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Legendre Polynomials Method for Solving a Class of Variable Order Fractional Differential Equation
Get PDFIn this paper, a numerical method based on the Legendre polynomials is presented for a class of variable order fractional differential equation. We adopt the Coimbra variable order fractional operator, which can be viewed as a Caputo-type definition. Three different kinds of operational matrixes with Legendre polynomials are derived. A truncated the Legendre polynomials series together with the products of several dependent matrixes are utilized to reduce the variable order fractional differential equation to a system of algebraic equations. The solution of this system gives the approximation solution for the truncated limited n. An error analysis technique is also given. Some examples are included to demonstrate the validity and applicability of the approach
Numerical Analysis for the Mooring System with Nonlinear Elastic Mooring Cables
Get PDFThis paper presents numerical analysis for the mooring system with nonlinear elastic mooring cables. The equation of motion for nonlinear elastic mooring cable is established by utilizing finite element method. A marine mooring system of floating rectangular box with nonlinear elastic cables is taken as an illustrative example. The dynamic analysis, static analysis, and uniformity analysis are carried out for the polyester mooring system and the results are compared with those of the steel wire and the chain mooring system. Results from the present study can provide valuable recommendations for the design and construction of the mooring system with nonlinear elastic mooring cables
Numerical Simulation of 3D Rough Surfaces and Analysis of Interfacial Contact Characteristics
Get PDFMechanical behaviors arising at the contact interface largely depend on its surface topographies, particularly when it comes to rough surfaces. A numerical simulation based on an appropriate characterization of rough surfaces especially in terms of three dimensional can be of great significance when it comes to capturing the deformation patterns of micro-scale contacts. In this paper, a simple and practical scheme is developed to generate 3D rough surfaces and to analyze and evaluate the contact characteristics. Firstly amplitude and spatial statistical characterizations of asperities are introduced to avert from the redundancy of topography data caused by traditional measuring methods. A calculation strategy is then proposed to transform varied white noise sequences into the Gaussian and non-Gaussian height sequences, in which operations like translating scaling and spatial reconfiguring are utilized to guarantee that the output first four moments are satisfied with the requirements given in advance. After that a more accurate FE model is developed to handle the problem that the asperities are so acute and tiny that can give rise to highly unstable data, in which a new meshing strategy is put forward to improve the mesh quality and solution efficiency. Finally, the simulation results are obtained through analyzing the contact characteristics of the established models. The unique feature of the proposed method is not only being capable of generating rough surfaces with any skewness and kurtosis in the whole skewnesskurtosis plane, but also narrowing down the errors of statistical characterization like the skewness and kurtosis to the level of 10-2 which is a highly accurate estimation in terms of the output first four moments and autocorrelation functions (ACFs). The proposed methods for numerical simulation of rough surfaces can provide massive and accurate surface topography data with a small amount of computational resources, based on which, FEM is adopted to precisely evaluate the contact characteristics including load-deformation relationship, the contact pressure distribution, the real contact area and the interfacial loading/unloading characteristics, and therefore is a good choice for the study on the contact characteristics of rough surfaces
An Incremental Kriging Method for Sequential Optimal Experimental Design
Get PDFKriging model, which provides an exact interpolation and minimizes the error estimates, is a highly-precise global approximation model in contrast with other traditional response surfaces. Therefore, sequential exploratory experimental design (SEED) with Kriging model is crucial for globally approximating a complex black-box function. However, the more sampling points are, the longer time it would take to update the Kriging model during sequential exploratory design. This paper, therefore, proposes a new construction method called incremental Kriging method (IKM) to improve the constructing efficiency with just a little and controllable loss of accuracy for Kriging model. The IKM, based on the matrix segmentation theory, is under the premise that the correlated parameter θ remains unchanged. Meantime, it utilizes the original model and incremental sampling data to quickly obtain an updated Kriging model. Fortunately, a large number of numerical tests showed that the stability of parameter θ would become better and better with the continual increase of the new sampling points in most instances. Even if there is a slight change for value θ, there is not obvious effect on the accuracy of Kriging model. Then, a new sequential incremental experimental design (SIED) algorithm based on IKM is presented to construct Kriging model steadily and effectively. At each sampling step, the SIED method finds an optimal sampling point which maximizes the mean square error of the current model. Meantime, it judges whether the θ should be changed according to the updating criterion. Kriging model will be updated by IKM when θ remains unchanged, or recreates the model with all the sampling points, otherwise. Finally, seven numerical tests and three engineering examples are given to illustrate the applicability, effectiveness and superiority of the proposed methods
A Meshfree Method For Mechanics and Conformational Change of Proteins and Their Assemblies
Get PDFMechanical properties of proteins play an important role in their biological function. For example, microtubules carry large loads to transport organelles inside the cell, and virus shells undergo changes in shape and mechanical properties during maturation which affect their infectivity. Various theoretical models including continuum elasticity have been applied to study these structural properties, and a significant success has been achieved. But, the previous frameworks lack a connection between the atomic and continuum descriptions. Here this is accomplished through the development of a meshfree framework based on reproducing kernel shape functions for the large deformation mechanics of protein structures. The framework is validated by comparing thermal fluctuations of small proteins against well established elastic network model. To demonstrate the usability of this framework, solutions to several other problems are presented. The response of virus shells to indentation under atomic force microscope tip is simulated and compared to the finite element results. Finally, the large scale conformational changes of viruses are analyzed by computing the deformations/strains associated with the conformational motions. Excellent agreement with the previously published results is observed while increasing the efficiency of numerical analysis. Furthermore, the results provide insights into the continuum behavior of proteins and the optimum amount of geometric details necessary for calculating their mechanical properties
A Computational Modeling Framework for Heat Transfer Processes in Laser-Induced Dermal Tissue Removal
Get PDFA widespread use of lasers is for the ablation of biological tissue, in particular for dermal applications involving the removal of cancerous tissue, skin spots, aged skin and wrinkles. For a laser to ablate tissue, the power intensity must be sufficiently high to induce vaporization/burning of the target material. However, if performed improperly, the process can cause excessive microscale thermal injuries to surrounding healthy tissue. This motivates the present work, which attempts to develop and assemble simple models for the primary heat transfer mechanisms that occur during the process. First, in order to qualitatively understand the system, the terms that contribute to achieving a target temperature are studied, accounting for: (a) incoming laser irradiance, (b) heat conduction to the body, (c) infrared radiation to the surroundings and (d) convection from a vacuum (needed in certain surgical procedures to collect unwanted debris). Thereafter, a computational framework is then developed, accounting for the previously mentioned terms and further including: (a) phase transformations, including latent heats of transformation, and (b) mass transport (losses) due to burning and ablation of the target tissue. The framework is, by design, straightforward to computationally implement, in order to be easily utilized by researchers in the field
A Double Iteration Process for Solving the Nonlinear Algebraic Equations, Especially for Ill-posed Nonlinear Algebraic Equations
Get PDFIn this paper, a novel double iteration process for solving the nonlinear algebraic equations is developed. In this process, the outer iteration controls the evolution path of the unknown vector x in the selected direction u which is determined from the inner iteration process. For the inner iteration, the direction of evolution u is determined by solving a linear algebraic equation: BTBu = BTF where B is the Jacobian matrix, F is the residual vector and the superscript ''T'' denotes the matrix transpose. For an ill-posed system, this linear algebraic equation is very difficult to solve since the resulting leading coefficient matrix is ill-posed in nature. We adopted the modified Tikhonov's regularization method (MTRM) developed by Liu (Liu, 2012) to solve the ill-posed linear algebraic equation. However, to exactly find the solution of the evolution direction u may consume too many iteration steps for the inner iteration process, which is definitely not economic. Therefore, the inner iteration process stops while the direction u makes the value of a0 being smaller than the selected margin ac or when the number of inner iteration steps exceeds the maximum tolerance Imax. For the outer iteration process, it terminates once the root mean square error for the residual is less than the convergence criterion ε or when the number of inner iteration steps exceeds the maximum tolerance Imax. Six numerical examples are given and it is found that the proposed method is very efficient especially for the nonlinear ill-posed systems
Solving the Cauchy Problem of the Nonlinear Steady-state Heat Equation Using Double Iteration Process
Get PDFIn this paper, the Cauchy inverse problem of the nonlinear steady-state heat equation is studied. The double iteration process is used to tackle this problem in which the outer loop is developed based on the residual norm based algorithm (RNBA) while the inner loop determines the evolution direction and the modified Tikhonov's regularization method (MTRM) developed by Liu (Liu, 2012) is adopted. For the conventional iteration processes, a fixed evolution direction such as F, B−1F, BTF or αF+(1-α)BTF is used where F is the residual vector, B is the Jacobian matrix, the superscript '-1' denotes the inverse, the superscript 'T' denotes the transpose of a matrix and α denotes the optimal coefficient. Unlike the conventional approaches, the current approach tries to find an appropriate direction from the initial guess BTF using the MTRM and the final evolution direction is determined once the value of a0 is less than the critical value ac. Since it may consume too much computation time for searching this appropriate evolution direction such that it makes this process computationally noneconomic, we terminate the inner iteration process as well as the whole process once the number of the iteration steps for the inner iteration exceeds a given maximum value, says Imax. Six examples are illustrated to show the validity of the current approach and results show that the proposed method is very efficient and accurate
A Novel Semi-Analytic Meshless Method for Solving Two- and Three-Dimensional Elliptic Equations of General Form with Variable Coefficients in Irregular Domains
Get PDFThe paper presents a new meshless numerical method for solving 2D and 3D boundary value problems (BVPs) with elliptic PDEs of general form. The coefficients of the PDEs including the main operator part are spatially dependent functions. The key idea of the method is the use of the basis functions which satisfy the homogeneous boundary conditions of the problem. This allows us to seek an approximate solution in the form which satisfies the boundary conditions of the initial problem with any choice of the free parameters. As a result we separate approximation of the boundary conditions and approximation of the PDE inside the solution domain. Numerical experiments are carried out for accuracy and convergence investigations. A comparison of the numerical results obtained in the paper with the exact solutions and with the data obtained with the use of other numerical techniques (Kansa's method, the method of particular solutions) is performed
Singular Boundary Method: Three Regularization Approaches and Exterior Wave Applications
Get PDFThis study investigates the singular boundary method (SBM) with three regularization approaches for solving 2D and 3D exterior wave problems. The singular boundary method is a recent meshless boundary collocation method, which introduces the concept of source intensity factors to eliminate the singularity of the fundamental solutions. Recently, three approaches, the inverse interpolation technique (IIT), the semi-analytical technique with boundary IIT (SAT1) and the semi-analytical technique with integral mean value (SAT2), have been proposed to determine the source intensity factors for removing the singularities of Helmholtz fundamental solutions at origin. This study compares numerical accuracy and stability of these three approaches on some benchmark examples under 2D and 3D exterior wave radiation and scattering problems. Numerical investigations show that SAT1>IIT>SAT2 in numercial accuracy and SAT2>SAT1>IIT in numerical stability. Then the SBM with SAT1 is applied to water wave-structure interaction and SH wave scattering problem. For water wave-structure interaction, numerical results show that both the porosity of the cylinder sidewall and the disorder arrangement have a great effect on the free-surface elevations in the vicinity of the wave structure. For SH wave scattering by a semi-circular hill, the focusing phenomenon is revisited