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Mechanical Response Analysis and Safety Assessment of Shallow-Buried Pipeline under the Influence of Mining
Get PDFIn accordance with the influence of underground mining on the deformation and failure of a shallow-buried gas pipeline, the pipe-soil interaction during mining is classified into two stages, namely coordinated deformation stage and partial hanging stage. According to the mechanical characteristics of the buried pipeline in each stage, the models of a) a beam on an elastic foundation, b) an elastic beam under uniform load, and c) a vertical and horizontal bending beam are introduced in a mining subsidence zone to mechanically analyze, respectively a) the pipeline in non-mining subsidence zone, b) the pipeline at the coordinated deformation stage, and c) the hanging stage. Mechanical model of segmented elastic beam of shallow-buried pipeline is established for each stage. Combined with the boundary conditions of each segmented elastic beam, the limiting condition of the pipe-soil interaction under the two stages is analyzed, and thus, the critical criterion of shallow-buried pipeline failure is obtained. Consequently, the mechanical analysis method on the deformation and failure of the shallow-buried pipeline is obtained. This method is used to analyze the practice of coal mining under a pipeline in Shanxi province, China. The analysis shows that the pipeline will likely fail in the pipe-soil partial hanging stage. Therefore, the pipe-soil interaction should be controlled within the coordinated deformation stage, in which the safe maximum length of working face should be limited to 435 m
A Smooth Discretization Bridging Finite Element and Mesh-free Methods Using Polynomial Reproducing Simplex Splines
Get PDFThis work sets forth a 'hybrid' discretization scheme utilizing bivariate simplex splines as kernels in a polynomial reproducing scheme constructed over a conventional Finite Element Method (FEM)-like domain discretization based on Delaunay triangulation. Careful construction of the simplex spline knotset ensures the success of the polynomial reproduction procedure at all points in the domain of interest, a significant advancement over its precursor, the DMS-FEM. The shape functions in the proposed method inherit the global continuity (Cp-1) and local supports of the simplex splines of degree p. In the proposed scheme, the triangles comprising the domain discretization also serve as background cells for numerical integration which here are near-aligned to the supports of the shape functions (and their intersections), thus considerably ameliorating an oft-cited source of inaccuracy in the numerical integration of mesh-free (MF) schemes. Numerical experiments show the proposed method requires lower order quadrature rules for accurate evaluation of integrals in the Galerkin weak form. Numerical demonstrations of optimal convergence rates for a few test cases are given and the method is also implemented to compute crack-tip fields in a gradient-enhanced elasticity model
A Wavelet Method for the Solution of Nonlinear Integral Equations with Singular Kernels
Get PDFIn this paper, we propose an efficient wavelet method for numerical solution of nonlinear integral equations with singular kernels. The proposed method is established based on a function approximation algorithm in terms of Coiflet scaling expansion and a special treatment of boundary extension. The adopted Coiflet bases in this algorithm allow each expansion coefficient being explicitly expressed by a single-point sampling of the function, which is crucially important for dealing with nonlinear terms in the equations. In addition, we use the technique of integration by parts to transform the original integral equations with non-smooth or singular kernels into regular ones with smooth kernels. Then, we incorporate the proposed function approximation algorithm into the Galerkin method for the solution of the transformed nonlinear integral equations. Numerical examples show that the proposed wavelet method is much more accurate and efficient than several others
Inverse Green Element Solutions of Heat Conduction Using the Time-Dependent and Logarithmic Fundamental Solutions
Get PDFThe solutions to inverse heat conduction problems (IHCPs) are provided in this paper by the Green element method (GEM), incorporating the logarithmic fundamental solution of the Laplace operator (Formulation 1) and the timedependent fundamental solution of the diffusion differential operator (Formulation 2). The IHCPs addressed relate to transient problems of the recovery of the temperature, heat flux and heat source in 2-D homogeneous domains. For each formulation, the global coefficient matrix is over-determined and ill-conditioned, requiring a solution strategy that involves the least square method with matrix decomposition by the singular value decomposition (SVD) method, and regularization by the Tikhonov regularization method. Comparisons of the two formulations are made using five numerical examples of transient IHCPs. Using the same spatial and temporal discretizations, the GEM with the logarithmic fundamental solution is generally more superior in accuracy and computational speed than the formulation with the time-dependent fundamental solution
Voxel-based Analysis of Electrostatic Fields in Virtual-human Model Duke using Indirect Boundary Element Method with Fast Multipole Method
Get PDFThe voxel-based indirect boundary element method (IBEM) combined with the Laplace-kernel fast multipole method (FMM) is capable of analyzing relatively large-scale problems. A typical application of the IBEM is the electric field analysis in virtual-human models such as the model called Duke provided by the foundation for research on information technologies in society (IT’IS Foundation). An important property of voxel-version Duke models is that they have various voxel sizes but the same structural feature. This property is useful for examining the O(N) and O(D2) dependencies of the calculation times and the amount of memory required by the FMM-IBEM, where N and D are the number of boundary elements and the reciprocal of the voxel-side length, respectively. In this paper, the dependencies were confirmed by analyzing Duke models with voxel-side lengths of 5.0, 2.0, 1.0, and 0.5 mm. The finest model had 2.2 billion voxels and 61 million square elements. In addition, a technique that improves the convergence performance of the linear equation solver by considering the non-uniqueness of the electric potential is proposed, and its effectiveness is demonstrated
An Approach with HaarWavelet Collocation Method for Numerical Simulations of Modified KdV and Modified Burgers Equations
Get PDFIn this paper, an efficient numerical schemes based on the Haar wavelet method are applied for finding numerical solution of nonlinear third-order modified Korteweg-de Vries (mKdV) equation as well as modified Burgers' equations. The numerical results are then compared with the exact solutions. The accuracy of the obtained solutions is quite high even if the number of calculation points is small
On the (Meshless Local Petrov-Galerkin) MLPG-Eshelby Method in Computational Finite Deformation Solid Mechanics - Part II
Get PDFThis paper presents a new method for the computational mechanics of large strain deformations of solids, as a fundamental departure from the currently popular finite element methods (FEM). The currently widely popular primal FEM: (1) uses element-based interpolations for displacements as the trial functions, and element-based interpolations of displacement-like quantities as the test functions; (2) uses the same type and class of trial & test functions, leading to a Galerkin approach; (3) uses the trial and test functions which are most often continuous at the inter-element boundaries; (4) leads to sparsely populated symmetric tangent stiffness matrices; (5) computes piecewise-linear predictor solutions based on the global weak-forms of the Newtonian Momentum Balance Laws for a Lagrangean Stress tensor, such as the symmetric Second Piola-Kirchhoff Stress tensor S [≡ JF−1·σ·F−t, where σ is the Cauchy Stress tensor and F the deformation gradient] in the initial or any other known reference configuration; and (6) computes a corrector solution, using Newton-Raphson or other Jacobian-inversion-free iterations, based on the global weak-forms of the Newtonian Momentum Balance Laws for the symmetric Cauchy Stress tensor σ in the current configuration. In a radical departure, the present approach blends the Energy-Conservation Laws of Noether and Eshelby, and the Meshless Local Petrov Galerkin (MLPG) Methods of Atluri, and is designated herein as the MLPG-Eshelby Method. In the MLPG-Eshelby Method, we: (1) use meshless node-based functions δX, for configurational changes of the undeformed configuration, as the trial functions; (2) meshless node-based functions δx, for configurational changes of the deformed configuration, as the test functions; (3) the trial functions δX and the test functions δx are necessarily different and belong to different classes of functions, thus naturally leading to a Petrov-Galerkin approach; (4) leads to sparsely populated unsymmetric tangent stiffness matrices; (5) the trial functions δX, as well as the test functions δx, may either be continuous or be discontinuous in their respective configurations; (6) generate piecewise-linear predictor solutions based on the local weak-forms of the Noether/Eshelby Energy Conservation Laws for the Lagrangean unsymmetric Eshelby Stress tensor T in the undeformed configuration [T = WI − P·F; where P = JF-1·σ is the first Piola-Kirchhoff Stress tensor, and W is the stress-work density per unit initial volume of the solid] and (7) generate corrector solutions, based on Newton-Raphson or Jacobian-inversion-free iterations, using the local weak-forms of the Noether/Eshelby Energy Conservation Laws in the current configuration, for a newly introduced Eulerean symmetric Stress tensor S˜ [which is the counter part of T] in the current configuration [S˜ = (W/J)I − σ, often called by chemists as the Chemical Potential Tensor]. It is shown in the present paper that the present MLPG-Eshelby Method, based on the meshless local weak-forms of the Noether/Eshelby Energy Conservation Laws, converges much faster and leads to much better accuracies than the currently popular FEM based on the global weak-forms of the Newtonian Momentum Balance Laws. The present paper is limited to hyperelasticity, while large strains of inelastic solids will be considered in our forthcoming papers
Forced Vibrations of a System Consisting of a Pre-strained Highly Elastic Plate under Compressible Viscous Fluid Loading
Get PDFThe forced vibration of the system consisting of the pre-stretched plate made of highly-elastic material and half-plane filled by barotropic compressible Newtonian viscous fluid is considered. It is assumed that this forced vibration is caused by the lineal located time-harmonic force acting on the free face plane of the plate. The motion of the pre-stretched plate is written by utilizing of the linearized exact equations of the theory of elastic waves in the initially stressed bodies, but the motion of the compressible viscous fluid is described by the linearized Navier-Stokes equations. The elastic relations of the plate material are described with the use of the harmonic potential. Moreover, it is assumed that the velocities and stresses of the constituents are continuous on the contact plane between the plate and fluid. The dimensionless parameters which characterize the compressibility, viscosity of the fluid and elastic constants of the plate material are introduced. The plane strain state in the plate is considered and the corresponding boundary- and contact-value problem is solved by employing exponential Fourier transformation with respect to the coordinate directed along the interface line and the inverse of this transformation is determined numerically by employing the Sommerfeld contour. Numerical results on the interface stresses and velocities and the influence of the problem parameters such as initial strains and thickness of the plate, the compressibility and viscosity of the plate, as well the magnitude of the frequency of the external forces on these results are presented and discussed. Numerical results are examined in the case where the fluid is Glycerin and the values of the elastic constants which enter into the mentioned above harmonic potential and the density of the plate material are taken as Lame’s constants and density of the Plexiglass (Lucite)
Activation Pattern of Nuclear Factor-kB in Skin after Mechanical Stretch – a Multiscale Modeling Approach
Get PDFThe activation of NF-kB is an important precursor in developing melanoma. However the role of mechanical stimulation in the NF-kB activation has not been studied. We used a multiscale computational modeling approach to investigate the role of mechanical stimulation and the skin tissue internal structures in the activation of NF-kB. Our model is made up of three levels – 1) the macro level where a FE model of the Zygomaticus major muscle was developed; 2) the meso level where a micro FE model of the skin block using a sample from human cadaver was developed; 3) the cell level where a system of ODEs were used to describe the activation of NF-kB under mechanical stimulation. Our results indicate that the activation of NF-kB is highly dependent on the skin geometry and internal structures as well as the type of mechanical stimulation. In particular, our results showed that the flattening of the dermal-epidermal junction due to aging may play an important role in the activation of NF-kB. The future works include the extending of the current multiscale model to the whole face to quantitatively analyze the influence mechanical stimulation and tissue properties on the activation of NF-kB pathways
Solution of Post-Buckling & Limit Load Problems, Without Inverting the Tangent Stiffness Matrix & Without Using Arc-Length Methods
Get PDFIn this study, the Scalar Homotopy Methods are applied to the solution of post-buckling and limit load problems of solids and structures, as exemplified by simple plane elastic frames, considering only geometrical nonlinearities. Explicitly derived tangent stiffness matrices and nodal forces of large-deformation planar beam elements, with two translational and one rotational degrees of freedom at each node, are adopted following the work of [Kondoh and Atluri (1986)]. By using the Scalar Homotopy Methods, the displacements of the equilibrium state are iteratively solved for, without inverting the Jacobian (tangent stiffness) matrix. It is well-known that, the simple Newton’s method (and the Newton-Raphson iteration method that is widely used in nonlinear structural mechanics), which necessitates the inversion of the Jacobian matrix, fails to pass the limit load as the Jacobian matrix becomes singular. Although the so called arc-length method can resolve this problem by limiting both the incremental displacements and forces, it is quite complex for implementation. Moreover, inverting the Jacobian matrix generally consumes the majority of the computational burden especially for large-scale problems. On the contrary, by using the presently developed Scalar Homotopy Methods, convergence near limit loads, and in the post-buckling region, can be easily achieved, without inverting the tangent stiffness matrix and without using complex arc-length methods. The present paper thus opens a promising path for conducting post-buckling and limit-load analyses of nonlinear structures. While the simple Williams’ toggle is considered as an illustrative example in this paper, extension to general finite element analyses of space frames, plates, shells and elastic-plastic solids will be considered in forthcoming studies