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The Generalized Tikhonov Regularization Method for High Order Numerical Derivatives
Get PDFNumerical differentiation is a classical ill-posed problem. The generalized Tikhonov regularization method is proposed to solve this problem. The error estimates are obtained for a priori and a posteriori parameter choice rules, respectively. Numerical examples are presented to illustrate the validity and effectiveness of this method
Solution of Two-Dimensional Viscous Flow in a Rectangular Domain by the Modified Decomposition Method
Get PDFIn this paper, the modified decomposition method (MDM) for solving the nonlinear two-dimensional viscous flow equations is presented. This study investigates the problem of laminar, isothermal, incompressible and viscous flow in a rectangular domain bounded by two moving porous walls, which enable the fluid to enter or exit during successive expansions or contractions. We first transform the original two-dimensional viscous flow problem into an equivalent fourth-order boundary value problem (BVP), then solve the problem by the MDM. The figures and tables clearly show high accuracy of the method to solve two-dimensional viscous flow
On the Numerical Solution of the Laplace Equation with Complete and Incomplete Cauchy Data Using Integral Equations
Get PDFWe consider the numerical solution of the Laplace equations in planar bounded domains with corners for two types of boundary conditions. The first one is the mixed boundary value problem (Dirichlet-Neumann), which is reduced, via a single-layer potential ansatz, to a system of well-posed boundary integral equations. The second one is the Cauchy problem having Dirichlet and Neumann data given on a part of the boundary of the solution domain. This problem is similarly transformed into a system of ill-posed boundary integral equations. For both systems, to numerically solve them, a mesh grading transformation is employed together with trigonometric quadrature methods. In the case of the Cauchy problem the Tikhonov regularization is used for the discretized system. Numerical examples are included both for the well-posed and ill-posed cases showing that accurate numerical solutions can be obtained with small computational effort
Direct Volume-to-Surface Integral Transformation for 2D BEM Analysis of Anisotropic Thermoelasticity
Get PDFAs has been well documented for the boundary element method (BEM), a volume integral is present in the integral equation for thermoelastic analysis. Any attempt to directly integrate the integral shall inevitably involve internal discretization that will destroy the BEM’s distinctive notion as a true boundary solution technique. Among the schemes to overcome this difficulty, the exact transformation approach is the most elegant since neither further approximation nor internal treatments are involved. Such transformation for 2D anisotropic thermoelasticity has been achieved by Shiah and Tan (1999) with the aid of domain mapping. This paper revisits this problem and presents a modified transformation for 2D anisotropic thermoelasticity, where no domain distortion is involved. Being defined in the original Cartesian coordinate system, the volume integral is analytically transformed to the boundary using the Stroh formalism. This transformation is favorable especially when the corresponding anisotropic field is directly calculated without resorting to the domain mapping technique. In the end, numerical examples are provided to show the validity of such a transformation
A (Constrained) Microstretch Approach in Living Tissue Modeling: a Numerical Investigation Using the Local Point Interpolation – Boundary Element Method
Get PDFExtended continuum mechanical approaches are now becoming increasingly popular for modeling various types of microstructured materials such as foams and porous solids. The potential advantages of the microcontinuum approach are currently being investigated in the field of biomechanical modeling. In this field, conducting a numerical investigation of the material response is evidently of paramount importance. This study sought to investigate the potential of the (constrained) microstretch modeling method. The problem’s field equations have been solved by applying a numerical approach combining the conventional isotropic boundary elements method with local radial point interpolation. Our resulting numerical examples demonstrated that the model is a good candidate for the mechanical modeling of living tissues
Frequency Domain Based Solution for Certain Class of Wave Equations: An exhaustive study of Numerical Solutions
Get PDFThe paper discusses the frequency domain based solution for a certain class of wave equations such as: a second order partial differential equation in one variable with constant and varying coefficients (Cantilever beam) and a coupled second order partial differential equation in two variables with constant and varying coefficients (Timoshenko beam). The exact solution of the Cantilever beam with uniform and varying cross-section and the Timoshenko beam with uniform cross-section is available. However, the exact solution for Timoshenko beam with varying cross-section is not available. Laplace spectral methods are used to solve these problems exactly in frequency domain. The numerical solution in frequency domain is done by discretisation in space by approximating the unknown function using spectral functions like Chebyshev polynomials, Legendre polynomials and also Normal polynomials. Different numerical methods such as Galerkin Method, Petrov- Galerkin method, Method of moments and Collocation method or the Pseudo-spectral method in frequency domain are studied and compared with the available exact solution. An approximate solution is also obtained for the Timoshenko beam with varying cross-section using Laplace Spectral Element Method (LSEM). The group speeds are computed exactly for the Cantilever beam and Timoshenko beam with uniform cross-section and is compared with the group speeds obtained numerically. The shear mode and the bending modes of the Timoshenko beam with uniform cross-section are separated numerically by applying a modulated pulse as the shear force and the corresponding group speeds for varying taper parameter m are obtained numerically by varying the frequency of the input pulse. An approximate expression for calculating group speeds corresponding to the shear mode and the bending mode, and also the cut-off frequency is obtained. Finally, we show that the cut-off frequency disappears for large m, for ε > 0 and increases for large m, for ε < 0
Meshless Local Petrov-Galerkin Mixed Collocation Method for Solving Cauchy Inverse Problems of Steady-State Heat Transfer
Get PDFIn this article, the Meshless Local Petrov-Galerkin (MLPG) Mixed Collocation Method is developed to solve the Cauchy inverse problems of Steady- State Heat Transfer In the MLPG mixed collocation method, the mixed scheme is applied to independently interpolate temperature as well as heat flux using the same meshless basis functions The balance and compatibility equations are satisfied at each node in a strong sense using the collocation method. The boundary conditions are also enforced using the collocation method, allowing temperature and heat flux to be over-specified at the same portion of the boundary. For the inverse problems where noise is present in the measurement, Tikhonov regularization method is used, to mitigate the inherent ill-posed nature of inverse problem, with its regularization parameter determined by the L-Curve method. Several numerical examples are given, wherein both temperature as well as heat flux are prescribed at part of the boundary, and the data at the other part of the boundary and in the domain have to be solved for. Through these numerical examples, we investigate the accuracy, convergence, and stability of the proposed MLPG mixed collocation method for solving inverse problems of Heat Transfer
Large Deformation Dynamic Three-Dimensional Coupled Finite Element Analysis of Soft Biological Tissues Treated as Biphasic Porous Media
Get PDFThe paper presents three-dimensional, large deformation, coupled finite element analysis (FEA) of dynamic loading on soft biological tissues treated as biphasic (solid-fluid) porous media. An overview is presented of the biphasic solidfluid mixture theory at finite strain, including inertia terms. The solid skeleton is modeled as an isotropic, compressible, hyperelastic material. FEA simulations include: (1) compressive uniaxial strain loading on a column of lung parenchyma with either pore air or water fluid, (2) out-of-plane pressure loading on a thin slab of lung parenchyma with either pore air or water fluid, and (3) pressure loading on a 1/8th symmetry vertebral disc (nucleus and annulus) with pore water. For the simulations, mixed formulation Q27P8 and stabilized Q8P8 finite elements are compared (“Q” indicates the number of solid skeleton displacement nodes, and “P” the number of pore fluid pressure nodes). The FEA results demonstrate the interplay of dynamics (wave propagation through solid skeleton and pore fluid), large deformations, effective stress and pore fluid pressure coupling, compressibility and viscosity of pore fluid, and three-dimensional effects for soft biological tissues treated as biphasic porous media
Particle-Based Moving Interface Method for The Study of the Interaction Between Soft Colloid Particles and Immersed Fibrous Network
Get PDFMany colloidal-sized particles encountered in biological and membranebased separation applications can be characterized as soft vesicles such as cells, yeast, viruses and surfactant micelles. The deformation of these vesicles is expected to critically affect permeation by accommodating pore shapes and sizes or enhancing the adhesion with a pore surface. Numerical and theoretical modelings will be critical to fully understand these processes and thus design novel filtration membranes that target, not only size, but deformability as a selection criterion. The present paper therefore introduces a multiscale strategy that enables the determination of the permeability of a fibrous network with respect to complex fluids loaded with vesicles. The contributions are two-fold. First, we introduce a particle-based moving interface method that can be used to characterized the severe deformation of vesicles interacting with an immersed fibrous network. Second, we present a homogenization strategy that permits the determination of a network permeability, based on the micromechanisms of vesicle deformation and permeation. As a proof of concept, we then investigate the role of vesicle-solvent surface tension on the permeation of both solvent and vesicle through a simple fiber network. We find that vesicles are always retarded relative to the continuum (or solvent) flow, and that the relative selectivity for the continuum versus the vesicle is inversely proportional to the capillary number
Numerical Solution of Fractional Fredholm-Volterra Integro-Differential Equations by Means of Generalized Hat Functions Method
Get PDFIn this paper, operational matrix method based on the generalized hat functions is introduced for the approximate solutions of linear and nonlinear fractional integro-differential equations. The fractional order generalized hat functions operational matrix of integration is also introduced. The linear and nonlinear fractional integro-differential equations are transformed into a system of algebraic equations. In addition, the method is presented with error analysis. Numerical examples are included to demonstrate the validity and applicability of the approach