed to restore the constraints to a predetermined accuracy, while the norm of the variations of the control and the parameter is minimized. The principal property of the algorithm is that it produces a sequence of feasible suboptimal solutions: the functions x(t), u(t), π obtained at the end of each cycle satisfy the constraints to a predetermined accuracy. Therefore, the functionals of any two elements of the sequence are comparable.The technique employed is of the hybrid type, in an attempt to combine some of the best features of both the indirect and direct approaches. While a predetermined number and sequence of subarcs are assumed (a feature of direct methods), enforcement of the state inequality constraint is obtained through a Valentine-type representation (a feature of indirect methods).By properly choosing the analytical form of certain non-differential constraints to be satisfied by the augmented control along each subarc composing the extremal arc (these nondifferential constraints are arrived at through the Valentine-type transformation), one ensures satisfaction of the state inequality constraint everywhere. Specifically, strict inequality is enforced for the subarcs internal to the state boundary and strict equality is enforced for the subarcs lying on the state boundary.To facilitate the numerical solution on digital computers, the actual time θ is replaced by the normalized time t, defined in such a way that each subarc composing the extremal arc has a normalized time length Δt = 1. In this way, variable-time corner conditions and variable-time terminal conditions are transformed into fixed-time corner conditions and fixed-time terminal conditions. The actual times θ₁, θ₂, τ at which (i) the state boundary is entered, (ii) the state boundary is exited, and (iii) the terminal manifold is reached are regarded to be components of the parameter π being optimized.The numerical examples illustrating the theory demonstrate the feasibility as well as the rapidity of convergence of the technique developed in this paper.     

Graphics processing unit (GPU) accelerated fast multipole BEM with level-skip M2L for 3D elasticity problems

In order to accelerate fast multipole boundary element method (FMBEM), in terms of the intrinsic parallelism of boundary elements and the FMBEM tree structure, a series of CUDA based GPU parallel algorithms for different parts of FMBEM with level-skip M2L for 3D elasticity are presented. A rigid body motion method (RBMM) for the FMBEM is proposed based on special displacement boundary conditions to deal with strongly singular integration and free term coefficients. The numerical example results show that our parallel algorithms obviously accelerates the FMBEM and can be used in large scale engineering problems with wide applications in the future.     

Plate bending by a boundary point method

The problem of linear elastic plate bending is solved by a boundary point method. Four fundamental homogeneous solutions for each of a number of sources, which are situated outside the plate, are superimposed and combined with appropriate particular solutions. Each source point is associated with a boundary point, which may be clamped or simply supported. At each boundary point, four edge conditions are enforced which allow the scalar coefficients, introduced in the superposition process, to be determined, and, hence, the plate displacement and stress solution to be obtained. Four non-rectangular example plates are considered, under uniform and hydrostatic loading, for which no internal sub-division is required, and it is demonstrated that accurate solutions may be obtained with 10–14 boundary points.     

Effect of sidewall boundary conditions on unsteady high speed cavity flow and acoustics

The effect of sidewall boundary conditions on the computed unsteady flow and sound pressure level is investigated in a transonic open cavity. The hybrid approach used for modeling turbulence combines a Reynolds averaged mode in the boundary layer, and a large eddy simulation mode in the massively separated flow region within the cavity to resolve the wide dynamic range involved. Computational results are presented for the instantaneous vorticity and for the sound pressure level spectra. Comparison of the results obtained using inviscid and periodic sidewall boundary conditions show the sensitivity of the computed SPL spectra and autocorrelation to the conditions enforced at the sidewalls. The computed SPL spectra are also compared with available experimental results, with LES computational results, and with prior investigations based on the same hybrid turbulence model without the wall function used in the current investigation. The comparisons show that the current results obtained using inviscid sidewall boundary conditions are closest to the experimental sound pressure level spectra and that agreement is achieved at considerable saving in required computational resources.     

Numerical study on mechanisms of second sweep and positive spikes in transitional flow on a flat plate

Ring-like vortex is a flow structure at late stages of a transitional boundary layer. Independent to the initial disturbance conditions corresponding to K- and N-scenarios of transition, the vortical structure shows some universal features. The nonlinear evolution of the ring-like vortices, detail flow structures around ring-like vortex and their effects on the surrounding flow were studied by direct numerical simulation with high order accuracy. A detailed enforced spatial transition on a flat-plate boundary layer in the compressible flow was studied. This study reveals the mechanism of the second sweep generation, mechanism of the positive spike formation and mechanism of high shear layer distribution.     

A Dual Consistent Finite Difference Method with Narrow Stencil Second Derivative Operators

We study the numerical solutions of time-dependent systems of partial differential equations, focusing on the implementation of boundary conditions. The numerical method considered is a finite difference scheme constructed by high order summation by parts operators, combined with a boundary procedure using penalties (SBP–SAT). Recently it was shown that SBP–SAT finite difference methods can yield superconvergent functional output if the boundary conditions are imposed such that the discretization is dual consistent. We generalize these results so that they include a broader range of boundary conditions and penalty parameters. The results are also generalized to hold for narrow-stencil second derivative operators. The derivations are supported by numerical experiments.     

基于无网格自然邻接点Petrov-Galerkin法求解带源参数瞬态热传导问题

基于无网格自然邻接点Petrov-Galerkin法,本文建立了一种求解带源参数瞬态热传导问题的新方法.为了克服移动最小二乘近似难以准确施加本质边界条件的缺点,采用了自然邻接点插值构造试函数.在局部多边形子域上采用局部Petrov-Galerkin方法建立瞬态热传导问题的积分弱形式.这些多边形子域可由Delaunay三角形创建.时间域则通过传统的两点差分法进行离散.最后通过算例验证了该数值算法的有效性和正确性.     

Direct Forcing for Lagrangian Rigid-Fluid Coupling

We propose a novel boundary handling algorithm for particle-based fluids. Based on a predictor-corrector scheme for both velocity and position, one- and two-way coupling with rigid bodies can be realized. The proposed algorithm offers significant improvements over existing penalty-based approaches. Different slip conditions can be realized and non-penetration is enforced. Direct forcing is employed to meet the desired boundary conditions and to ensure valid states after each simulation step. We have performed various experiments in 2D and 3D. They illustrate one- and two-way coupling of rigid bodies and fluids, the effects of hydrostatic and dynamic forces on a rigid body as well as different slip conditions. Numerical experiments and performance measurements are provided.     

Boundary conditions for incompressible flows

A general framework is presented for the formulation and analysis of rigid no-slip boundary conditions for numerical schemes for the solution of the incompressible Navier-Stokes equations. It is shown that fractional-step (splitting) methods are prone to introduce a spurious numerical boundary layer that induces substantial time differencing errors. High-order extrapolation methods are analyzed to reduce these errors. Both improved pressure boundary condition and velocity boundary condition methods are developed that allow accurate implementation of rigid no-slip boundary conditions.     

First vorticity–velocity–pressure numerical scheme for the Stokes problem

We consider the bidimensional Stokes problem for incompressible fluids and recall the vorticity, velocity and pressure variational formulation, which was previously proposed by one of the authors, and allows very general boundary conditions. We develop a natural implementation of this numerical method and we describe in this paper the numerical results we obtain. Moreover, we prove that the low degree numerical scheme we use is stable for Dirichlet boundary conditions on the vorticity. Numerical results are in accordance with the theoretical ones. In the general case of unstructured meshes, a stability problem is present for Dirichlet boundary conditions on the velocity, exactly as in the stream function-vorticity formulation. Finally, we show on some examples that we observe numerical convergence for regular meshes or embedded ones for Dirichlet boundary conditions on the velocity.