Combining differential quadrature method with simulation technique to solve non‐linear differential equations

Nowadays, most of the ordinary differential equations (ODEs) can be solved by modelica‐based approaches, such as Matlab/Simulink, Dymola and LabView, which use simulation technique (ST). However, these kinds of approaches restrict the users in the enforcement of conditions at any instant of the time domain. This limitation is one of the most important drawbacks of the ST. Another method of solution, differential quadrature method (DQM), leads to very accurate results using only a few grids on the domain. On the other hand, DQM is not flexible for the solution of non‐linear ODEs and it is not so easy to impose multiple conditions on the same location. For these reasons, the author aims to eliminate the mentioned disadvantages of the simulation technique (ST) and DQM using favorable characteristics of each method in the other. This work aims to show how the combining method (CM) works simply by solving some non‐linear problems and how the CM gives more accurate results compared with those of other methods. Copyright © 2008 John Wiley & Sons, Ltd.     

A localized differential quadrature (LDQ) method and its application to the 2D wave equation

 Differential Quadrature (DQ) is a numerical technique of high accuracy, but it is sensitive to grid distribution and requires that the number of grid points cannot be too large. These two requirements greatly restrict wider applications of DQ method. Through a simplified stability analysis in this paper, it is concluded that these two limitations are due to stability requirements. This analysis leads us to propose to localize differential quadrature to a small neighbourhood so as to keep the balance of accuracy and stability. The derivatives at a grid point are approximated by a weighted sum of the points in its neighbourhood rather than of all grid points. The method is applied to the one- and two-dimensional wave equations. Numerical examples show the present method produces very accurate results while maintaining good stability. The proposed method enables us to solve more complicated problems and enhance DQ's flexibility significantly. Received: 23 October 2001 / Accepted: 3 July 2002     

Stochastic simulation method for a 2D elasticity problem with random loads

We develop a stochastic simulation method for a numerical solution of the Lamé equation with random loads. To treat the general case of large intensity of random loads, we use the Random Walk on Fixed Spheres (RWFS) method described in our paper [Sabelfeld KK, Shalimova IA, Levykin AI. Discrete random walk over large spherical grids generated by spherical means for PDEs. Monte Carlo Methods and Applications 2006; 12(1): 55–93]. The vector random field of loads which stands on the right-hand side of the system of elasticity equations is simulated by the Randomization Spectral method presented in [Sabelfeld KK. Monte Carlo methods in boundary value problems. Berlin (Heidelberg, New York): Springer-Verlag; 1991] and recently revised and generalized in [Kurbanmuradov O, Sabelfeld KK. Stochastic spectral and Fourier-wavelet methods for vector Gaussian random field. Monte Carlo Methods and Applications 2006; 12(5–6): 395–445]. Comparative analysis of the RWFS method and an alternative direct evaluation of the correlation tensor of the solution is made. We derive also a closed boundary value problem for the correlation tensor of the solution which is applicable in the case of inhomogeneous random loads. Calculations of the longitudinal and transverse correlations are presented for a domain which is a union of two arbitrarily overlapped discs. We also discuss a possibility to solve an inverse problem of the determination of the elastic constants from the known longitudinal and transverse correlations of the loads, and give some relevant numerical illustrations.     

Development of novel pH-sensitive nanoparticles loaded hydrogel for transdermal drug delivery

Objective: Difference of pH that exists between the skin surface and blood circulation can be exploited for transdermal delivery of drug molecules by loading drug into pH-sensitive polymer. Eudragit S100 (ES100), a pH-sensitive polymer having dissolution profile above pH 7.4, is used in oral, ocular, vaginal and topical delivery of drug molecules. However, pH-sensitive potential of this polymer has not been explored for transdermal delivery. The aim of this research work was to exploit the pH-sensitive potential of ES100 as a nanocarrier for transdermal delivery of model drug, that is, Piroxicam.

Methods: Simple nanoprecipitation technique was employed to prepare the nanoparticles and response surface quadratic model was applied to get an optimized formulation. The prepared nanoparticles were characterized and loaded into Carbopol 934 based hydrogel. In vitro release, ex vivo permeation and accelerated stability studies were carried out on the prepared formulation.

Results: Particles with an average size of 25–40?nm were obtained with an encapsulation efficiency of 88%. Release studies revealed that nanoparticles remained stable at acidic pH while sustained release with no initial burst effect was observed at pH 7.4 from the hydrogel. Permeation of these nanocarriers from hydrogel matrix showed significant permeation of Piroxicam through mice skin.

Conclusion: It can be concluded that ES100 based pH-sensitive nanoparticles have potential to be delivered through transdermal route.     

On the random differential quadrature (RDQ) method: consistency analysis and application in elasticity problems

Differential Quadrature (DQ) is an efficient derivative approximation technique but it requires a regular domain with uniformly arranged nodes. This restricts its application for a regular domain only discretized by the field nodes in a fixed pattern. In the presented random differential quadrature (RDQ) method however this restriction of the DQ method is removed and its applicability is extended for a regular domain discretized by randomly distributed field nodes and for an irregular domain discretized by uniform or randomly distributed field nodes. The consistency analysis of the locally applied DQ method is carried out, based on it approaches are suggested to obtain the fast convergence of function value by the RDQ method. The convergence studies are carried out by solving 1D, 2D and elasticity problems and it is concluded that the RDQ method can effectively handle regular as well as irregular domains discretized by random or uniformly distributed field nodes.     

Stability and accuracy of the iterative differential quadrature method

In this paper the stability and accuracy of an iterative method based on differential quadrature rules will be discussed. The method has already been proposed by the author in a previous work, where its good performance has been shown. Nevertheless, discussion about stability and accuracy remained open. An answer to this question will be provided in this paper, where the conditional stability of the method will be pointed out, in addition to an examination of the possible errors which arise under certain conditions. The discussion will be preceded by an overview of the method and its foundations, i.e. the differential quadrature rules, and followed by a numerical case which shows how the method behaves when applied to reduce continuous systems to two‐degree‐of‐freedom systems in the non‐linear range. In particular, here the case of oscillators coupled in non‐linear terms will be treated. The numerical results, used to draw Poincaré maps, will be compared with those obtained by using the Runge–Kutta method with a high precision goal. Copyright © 2003 John Wiley & Sons, Ltd.     

Fundamental frequency analysis of laminated rectangular plates by differential quadrature method

In this paper a differential quadrature method is presented for computation of the fundamental frequency of a thin laminated rectangular plate. The partial differential equations of motion for free vibration are solved for the boundary conditions by approximating them by substituting weighted polynomials functions for the differential operator. By doing this, the coupled partial differential equations of motion are reduced to sets of homogeneous algebraic equations. These sets of homogeneous algebraic equations are combined to give a set of general eigenvalue equations for the problem. Three types of laminated plate problems, which include symmetric, antisymmetric cross-ply, and symmetric, balanced angle-ply laminates, are analysed by the method and the results obtained are compared with solutions reported in the literature for other numerical methods. The effects of the level of discretization on the accuracy and rate of convergence of the results are also discussed. The method presented gives accurate results and is found to use not much computer time.     

三维壁板气动弹性颤振研究的微分求积方法

考虑几何非线性,采用活塞理论计算气动力,基于VonKarman薄板理论和线弹性应力应变关系,建立了三维薄板气动弹性微分方程,采用一种全新的方法即微分求积方法对方程进行了离散,并建立了气动弹性微分方程的微分求积格式,采用Lyapunov间接法确定了系统颤振边界,并分析了系统参数对颤振边界的影响,最后采用数值方法分析了各种系统参数对壁板颤振幅值的影响,得到了一些有意义的结果。     

截锥壳颤振分析的微分求积法

引入微分求积法(Differential Quadrature Method,简称DQM)对截锥壳气动弹性方程离散,采用一阶活塞理论气动力,运用特征值分析方法求解系统的颤振临界动压。研究了半顶角、径厚比、长径比等几何参数对颤振临界动压的影响。结果表明,DQM求解截锥壳气动弹性方程具有良好的精度和计算效率,结构产生1阶~2阶耦合型颤振的最低临界动压对应的周向波数较大,并因几何参数而异;颤振临界动压参数随半顶角的增大而减小,随着径厚比的增大而增大,随长径比的增大而减小。     

Nonlinear bending analysis of orthotropic rectangular plates by the method of differential quadrature

The behavior of thin, rectangular, orthotropic elastic plates, with immovable edges and undergoing large deflections, is investigated by the numerical technique of differential quadrature. Approximate results are obtained, using the Newton-Raphson method and, alternatively, a finite-difference-based method to solve the nonlinear systems of equations. Bending stresses, membrane stresses, and deflections are calculated for plates with fully clamped and simply supported flexural edge conditions under uniform pressure loading. Results are compared with existing analytical, numerical, and experimental ones. The present method gives good accuracy and is computationally efficient.     

Solution of anisotropic nonuniform plate problems by the differential quadrature finite difference method

The differential quadrature finite difference method (DQFDM) has been proposed by the author. The finite difference operators are derived by the differential quadrature (DQ). They can be obtained by using the weighting coefficients for DQ discretizations. The derivation is straight and easy. By using different orders or the same order but different grid DQ discretizations for the same derivative or partial derivative, various finite difference operators for the same differential or partial differential operator can be obtained. Finite difference operators for unequally spaced and irregular grids can also be generated through the use of generic differential quadrature (GDQ). The derivation of higher order finite difference operators is also easy. The DQFDM is used to solve anisotropic nonuniform plate problems. Numerical results are presented. They demonstrate the DQFDM. Received: 11 January 2000     

The Timoshenko beam model of the differential quadrature element method

A new numerical approach for solving Timoshenko beam problems is proposed. The approach uses the differential quadrature method (DQM) to discretize the Timoshenko beam equations defined on all elements, the transition conditions defined on the inter-element boundary of two adjacent elements and the boundary conditions of Timoshenko beam structures. The resulting overall discrete equation can be solved by using a solver of the linear algebra. Numerical results of the DQEM Timoshenko beam model are presented. They demonstrate the DQEM numerical method.     

Solution of the Poisson equation by differential quadrature

The method of differential quadrature is demonstrated by solving the two-dimensional Poisson equation. The results for three test problems are compared with the exact analytical solutions and the numerical solutions obtained by others for the Galerkin, the control-volume and the five-point finite difference methods. The method of differential quadrature leads to more accurate results for comparable levels of computational effort.     

A meshless collocation method based on the differential reproducing kernel interpolation

A differential reproducing kernel (DRK) interpolation-based collocation method is developed for solving partial differential equations governing a certain physical problem. The novelty of this method is that we construct a set of differential reproducing conditions to determine the shape functions of derivatives of the DRK interpolation function, without directly differentiating the DRK interpolation function. In addition, the shape function of the DRK interpolation function at each sampling node is separated into a primitive function processing Kronecker delta properties and an enrichment function constituting reproducing conditions, so that the nodal interpolation properties are satisfied. A point collocation method based on the present DRK interpolation is developed for the analysis of one-dimensional bar problems, two-dimensional potential problems, and plane problems of elastic solids. It is shown that the present DRK interpolation-based collocation method is indeed a truly meshless approach, with excellent accuracy and fast convergence rate.     

The meshless regular hybrid boundary node method for 2D linear elasticity

The meshless Regular Hybrid Boundary Node Method (RHBNM) is a promising method for solving boundary value problems, and is further developed and numerically implemented for 2D linear elasticity in this paper. The present method is based on a modified functional and the Moving Least Squares (MLS) approximation, and exploits the meshless attributes of the MLS and the reduced dimensionality advantages of the BEM. As a result, the RHBNM is truly meshless, i.e. it only requires nodes constructed on the surface, and absolutely no cells are needed either for interpolation of the solution variables or for the boundary integration. All integrals can be easily evaluated over regular shaped domains and their boundaries.Numerical examples show that the high convergence rates with mesh refinement and the high accuracy with a small node number is achievable. The treatment of singularities and further integrations required for the computation of the unknown domain variables, as in the conventional BEM, can be avoided.     

Application of generalized differential quadrature method to structural problems

This paper presents the first endeavour to exploit a generalized differential quadrature method as an accurate, efficient and simple numerical technique for structural analysis. Firstly, drawbacks existing in the method of differential quadrature (DQ) are evaluated and discussed. Then, an improved and simpler generalized differential quadrature method (GDQ) is introduced to overcome the existing drawback and to simplify the procedure for determining the weighting coefficients. Subsequently, the generalized differential quadrature is systematically employed to solve problems in structural analysis. Numerical examples have shown the superb accuracy, efficiency, convenience and the great potential of this method.     

Modeling and simulation of osteoporosis and fracture of trabecular bone by meshless method

Osteoporosis is a skeletal disease characterized by a decrease in bone strength as a result of a decrease of bone mass and a deterioration of bone microstructure. In this work, the imaging data of a CT scanned human femoral neck trabecular bone is directly converted into a meshless model. A model is developed to analyze osteoporosis process. A fracture criterion and the corresponding post-failure are proposed for trabecular bone. The fracture process is modeled and simulated. The simulations show that the fracture stress is not a monotonically decreasing function in the process of fracture, and the microstructure of trabecular bone has a positive effect in preventing progressive failure. The approach in this work may be used to understand the osteoporosis-related fracture and the bone density–strength relationship, and to serve as a way for prognosis of osteoporosis.     

Imposition of boundary conditions by modifying the weighting coefficient matrices in the differential quadrature method

One of the important issues in the implementation of the differential quadrature method is the imposition of the given boundary conditions. There may be multiple boundary conditions involving higher‐order derivatives at the boundary points. The boundary conditions can be imposed by modifying the weighting coefficient matrices directly. However, the existing method is not robust and is known to have many limitations. In this paper, a systematic procedure is proposed to construct the modified weighting coefficient matrices to overcome these limitations. The given boundary conditions are imposed exactly. Furthermore, it is found that the numerical results depend only on those sampling grid points where the differential quadrature analogous equations of the governing differential equations are established. The other sampling grid points with no associated boundary conditions are not essential. Copyright © 2002 John Wiley & Sons, Ltd.     

A new method for meshless integration in 2D and 3D Galerkin meshfree methods

A method for the evaluation of regular domain integrals without domain discretization is presented. In this method, a domain integral is transformed into a boundary integral and a 1D integral. The method is then utilized for the evaluation of domain integrals in meshless methods based on the weak form, such as the element-free Galerkin method and the meshless radial point interpolation method. The proposed technique results in truly meshless methods with better accuracy and efficiency in comparison with their original forms. Some examples, including linear and large-deformation problems, are also provided to demonstrate the usefulness of the proposed method.     

Static and free vibrational analysis of beams and plates by differential quadrature method

Summary The new approach for application of boundary conditions in the differential quadrature (DQ) method, proposed earlier by the present authors, is extended to generalized force boundary conditions in two dimensions. A variety of problems is then analyzed by theDQ method with the new approach for application of boundary conditions, such as deflections of beams and circular and rectangular plates under nonuniformly distributed loadings, deflection of a rectangular plate on a Winkler foundation, and buckling and free vibrational analyses of circular plates. It is found that the present method gives good accuracy and is computationally efficient. Exact solutions can be obtained by theDQ method if analytical solutions are polynomials and the method is insensitive to the spacing of grid points for the cases considered.