Houbolt Finite-difference Scheme Research Articles

A fundamental sequences method with time-reduction for one-dimensional lateral Cauchy problems

A fundamental sequences method is derived for the numerical solution of an ill-posed one-dimensional lateral Cauchy problem for a hyperbolic damped wave equation, including as a special case the parabolic heat equation. Either the Laguerre transform or the Houbolt finite difference scheme is applied to reduce the time-dependent lateral Cauchy problem to a sequence of second-order ordinary differential equations (ODEs) with function values and derivatives specified at the right endpoint of a finite space interval. A set of fundamental solutions is constructed, termed a fundamental sequence, to the differential equations. The solution of the obtained ODEs is approximated by a linear combination of elements in the fundamental sequence. Source points are placed outside of the solution interval in space, and by collocating at the endpoints of this interval a sequence of linear equations is obtained for finding the unknown coeficients. Tikhonov regularization is used to render a stable solution to the obtained systems of linear equations. Numerical results both for the parabolic and hyperbolic case confirm the efficiency of the proposed method including noisy data. The presented results complement the higher-dimensional case initiated in our previous researches.

Singular boundary method using dual reciprocity for two-dimensional transient diffusion

In this paper, a meshless numerical algorithm is developed for the solution of two-dimensional diffusion equations with different boundary conditions. The proposed numerical method, which is truly meshless and quadrature-free, is based on the time-marching finite difference scheme, the method of the particular solutions and the singular boundary method. The diffusion equation is transformed into a Poisson-type or a modified Helmholtz equation after the time domain is discretized by the Euler or Houbolt finite difference scheme.

Cross-flow vortex-induced vibration of a flexible riser transporting an internal flow from subcritical to supercritical

This study aims to model and explore the IFE (Internal Flow Effect) on the CF (Cross-Flow) VIV (Vortex-Induced Vibration) of a flexible riser transporting an axial single-phase internal flow. The simulation model employs the nonlinear equations describing the coupling of axial and transverse vibrations of a fluid-conveying pipe as the structural model, and adopts a distribution of van der Pol oscillators to create the VIV effect. The governing equations can be solved via a Galerkin-based multi-mode approach combined with the Houbolt's finite difference scheme. The developed code has been validated for VIV effect and IFE from subcritical to supercritical region. The Argand diagram of a flexible riser is plotted at first, and the varying natural frequencies with the increase of internal flow velocity and the critical internal flow velocity can be obtained. Then simulations of the flexible riser at two uniform currents with the increase of internal flow velocity in a transition range from being subcritical to supercritical are conducted. The IFE on CF VIVs are examined by the space-time modifications of riser responses and dominant vibration frequency for which the mode switching and sharing can be identified. It has demonstrated that internal flow influences the vibration amplitude and the dominant vibration frequency. Internal flow can trigger new natural modes and switch the role of the most predominant one. Moreover, a buckling-flutter coupled instability is captured where the riser is experiencing a static divergence and a Hopf bifurcation via the first natural mode.

Three-dimensional analysis for functionally graded piezoelectric semiconductors

This article presents numerical analyses of functionally graded piezoelectric beam with piezoelectric semiconducting material properties using meshless method. Unlike piezoelectric dielectric materials, electron density and electric current are additionally considered in constitutive equations for piezoelectric semiconductors. Mutual coupling of elastic displacements, electric potentials, and electron density increases the complexity of the analyzed problem. For the solution of the set of partial differential equations with non-constant coefficients, the local radial basis function collocation method is proposed in this work. Approximating the spatial variations of all physical fields in the partial differential equations by the multiquadric radial basis function entails an ensuing system for time-dependent problems solved by the Houbolt finite-difference scheme as a time-stepping method. The presented local radial basis function collocation method is verified using the corresponding results obtained using the finite element method. The influence of material parameter gradation and initial electron density is then investigated, along with transient analyses.

Modeling of porous piezoelectric structures by the meshless local Petrov-Galerkin method

ABSTRACTA meshless method based on the local Petrov-Galerkin approach is proposed to solve initial-boundary value problems of porous piezoelectric solids. Constitutive equations for porous piezoelectric materials possess a coupling between mechanical displacements and electric intensity vectors in both solid and fluid phases. Stationary and transient 2-D and 3-D axisymmetric problems are considered in this article. Nodal points are spread on the problem domain, and each node is surrounded by a small circle for simplicity. The spatial variation of displacements and electric potentials for both phases is approximated by the moving least-squares scheme. After performing the spatial integration, one obtains a system of ordinary differential equations for certain nodal unknowns. The resulting system is solved numerically by the Houbolt finite-difference scheme as a time stepping method. The proposed method is applied to bending problems associated with a porous piezoelectric 2-D plate and 3-D axisymmetric cylinder under simply supported and clamped boundary conditions.

Analyses of Circular Magnetoelectroelastic Plates with Functionally Graded Material Properties

A meshless method based on the local Petrov–Galerkin approach is proposed for plate bending analysis with material containing functionally graded magnetoelectroelastic properties. Material properties are considered to be continuously varying along the plate thickness. Axial symmetry of geometry and boundary conditions for a circular plate reduces the original 3D boundary value problem into a 2D problem in axial cross section. Both stationary and transient dynamic conditions for a pure mechanical load are considered in this article. The local weak formulation is employed on circular subdomains in the axial cross section. Subdomains surrounding nodes are randomly spread over the analyzed domain. The test functions are taken as unit step functions in derivation of the local integral equations (LIEs). The moving least-squares (MLS) method is adopted for the approximation of the physical quantities in the LIEs. After performing the spatial integrations, one obtains a system of ordinary differential equations for certain nodal unknowns. That system is solved numerically by the Houbolt finite-difference scheme as a time-stepping method.

Meshless analysis of piezoelectric sensor embedded in composite floor panel

Piezoelectric materials are the key element in various sensors and sensory devices used in industry and research. In this article, we use the meshless local Petrov–Galerkin method to analyze a three-dimensional piezoelectric sensor that is embedded in a composite floor panel. Temporal variation of panel deformation is determined analytically and then prescribed as a boundary condition for the sensor. In the proposed formulation, quasi-static governing equations for the electric field and elastodynamic equations for mechanical fields are coupled together. Local integral equations are derived from the local weak form of the governing equations, using a Heaviside step function as the test function. Nodal points are distributed in the analyzed domain, and each node is the center of a small subdomain of spherical shape. The spatial variations of the displacement and electric potential are approximated by the moving least squares scheme. A system of ordinary differential equations is obtained after evaluation of all spatial integrals. The Houbolt finite-difference scheme is applied to solve this system of ordinary differential equations as a time-stepping method. The temporal variation of the induced electric field is finally obtained. It is shown that significant peak amplitudes of the electric field are detected at the top of the sensor.

Bending analyses of 1D orthorhombic quasicrystal plates

The meshless Petrov–Galerkin method (MLPG) is applied to plate bending analysis in 1D orthorhombic quasicrystals (QCs) under static and transient dynamic loads. The Bak and elasto-hydrodynamic models are applied for phason governing equation in the elastodynamic case. The phason displacement for the orthorhombic QC in the first-order shear deformation plate theory depends only on the in-plane coordinates on the mean plate surface. Nodal points are randomly distributed over the mean surface of the considered plate. Each node is the center of a circle surrounding this node. The coupled governing partial differential equations are satisfied in a weak-form on small fictitious subdomains. The spatial variations of the phonon and phason displacements are approximated by the moving least-squares (MLS) scheme. After performing the spatial MLS approximation, a system of ordinary differential equations (ODEs) for nodal unknowns is obtained. The system of the ODEs of the second order is solved by the Houbolt finite-difference scheme. Our numerical examples demonstrate clearly the effect of the coupling parameter on both static and dynamic phonon/phason deflections.

Crack analysis in decagonal quasicrystals by the MLPG

A meshless method based on the local Petrov-Galerkin approach is proposed to solve initial-boundary-value crack problems in decagonal quasicrystals. These quasicrystals belong to the class of two-dimensional (2-d) quasicrystals, where the atomic arrangement is quasiperiodic in a plane, and periodic in the perpendicular direction. The ten-fold symmetries occur in these quasicrystals. The 2-d crack problem is described by a coupling of phonon and phason displacements. Both stationary governing equations and dynamic equations represented by the Bak’s model with oscillations for phasons are analyzed here. Nodal points are spread on the analyzed domain, and each node is surrounded by a small circle for simplicity. The spatial variation of phonon and phason displacements is approximated by the moving least-squares scheme. After performing the spatial integrations, one obtains a system of ordinary differential equations for certain nodal unknowns. That system is solved numerically by the Houbolt finite-difference scheme as a time-stepping method.

Analyses of functionally graded plates with a magnetoelectroelastic layer

A meshless local Petrov–Galerkin (MLPG) method is presented for the analysis of functionally graded material (FGM) plates with a sensor/actuator magnetoelectroelastic layer localized on the top surface of the plate. The Reissner–Mindlin shear deformation theory is applied to describe the plate bending problem. The expressions for the bending moment, shear force and normal force are obtained by integration through the FGM plate and magnetoelectric layer for the corresponding constitutive equations. Then, the original three-dimensional (3D) thick-plate problem is reduced to a two-dimensional (2D) problem. Nodal points are randomly distributed over the mean surface of the considered plate. Each node is the center of a circle surrounding the node. The weak-form on small subdomains with a Heaviside step function as the test function is applied to derive local integral equations. After performing the spatial MLS approximation, a system of ordinary differential equations of the second order for certain nodal unknowns is obtained. The derived ordinary differential equations are solved by the Houbolt finite-difference scheme. Pure mechanical loads or electromagnetic potentials are prescribed on the top of the layered plate. Both stationary and transient dynamic loads are analyzed.

Meshless Modelling of Laminate Mindlin Plates under Dynamic Loads

Collocation method and Galerkin method have been dominant in the existing meshless methods. A meshless local Petrov-Galerkin (MLPG) method is applied to solve laminate plate problems described by the Reissner-Mindlin theory for transient dynamic loads. The Reissner-Mindlin theory reduces the original three-dimensional (3-D) thick plate problem to a two-dimensional (2-D) problem. The bending moment and the shear force expressions are obtained by integration through the laminated plate for the considered constitutive equations in each lamina. The weak-form on small subdomains with a Heaviside step function as the test functions is applied to derive local integral equations. After performing the spatial MLS approximation, a system of ordinary differential equations of the second order for certain nodal unknowns is obtained. The derived ordinary differential equations are solved by the Houbolt finite-difference scheme as a time-stepping method.

Analysis of an interface crack between two dissimilar piezoelectric solids

A meshless method based on the local Petrov–Galerkin approach is proposed, to solve the interface crack problem between two dissimilar piezoelectric solids. Permeable and impermeable electrical boundary conditions are considered on the crack-faces. Quasi-static governing equations for the electrical fields and elastodynamic equations with an inertial term for the 2-D mechanical fields are considered. Nodal points are spread on the analyzed domain, and each node is surrounded by a small circle for simplicity. A Heaviside step function as the test functions is applied in the weak-form on the local subdomain. The local integral equations are derived. The spatial variations of the displacements and electric potential are approximated by the Moving Least-Squares (MLS) scheme. After performing the spatial integrations, one obtains a system of ordinary differential equations for certain nodal unknowns. The system of the ordinary differential equations of the second order resulting from the equations of motion is solved by the Houbolt finite-difference scheme as a time-stepping method.

Analysis of the bending of circular piezoelectric plates with functionally graded material properties by a MLPG method

A meshless method based on the local Petrov–Galerkin approach is proposed for plate bending analysis with functionally graded piezoelectric material properties. Material properties are considered to be continuously varying along the plate thickness. The axial symmetry of geometry and boundary conditions for a circular plate reduces the original three-dimensional (3-D) boundary value problem into a two-dimensional (2-D) problem on the axial cross section. Both mechanical and thermal loads with stationary and transient dynamic conditions are considered in this paper. The local weak formulation is employed on circular subdomains on the axial cross section. Subdomains surrounding nodes are randomly spread over the analyzed domain. The test functions are taken as unit step functions in the derivation of the local integral equations (LIEs). The moving least-squares (MLS) method is adopted for the approximation of the physical quantities in the LIEs. After performing the spatial integrations, one obtains a system of ordinary differential equations for certain nodal unknowns. That system is solved numerically by the Houbolt finite-difference scheme as a time-stepping method.

Semi-permeable crack analysis in magnetoelectroelastic solids

This paper discusses various electromagnetic boundary conditions on the crack-faces in two-dimensional magnetoelectroelastic materials. For this purpose, a meshless method based on the local Petrov–Galerkin approach is developed to solve the initial-boundary value problems of two-dimensional cracked magnetoelectroelastic solids with nonlinear electrical and magnetic boundary conditions on the crack-faces. A Heaviside step function as the test function is applied in the weak form to derive local integral equations. Nodal points are spread on the analyzed domain and each node is surrounded by a small circle for simplicity. The spatial variations of the displacements, electric and magnetic potentials are approximated by the moving least-squares (MLS) scheme. After performing the spatial integrations, one obtains a system of ordinary differential equations for certain nodal unknowns. That system is solved numerically by the Houbolt finite-difference scheme as a time-stepping method. An iterative solution algorithm is developed to consider nonlinear electromagnetic crack-face boundary conditions.

Analysis of Thermo-Piezoelectricity Problems by Meshless Method

In this paper meshless method based on the local Petrov-Galerkin approach is pesented for the solution of boundary value problems for coupled thermo-electro-mechanical fields. Transient dynamic governing equations are considered in analysis of the problems. Material properties of piezoelectric materials are influenced by a thermal field. It is leading to an induced nonhomogeneity and the governing equations are more complicated compared to a homogeneous counterpart. Two-dimensional analyzed domain is divided into small circular subdomains surrounding nodes that are randomly spread over the whole domain. A unit step function is used as the test functions in the local weak-form. The derived local integral equations (LIEs) have boundary-domain integral form. The moving least-squares (MLS) method is adopted for the approximation of the physical quantities in the LIEs and afterwards to obtain a system of ordinary differential equations (ODE) for unknown nodal quantities. To solve this system of ODE, Houbolt finite-difference scheme is applied as a time-stepping method.

Fracture analysis in continuously nonhomogeneous magneto-electro-elastic solids under a thermal load by the MLPG

A meshless method based on the local Petrov–Galerkin approach is proposed, to solve initial-boundary value problems of magneto-electro-elastic solids with continuously varying material properties. Stationary and transient thermal problems are considered in this paper. The mechanical 2-D fields are described by the equations of motion with an inertial term. Nodal points are spread on the analyzed domain, and each node is surrounded by a small circle for simplicity. The spatial variation of displacements, electric and magnetic potentials is approximated by the moving least-squares (MLS) scheme. After performing the spatial integrations, one obtains a system of ordinary differential equations for certain nodal unknowns. That system is solved numerically by the Houbolt finite-difference scheme as a time stepping method.

The time-marching method of fundamental solutions for wave equations

In this paper, a meshless numerical algorithm is developed for the solution of multi-dimensional wave equations with complicated domains. The proposed numerical method, which is truly meshless and quadrature-free, is based on the Houbolt finite difference (FD) scheme, the method of the particular solutions (MPS) and the method of fundamental solutions (MFS). The wave equation is transformed into a Poisson-type equation with a time-dependent loading after the time domain is discretized by the Houbolt FD scheme. The Houbolt method is used to avoid the difficult problem of dealing with time evolution and the initial conditions to form the linear algebraic system. The MPS and MFS are then coupled to analyze the governing Poisson equation at each time step. In this paper we consider six numerical examples, namely, the problem of two-dimensional membrane vibrations, the wave propagation in a two-dimensional irregular domain, the wave propagation in an L-shaped geometry and wave vibration problems in the three-dimensional irregular domain, etc. Numerical validations of the robustness and the accuracy of the proposed method have proven that the meshless numerical model is a highly accurate and efficient tool for solving multi-dimensional wave equations with irregular geometries and even with non-smooth boundaries.

Elastic analysis in 3D anisotropic functionally graded solids by the MLPG

Elastic analysis in 3D anisotropic functionally graded solids by the MLPG J. Sladek1, V. Sladek1, P. Solek2 Summary Functionally graded materials are multi-phase materials with the phase volume fractions varying gradually in space, in a pre-determined profile. This results in continuously graded mechanical properties at the (macroscopic) structural scale. Often, these spatial gradients in material behaviour render FGMs as superior to conventional composites. FGMs possess some advantages over conventional composites because of their continuously graded structures and properties. Due to the high mathematical complexity of the initial-boundary value problems, analytical approaches for elastic analyses of FGMs are restricted to simple geometries and boundary conditions. The elastic analysis in FGM demands an accurate and efficient numerical method. In spite of the great success of the finite and boundary element methods as effective numerical tools for the solution of boundary value problems on complex domains, there is still a growing interest in development of new advanced methods. Many meshless formulations are becoming to be popular due to their high adaptivity and a low cost to prepare input data for numerical analysis. The meshless local Petrov-Galerkin (MLPG) method is the fundamental base for derivations of many meshless formulations, since trial and test functions are chosen from different functional spaces. In this paper, the Heaviside step function is used as the test function. It yields a pure boundary integral formulation on local sub-domains for anisotropic elastostatics, while in elastodynamics an additional domain integral of inertia terms is involved. Small spherical subdomains surrounding the nodes are randomly distributed over the analyzed domain. The spatial variation of the displacement is approximated by the moving least-square (MLS) scheme. After performing the spatial integrations, one obtains the system ordinary differential equations for certain nodal unknowns. That system is solved numerically by the Houbolt finite difference scheme as a time stepping method. Alternatively, the Laplace transform is applied to eliminate the time variable. Then, the local integral equations are derived for Laplace transforms. Several quasi-static boundary value problems have to be solved for various values of the Laplace transform parameter. The Stehfest inversion method is applied to obtain the time-dependent solutions. The integral equations have a very simple nonsingular form. Moreover, both the boundary and domain integrations can be easily carried out on spherical

Meshless local boundary integral equation method for 2D elastodynamic problems

AbstractA new meshless method for solving transient elastodynamic boundary value problems, based on the local boundary integral equation (LBIE) method and the moving least squares approximation (MLS), is proposed in this paper. The LBIE with the MLS is applied to both transient and steady‐state (Laplace transformed) elastodynamics. Applying the MLS approximation for spatially dependent terms in the first approach, the LBIEs are transformed into a system of ordinary differential equations for nodal unknowns. This system of ordinary differential equations is solved by the Houbolt finite difference scheme. In the second formulation, the time variable is eliminated by using the Laplace transformation. Unknown Laplace transforms of displacements and traction vectors are computed from the LBIEs with the MLS approximation. The time‐dependent values are obtained by the Durbin inversion technique. Copyright © 2003 John Wiley & Sons, Ltd.