Equivalence of continuum and discrete analytic sensitivity methods for nonlinear differential equations

This paper presents results for the study of equivalence between the total form continuum sensitivity equation (CSE) method and the discrete analytic method of shape design sensitivity analysis. For the discrete analytic method, the sensitivity equations are obtained by taking analytic derivatives of the discretized equilibrium equations with respect to the shape design parameters. For the CSE method, the equilibrium equations are firstly differentiated to form a set of linear continuous sensitivity equations and then discretized for solving the shape sensitivities. The sensitivity equations can be derived by taking the material derivatives (total form) or the partial derivatives (local form) of the equilibrium equations. The total form CSE method is shown for the first time to be equivalent, after finite element discretization, to the discrete analytic method for nonlinear second-order differential equations of a particular form with design dependent loads when they use the same: (1) finite element discretization, (2) numerical integration of element matrices, (3) design velocity fields that are linear with respect to the design variable and (4) shape functions for domain transformation and for design velocity field calculations. The shape sensitivity equations for three-dimensional geometric nonlinear elastic structures and linear potential flow are derived by using both total form CSE and discrete analytic method to show the equivalence of the two methods for these specific examples. The accuracy of shape sensitivity analysis is verified by potential flow around an airfoil and a joined beam with an airfoil under gust load. The results show that analytic sensitivity results are consistent with the complex step results.     

基于微分/代数方程的多体系统动力学设计灵敏度分析的伴随变量方法

基于多体系统动力学微分/代数方程数学模型和通用积分形式的目标函数,建立了多体系统动力学设计灵敏度分析的伴随变量方法,避免了复杂的设计灵敏度计算,对于设计变量较多的多体系统灵敏度分析具有较高的计算效率.文中给出了通用公式以及具体的计算过程和验证方法,并将目标函数及其导数积分形式的计算转化为微分方程的初值问题,进一步提高了计算效率和精度.文末通过一曲柄-滑块机构算例对算法的有效性进行了验证.     

一种改进的基于深度神经网络的偏微分方程求解方法

偏微分方程求解是计算流体力学等科学与工程领域中数值分析的计算核心。由于物理的多尺度特性和对离散网格质量的敏感性,传统的数值求解方法通常包含复杂的人机交互和昂贵的网格剖分开销,限制了其在许多实时模拟和优化设计问题上的应用效率。提出了一种改进的基于深度神经网络的偏微分方程求解方法TaylorPINN。该方法利用深度神经网络的万能逼近定理和泰勒公式的函数拟合能力,实现了无网格的数值求解过程。在Helmholtz、Klein-Gordon和Navier-Stokes方程上的数值实验结果表明,TaylorPINN能够很好地拟合计算域内时空点坐标与待求函数值之间的映射关系,并提供了准确的数值预测结果。与常用的基于物理信息神经网络方法相比,对于不同的数值问题,TaylorPINN将预测精度提升了3~20倍。     

Second order adjoint sensitivity analysis of multibody systems described by differential–algebraic equations

The optimization strategies employing second order sensitivity information has higher accuracy, but its computation is complex. In this paper, an adjoint variable method applied for the second order design sensitivity analysis of multibody design problems is developed. Based on Lagrange equations of multibody system dynamics, a general objective function, constraint conditions, initial and end conditions, the adjoint variable equations for first order sensitivity analysis and design sensitivity formulations are derived firstly. Then, second order sensitivity analysis formulations, as well as the detailed computation steps, are given based on the previous results. For simplification, the second derivative of the objective function with respect to design variables is translated into an initial value problem of an ordinary differential equation with one variable. Finally, a numerical example of slider–crank mechanism validates the accuracy and efficiency of the method for second order sensitivity analysis.     

基于隐式微分/代数方程的多体系统动力学设计灵敏度分析方法

基于一般性的积分型目标函数、隐式相容初始条件及终止时刻表达式,系统建立了含设计参数的用隐式微分／代数方程表达的多体系统动力学设计灵敏度分析的直接微分方法和伴随变量方法．为降低目标函数及其对设计变量导数的计算复杂性。将其积分形式的计算转化为微分形式．所得到的结果可方便地应用于高效的间接最优化设计方法．最后通过采用绝对坐标建模的平面两连杆机械臂模型对该方法进行了验证．     

Review of options for structural design sensitivity analysis. Part 1: Linear systems

The paper reviews options for structural design sensitivity analysis, including global finite differences, continuum derivatives, discrete derivatives, and computational or automated differentiation. The objective is to put these different approaches to design sensitivity analysis in the context of accuracy and consistency, computational cost, and implementation options and effort. Linear static analysis and transient dynamic analysis are reviewed. In a separate appendix, special attention is paid to the semi-analytical method. A future paper will address design sensitivity analysis in nonlinear structural problems.     

Analytical and numerical solutions of the unsteady 2D flow of MHD fractional Maxwell fluid induced by variable pressure gradient

The nonlocal property of the fractional derivative can supply more precise mathematical models for depicting flow dynamics of complex fluid which cannot be modelled appropriately by normal integer order differential equations. This paper studies the analytical and numerical methods of unsteady 2D flow of Magnetohydrodynamic (MHD) fractional Maxwell fluid in a rectangular pipe driven by variable pressure gradient. The governing equation is formulated with Caputo time dependent fractional derivatives whose orders are distributed in interval (0, 2). A challenge is to firstly obtain the exact solution by combining modified separation of variables method with Mikusiński-type operational calculus. Meanwhile, the numerical solution is also obtained by the implicit finite difference method whose validity has been confirmed by the comparison with the exact solution constructed. Different to the most classical works, both the stability and convergence analysis of two-dimensional multi-term time fractional momentum equation are derived. Based on numerical analysis, the results show that the velocity increases with the rise of the fractional parameter and relaxation time. While an increase in the values of Hartmann number leads to a slower velocity in the rectangular pipe.     

Local Discontinuous Galerkin Methods for Partial Differential Equations with Higher Order Derivatives

In this paper we review the existing and develop new local discontinuous Galerkin methods for solving time dependent partial differential equations with higher order derivatives in one and multiple space dimensions. We review local discontinuous Galerkin methods for convection diffusion equations involving second derivatives and for KdV type equations involving third derivatives. We then develop new local discontinuous Galerkin methods for the time dependent bi-harmonic type equations involving fourth derivatives, and partial differential equations involving fifth derivatives. For these new methods we present correct interface numerical fluxes and prove L ² stability for general nonlinear problems. Preliminary numerical examples are shown to illustrate these methods. Finally, we present new results on a post-processing technique, originally designed for methods with good negative-order error estimates, on the local discontinuous Galerkin methods applied to equations with higher derivatives. Numerical experiments show that this technique works as well for the new higher derivative cases, in effectively doubling the rate of convergence with negligible additional computational cost, for linear as well as some nonlinear problems, with a local uniform mesh.     

Two-Order Approximate and Large Stepsize Numerical Direction Based on the Quadratic Hypothesis and Fitting Method

Many effective optimization algorithms require partial derivatives of objective functions, while some optimization problems’ objective functions have no derivatives. According to former research studies, some search directions are obtained using the quadratic hypothesis of objective functions. Based on derivatives, quadratic function assumptions, and directional derivatives, the computational formulas of numerical first-order partial derivatives, second-order partial derivatives, and numerical second-order mixed partial derivatives were constructed. Based on the coordinate transformation relation, a set of orthogonal vectors in the fixed coordinate system was established according to the optimization direction. A numerical algorithm was proposed, taking the second order approximation direction as an example. A large stepsize numerical algorithm based on coordinate transformation was proposed. Several algorithms were validated by an unconstrained optimization of the two-dimensional Rosenbrock objective function. The numerical second order approximation direction with the numerical mixed partial derivatives showed good results. Its calculated amount is 0.2843% of that of without second-order mixed partial derivative. In the process of rotating the local coordinate system 360°, because the objective function is more complex than the quadratic function, if the numerical direction derivative is used instead of the analytic partial derivative, the optimization direction varies with a range of 103.05°. Because theoretical error is in the numerical negative gradient direction, the calculation with the coordinate transformation is 94.71% less than the calculation without coordinate transformation. If there is no theoretical error in the numerical negative gradient direction or in the large-stepsize numerical optimization algorithm based on the coordinate transformation, the sawtooth phenomenon occurs. When each numerical mixed partial derivative takes more than one point, the optimization results cannot be improved. The numerical direction based on the quadratic hypothesis only requires the objective function to be obtained, but does not require derivability and does not take into account truncation error and rounding error. Thus, the application scopes of many optimization methods are extended.      

Continuum shape sensitivity analysis for aeroelastic gust using an arbitrary lagrangian-eulerian reference frame

Continuum Sensitivity Analysis (CSA), a method to determine response derivatives with respect to design variables, is derived here for the first time in an arbitrary Lagrangian-Eulerian (ALE) reference frame. CSA differentiates nonlinear governing system of equations to arrive at a linear system of partial differential continuum sensitivity equations (CSEs), here, for fluid-structure interaction (FSI). The CSEs and associated sensitivity boundary conditions are derived here for the first time for FSI, using the boundary velocity formulation, carefully distinguishing design velocity from flow velocity and ALE mesh velocity. Whereas boundary conditions must be differentiated using the material (total) derivative, it is sometimes advantageous to derive the CSEs using local (partial) derivatives. The benefit is that geometric sensitivity, known as design velocity, may not be required in the domain. It is shown here that this advantage is realized when the ALE frame undergoes only the rigid body motion associated with the structure to which it is attached. It is further shown that the advantage is not realized when the ALE mesh deforms due to the flexible motion of the fluid-structure interface. The equations for the transient gust response of a two-dimensional airfoil in compressible flow, flexibly attached to a rigid body mass, are presented as a model problem to illustrate a detailed derivation.     

The numerical solution of nonlinear generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations via the meshless method of integrated radial basis functions

In this paper, a numerical technique is proposed for solving the nonlinear generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations. The used numerical method is based on the integrated radial basis functions (IRBFs). First, the time derivative has been approximated using a finite difference scheme. Then, the IRBF method is developed to approximate the spatial derivatives. The two-dimensional version of these equations is solved using the presented method on different computational geometries such as the rectangular, triangular, circular and butterfly domains and also other irregular regions. The aim of this paper is to show that the integrated radial basis function method is also suitable for solving nonlinear partial differential equations. Numerical examples confirm the efficiency of the proposed scheme.

     

Hybrid laplace transform/weighting function scheme for transient multidimensional conservation equations

A hybrid Laplace transform/weighting function scheme is developed for solving time-dependent multidimensional conservation equations. The new method removes the time derivatives from the governing differential equations using the Laplace transform and solves the associated equation with the weighting function scheme. The similarity transform method is used to treat the complex coefficient system of the equations, which allows the simplest form of complex number functions to be obtained, and then to use the partial fractions method or a numerical method to invert the Laplace transform and transform the functions to the physical plane. Three different examples have been analyzed by the present method. The present method solutions are compared in tables with the exact solutions and those obtained by the other numerical methods. It is found that the present method is a reliable and efficient numerical tool.     

Smoothed particle hydrodynamics: Applications to heat conduction

In this paper, we modify the numerical steps involved in a smoothed particle hydrodynamics (SPH) simulation. Specifically, the second order partial differential equation (PDE) is decomposed into two first order PDEs. Using the ghost particle method, consistent estimation of near-boundary corrections for system variables is also accomplished. Here, we focus on SPH equations for heat conduction to verify our numerical scheme. Each particle carries a physical entity (here, this entity is temperature) and transfers it to neighboring particles, thus exhibiting the mesh-less nature of the SPH framework, which is potentially applicable to complex geometries and nanoscale heat transfer. We demonstrate here only 1D and 2D simulations because 3D codes are as simple to generate as 1D codes in the SPH framework. Our methodology can be extended to systems where the governing equations are described by PDEs.     

River mass transport linear-quadratic finite element model (RIMTRA)

A linear-quadratic finite element model, RIMTRA for predicting the transport of a single contaminant or BOD-DOD interactions in a polluted river is presented. Streamflow condition is one of steady and uniform or non-uniform flow. Galerkin finite element method with linear and quadratic basis functions; forward differencing for time derivative, and time-weighting factor for implicit control were adopted for solution of the governing differential equations. Model verification and performance evaluation resulted in two test problems: (1) a case of pure advection mass transport in which the effects of varying values of grid spacing and time steps on numerical dispersion were evaluated. Also, predicted results were compared with those of five finite difference codes in order to assess the performance of the proposed model; and (2) BOD-DOD profile computation with predicted results compared to analytical solutions. The results show great promise for the proposed RIMTRA model.     

Adaptivity in 3D image processing

We present an adaptive numerical scheme for computing the nonlinear partial differential equations arising in 3D image multiscale analysis. The scheme is based on a semi-implicit scale discretization and on an adaptive finite element method in 3D-space. Successive coarsening of the computational grid is used for increasing the efficiency of the numerical procedure. L_∞-stability of the semi–discrete scheme is proved and computational results related to 3D nonlinear image filtering are discussed. Received: 15 December 1999 / Accepted: 8 June 2001     

An efficient parallel algorithm with application to computational fluid dynamics

When solving time-dependent partial differential equations on parallel computers using the nonoverlapping domain decomposition method, one often needs numerical boundary conditions on the boundaries between subdomains. These numerical boundary conditions can significantly affect the stability and accuracy of the final algorithm.In this paper, a stability and accuracy analysis of the existing methods for generating numerical boundary conditions will be presented, and a new approach based on explicit predictors and implicit correctors will be used to solve convection-diffusion equations on parallel computers, with application to aerospace engineering for the solution of Euler equations in computational fluid dynamics simulations. Both theoretical analyses and numerical results demonstrate significant improvement in stability and accuracy by using the new approach.     

An extension of MacCormack's method for flows with higher-order equations and in different configurations

The numerical scheme for the computation of a shock discontinuity developed by MacCormack has been extended to solve a number of differential equations, including cases explicitly containing higher-order derivatives: (1) Korteweg-de Vries equation with a term of third-order derivative, (2) a system of nonlinear equations governing nonsteady one-dimensional plasma flow in cylindrical coordinate, (3) equations of solar wind. Comparisons with previous results are made, if available, to illustrate the advantages of the present method. The question of convergence of the numerical calculation is discussed.     

Nonlinear dynamic analysis of space trusses

A computational procedure is presented for predicting the dynamic response of space trusses with both geometric and material nonlinearities. A mixed formulation is used with the fundamental unknowns consisting of member forces, nodal velocities and nodal displacements. The governing equations consist of a mixed system of algebraic and differential equations. The temporal integration of the differential equations is performed by using an explicit half-station leap-frog method. The advantages of the proposed computational procedure over explicit methods used with the displacement formulation are discussed. The high accuracy of the procedure is demonstrated by means of numerical examples of plane and space trusses. The constitutive relations in these examples are assumed, for convenience, to be represented by the Ramberg-Osgood polynomials. Comparison is also made with solutions obtained by using implicit multistep temporal integration schemes.     

Fourth-order central compact scheme for the numerical solution of incompressible Navier–Stokes equations

This paper provides an implicit central compact scheme for the numerical solution of incompressible Navier–Stokes equations. The solution procedure is based on the artificial compressibility method that transforms the governing equations into a hyperbolic-parabolic form. A fourth-order central compact scheme with a sixth-order numerical filtering is used for the discretization of convective terms and fourth-order central compact scheme for the viscous terms. Dual-time stepping approach is applied to time discretization with backward Euler difference scheme to the pseudo-time derivative, and three point second-order backward difference scheme to the physical time derivative. An approximate factorization-based alternating direction implicit scheme is used to solve the resulting block tridiagonal system of equations. The accuracy and efficiency of the proposed numerical method is verified by simulating several two-dimensional steady and unsteady benchmark problems.     

Shape design sensitivity analysis of nonlinear structural systems

A unified approach is presented for shape design sensitivity analysis of nonlinear structural systems that include trusses and beams. Both geometric and material nonlinearities are considered. Design variables that specify the shape of components of built-up structures are treated, using the continuum equilibrium equations and the material derivative concept. To best utilize the basic character of the finite element method, shape design sensitivity information is expressed as domain integrals. For numerical evaluation of shape design sensitivity expressions, two alternative methods are presented: the adjoint variable and direct differentiation methods. Advantages and disadvantages of each method are discussed. Using the domain formulation of shape design sensitivity analysis, and the adjoint variable and direct differentiation methods, design sensitivity expressions are derived in the continuous setting in terms of shape design variations. A numerical method to implement the shape design sensitivity analysis, using established finite element codes, is discussed. Unlike conventional methods, the current approach does not require differentiation of finite element stiffness and mass matrices.