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Analytical Solution of Stokes Flow in a Driven Cavity Using the Natural Boundary Element Method
Get PDFIn this paper, the natural boundary element method is used to solve two-dimensional steady-state incompressible Stokes flows in a driven cavity. The analytical functions are expressed for the Stokes problem in an exterior circular domain under single value conditions, which satisfy the Stokes equations’ solutions in the form of complex functions. In order to obtain a uniform integral formula, the velocities on the boundary are expanded into Laurent series, and then compared with the analytical solutions obtained as described above. In this manner, the coefficients of the analytical solutions in the form of complex function are further confirmed. According to the formulae of Fourier series and convolutions, the boundary integral formulae related only to boundary velocities are obtained for the Stokes problems in an exterior circular domain. Similarly, the boundary integral formulae are investigated for the interior circular domain. The formulae are applied to the Stokes flows in some circular and annular cavities, and the obtained results are compared with those produced by previous works. The results prove that the current technique is in very good agreement with previous investigations, and that the natural boundary element method is an accurate and flexible method for the solution of Stokes flows
Establishment and Stability Analysis of a Hybrid Viscoelastic Model Based on Meshless for Surgical Robot System
Get PDFAiming at the shortcomings of mass-spring model, this paper, on the basis of preliminary studies, established a new viscoelastic soft tissue model based on meshless structure. The model is consisted of a large quantity of filled spheres, with every three spheres being connected by a spring and a Kelvin structure, which can further enhance the real-time virtual simulation operability while ensure the viscoelasticity of basic model. The stress relaxation and creep equation of the model can be derived from formula derivation. Through setting different parameters to the filled spheres, this model, with certain universal property, can create a virtual liver, kidney and skin soft tissue. By utilizing force feedback devices to handle different virtual soft tissue thus to get virtual simulation experimental data, then compare them with the actual characteristics data of soft tissue in published literature, and experiments have shown that this model possesses good stress-strain, stress relaxation and creep properties. Finally, this paper compared mass-spring model for the stress-strain and cutting simulation experiments, which further confirmed the effectiveness and stability of the proposed model
Algebraic Multigrid Methods Based on Generic Approximate Banded Inverse Matrix Techniques
Get PDFSince the introduction of the Algebraic MultiGrid algorithm (AMG) over twenty years ago, significant progress has been made in improving the coarsening and the convergence behavior of the method. In this paper, an AMG method is introduced that utilizes a new generic approximate inverse algorithm as a smoother in conjunction with common coarsening techniques, such as classical Ruge-Stüben coarsening, CLJP and PMIS coarsening. The proposed approximate inverse scheme, namely Generic Approximate Banded Inverse (GenAbI), is a banded approximate inverse based on Incomplete LU factorization with zero fill–in (ILU(0)). The new class of Generic Approximate Banded Inverse can be computed for any sparsity pattern of the coefficient matrix, in an analogous way as the explicit approximate inverse, yielding a suitable smoother to be used in conjunction with an Algebraic Multigrid method. The proposed smoother is parameterized and thus by increasing the “retention” parameter the smoothing scheme becomes more effective in terms of required number of cycles for convergence. Finally, the applicability and effectiveness of the proposed AMG method along with implementation issues, based on the Generic Approximate Banded Inverse matrix, is demonstrated by solving two and three dimensional problems and numerical results on the convergence behavior and convergence factor are given
A Projection Method for the Monolithic Interaction System of an Incompressible Fluid and a Structure using a New Algebraic Splitting
Get PDFIn this study, a projection method for the monolithic interaction system of an incompressible fluid and a structure using a new algebraic splitting is proposed. The proposed method splits the monolithic equation system into the equilibrium equations and the pressure Poisson equation (PPE) algebraically using the intermediate velocity in the nonlinear iterations. Since the proposed equilibrium equation satisfies the interface condition, the proposed method is strongly coupled. Moreover, the proposed PPE enforces the incompressibility constraint. Different from previous studies, the proposed algebraic splitting never generates any Schur complement. The proposed method is applied to a channel with a flexible flap, which is one of typical test problems, where its superior computational efficiencies are demonstrated
An Improved Isogeometric Boundary Element Method Approach in Two Dimensional Elastostatics
Get PDFThe NURBS based isogeometric analysis offers a novel integration between the CAD and the numerical structural analysis codes due to its superior capacity to describe accurately any complex geometry. Since it was proposed in 2005, the approach has attracted rapidly growing research interests and wide applications in the Finite Element context. Only recently, in 2012, it was successfully tested together with the Boundary Element Method. The combination of the isogeometric approach and the Boundary Element Method is efficient since both the NURBS geometrical representation and the Boundary Element Method deal with quantities entirely on the boundary of the problem. Actually, there are still some difficulties in imposing generic boundary conditions, mainly due to the fact that the NURBS basis functions are not interpolatory functions. In this work it is shown that the direct imposition of the inhomogeneous generic boundary conditions to the NURBS control points may lead to significant errors. Consequently an improved formulation is proposed that relates the boundary conditions to the governing unknown variables by developing a transformation strategy. Several elasticity problems evince that higher solution accuracy can be achieved by the present formulation
The Boundary Integral Equation for 3D General Anisotropic Thermoelasticity
Get PDFGreen’s functions, or fundamental solutions, are necessary items in the formulation of the boundary integral equation (BIE), the analytical basis of the boundary element method (BEM). In the formulation of the BEM for 3D general anisotropic elasticity, considerable attention has been devoted to developing efficient algorithms for computing these quantities over the years. The mathematical complexity of this Green’s function has also posed an obstacle in the development of this numerical method to treat problems of 3D anisotropic thermoelasticity. This is because thermal effects manifest themselves as an additional domain integral in the integral equation; this has implications for the numerical modeling in BEM. Difficulties in deriving a true BIE arise, unless some simple representations of the thermal effects are used, such as in the dual reciprocity approach. These approximation schemes, however, have some serious limitations. An integral transformation method to obtain an exact BIE has been successfully employed, but only for isotropy and two-dimensional (2D) general anisotropy. The extension of this scheme to three-dimensional (3D) general anisotropy has remained a very serious challenge. This paper reports on the progress towards this end. By following the same steps as for 2D general anisotropy, and using a double-Fourier series representation of the Green’s function first proposed by the authors recently, a true BIE is derived for 3D general anisotropic thermoelasticity. Some numerical results are presented to demonstrate the success of this derivation
A Solution Procedure for a Vibro-Impact Problem under Fully Correlated Gaussian White Noises
Get PDFThis study is concerned with a solution procedure to obtain the probability density function (PDF) of a vibro-impact Duffing oscillator under fully correlated external and parametric Gaussian white noises. The proposed solution procedure consists of three steps. In the first step, the Zhuravlev non-smooth coordinate transformation is adopted to introduce an additional impulsive damping term, in which the original vibro-impact oscillator is converted into a new oscillator without any barrier. After that, the PDF of the new oscillator is obtained by solving the Fokker-Planck equation with the exponential-polynomial closure method. Last, the PDF of the original oscillator is formulated in terms of the methodology on seeking the PDF of a function of random variables. A numerical analysis on four different cases is conducted to examine the effectiveness of the proposed solution procedure. Comparison with the simulated result shows that the proposed solution procedure can provide a satisfactory PDF solution for the four cases. The tail region of the PDF solution is also approximated well. The numerical analysis also shows that the change of parametric excitation has a significant effect on the PDF distributions of displacement and velocity
A Global Numerical Analysis of the “Central Incisor / Local Maxillary Bone” System using a Meshless Method
Get PDFIn this work the maxillary central incisor is numerically analysed with an advance discretization technique – Natural Neighbour Radial Point Interpolation Method (NNRPIM). The NNRPIM permits to organically determine the nodal connectivity, which is essential to construct the interpolation functions. The NNRPIM procedure, based uniquely in the computational nodal mesh discretizing the problem domain, allows to obtain autonomously the required integration mesh, permitting to numerically integrate the differential equations ruling the studied physical phenomenon.
A numerical analysis of a tooth structure using a meshless method is presented for the first time. A two-dimensional model of the maxillary central incisor, based on the clinical literature, is established and two distinct analyses are performed. First, a complete elasto-static analysis of the incisor/maxillary structure using the NNRPIM is evaluated and then a non-linear iterative bone tissue remodelling analysis of the maxillary bone, surrounding the central incisive, is performed. The obtained NNRPIM solutions are compared with other numerical methods solutions available in the literature and with clinical cases. The results show that the NNRPIM is a suitable numerical method to analyse numerically dental biomechanics problems
Mechanical Modelling of Confined Cell Migration Across Constricted-curved Micro-channels
Get PDFConfined migration is a crucial phenomenon during embryogenesis, immune response and cancer. Here, a two-dimensional finite element model of a HeLa cell migrating across constricted–curved micro-channels is proposed. The cell is modelled as a continuum with embedded cytoplasm and nucleus, which are described by standard Maxwell viscoelastic models. The decomposition of the deformation gradient is employed to define the cyclic active strains of protrusion and contraction, which are synchronized with the adhesion forces between the cell and the substrate. The micro-channels are represented by two rigid walls and exert an additional viscous force on the cell boundaries. Five configurations have been tested: 1) top constriction, 2) top-bottom constriction, 3) shifted top-bottom constriction, 4) embedded obstacle and 5) bending micro-channel. Additionally, for the first four micro-channels both sub-cellular and sub-nuclear constrictions have been obtained, while for the fifth micro-channel three types of bending have been investigated (‘curved’, ‘sharp’ and ‘sharper’). For each configuration, several parameters such as the cell behaviour, the covered distance, the migration velocity, the ratio between the cell and the nucleus area as well as the cell-substrate and cell-channel surfaces forces have been evaluated. The results show once more the fundamental role played by mechanics of both the cell and the environment
Change of Scale Strategy for the Microstructural Modelling of Polymeric Rohacell Foams
Get PDFIn this paper a numerical model dedicated to the simulation of the mechanical behaviour of polymeric Rohacell foams is presented. The finite elements model is developed at the scale of the microstructure idealized by a representative unit cell: the truncated octahedron. Observations made on micrographs of Rohacell lead to mesh this representative unit cell as a lattice of beam elements. Each beam is assigned a brittle linear elastic mechanical behaviour in tension and an elastoplastic behaviour in compression. The plasticity in compression is introduced as a way to mimic the buckling of the edges of the cells observed in experimental crushing tests. A contact law introduced between the beams stands for densification. A change in scale is then realized by increasing the length of the edges of the unit cell. Several computations show the ability of the proposed approach to preserve the physical degradation phenomena and the loads while drastically decreasing the computational time