Unknown Field Variables Research Articles

Numerical MLPG Analysis of Piezoelectric Sensor in Structures

Abstract The paper deals with a numerical analysis of the electro-mechanical response of piezoelectric sensors subjected to an external non-uniform displacement field. The meshless method based on the local Petrov-Galerkin (MLPG) approach is utilized for the numerical solution of a boundary value problem for the coupled electro-mechanical fields that characterize the piezoelectric material. The sensor is modeled as a 3-D piezoelectric solid. The transient effects are not considered. Using the present MLPG approach, the assumed solid of the cylindrical shape is discretized with nodal points only, and a small spherical subdomain is introduced around each nodal point. Local integral equations constructed from the weak form of governing PDEs are defined over these local subdomains. A moving least-squares (MLS) approximation scheme is used to approximate the spatial variations of the unknown field variables, and the Heaviside unit step function is used as a test function. The electric field induced on the sensor is studied in a numerical example for two loading scenarios.

A new trigonometric zigzag theory for buckling and free vibration analysis of laminated composite and sandwich plates

In this work, a new trigonometric zigzag theory is proposed for the analysis of laminated composite and sandwich plates. The theory is based upon shear strain shape function assuming non-linear distribution of transverse shear stresses. It satisfies the necessary conditions of inter-laminar stress continuity at the layer interfaces as well as the condition of zero transverse shear stresses at the top and bottom surfaces of the plates. The theory has the same number of unknown field variables as that of FSDT and the number of unknowns is layer independent which makes the solution computationally more efficient. A C0 continuous isoparametric serendipity element is employed to solve the discrete eigenvalue equations arising in both the problems. The accuracy and the efficiency of the theory are thus demonstrated through numerical experiments on the free vibration and stability analysis of laminated composite and sandwich plates with different modular ratio, aspect ratio, span to thickness ratio, loading and boundary conditions, ply orientations, lay-up number, eigen modes, etc. Higher modes with corresponding mode shapes of vibration and buckling are also represented for laminated composite and sandwich plates. An elegant performance of the proposed theory is observed when it is validated with relevant numerical examples.

A 3D shell-like approach using element-free Galerkin method for analysis of thin and thick plate structures

A new efficient meshless method based on the element-free Galerkin method is proposed to analyze the static deformation of thin and thick plate structures in this paper. Using the new 3D shell-like kinematics in analogy to the solid-shell concept of the finite element method, discretization is carried out by the nodes located on the upper and lower surfaces of the structures. The approximation of all unknown field variables is carried out by using the moving least squares (MLS) approximation scheme in the in-plane directions, while the linear interpolation is applied through the thickness direction. Thus, different boundary conditions are defined only using displacements and penalty method is used to enforce the essential boundary conditions. The constrained Galerkin weak form, which incorporates only displacement degrees of freedom (d.o.f.s), is derived. A modified 3D constitutive relationship is adopted in order to avoid or eliminate some self-locking effects. The numeric efficiency of the proposed meshless formulation is illustrated by the numeric examples.

Particular Solutions of a Two-Dimensional Infinite Wedge for Various Boundary Conditions With Weak Singularity

The particular solutions of a two-dimensional infinite wedge for various boundary conditions with lnr weak singularity have been investigated in this paper. The relations of the weak singularities and the discontinuities of the first kind of the boundary variables at a corner of a two-dimensional elastic body have been established. By using the relations, the singular behaviors of the unknown boundary variables at a corner of an elastic body can be obtained before solving the boundary value problem by using the boundary element method (BEM). Especially, if the boundary conditions at a corner are displacements prescribed, the values of the unknown tractions at the corner can be determined in advance. Thus, the difficulty related to the multivalued tractions at a corner in BEM analysis for problems with boundary displacements prescribed has been overcome completely. In addition, more appropriate shape functions for the unknown boundary field variables of a corner element can be constructed, and the accuracy of the BEM may be greatly increased.

A meshless local boundary integral equation method for two-dimensional steady elliptic problems

A novel meshless local boundary integral equation (LBIE) method is proposed for the numerical solution of two-dimensional steady elliptic problems, such as heat conduction, electrostatics or linear elasticity. The domain is discretized by a distribution of boundary and internal nodes. From this nodal points’ cloud a “background” mesh is created by a triangulation algorithm. A local form of the singular boundary integral equation of the conventional boundary elements method is adopted. Its local form is derived by considering a local domain of each node, comprising by the union of neighboring “background” triangles. Therefore, the boundary shape of this local domain is a polygonal closed line. A combination of interpolation schemes is taken into account. Interpolation of boundary unknown field variables is accomplished through boundary elements’ shape functions. On the other hand, the Radial Basis Point Interpolation Functions method is employed for interpolating the unknown interior fields. Essential boundary conditions are imposed directly due to the Kronecker delta-function property of the boundary elements’ interpolation functions. After the numerical evaluation of all boundary integrals, a banded stiffness matrix is constructed, as in the finite elements method. Several potential and elastostatic benchmark problems in two dimensions are solved numerically. The proposed meshless LBIE method is also compared with other numerical methods, in order to demonstrate its efficiency, accuracy and convergence.

Investigation of Single Pass Shape Rolling Using an Upper Bound Method

This paper presents a generalized analytical upper bound method for the study of external shape, pressure, and torque in the single pass rolling of shaped sections. Generalized kinematically admissible velocity fields have been calculated from an assumed deforming geometry, in turn mathematically developed from a new parameterization of curves for the stream line flow of the material. An upper bound on rolling power was established based on the calculated velocity fields. Unknown variables in the velocity field were determined by minimizing rolling power with respect to unknown velocity field variables, yielding an upper limit to the actual power required as well as rolling pressure and torque. As an applied example, the single oval-to-round and rectangle-to-diamond roll passes have been chosen and analyzed. Velocity fields and power relations were obtained for each pass and computer analysis was carried out to analyze and simulate the process of shape rolling. External shape, average roll torque, and rolling pressure data from the analysis were compared with other worker’s analytical solutions, numerical analyses, and experimental data. The results were found to be in good agreement with previous research and the method was shown to be quicker and easier to use.

Application of relations of singularity intensities of tangent derivatives of boundary displacements and tractions to BEM

In order to obtain a highly accurate numerical solution for a two-dimensional elasticity problem with singular boundary conditions on the smooth surface of an elastic body by boundary element method (BEM), the continuous or the singular requirements of the boundary field variables and their derivatives at element intersections have to be satisfied. The singularity intensities of the unknown boundary field variables have been determined through the theoretical relations of singularity intensities of tangent derivatives of boundary displacements and tractions, a priori. The continuous or singular requirements of boundary field variables and their derivatives at element intersections are automatically satisfied by using single node quadratic element (SNQE) developed in this paper. An example problem with singular boundary conditions on the surface of a semi-infinite-plane is numerically studied by BEM by using traditional three-nodes isoparametric elements and SNQEs. Numerical results show that the computation precisions at singular boundary points are greatly increased by using SNQEs.

Theoretical relations of boundary displacement derivatives and tractions at singular boundary point for 2D isotropic elastic problems

Although boundary displacement and traction are independent field variables in boundary conditions of an elasticity problem at a non-singular boundary point, there exist definite relations of singularity intensities between boundary displacement derivatives and tractions at a singular boundary point. The analytical forms of the relations at a singular smooth point for 2D isotropic elastic problems have been established in this work. By using the relations, positions of the singular boundary points and the corresponding singularity intensities of the unknown boundary field variables can be determined a priori. Therefore, more appropriate shape functions of the unknown boundary field variables in singular elements can be constructed. A numerical example shows that the accuracy of the BEM analysis using the developed theory is greatly increased.

A CG-FFT approach to the solution of a stress-velocity formulation of three-dimensional elastic scattering problems

In this paper we introduce a Conjugate Gradient Fast Fourier Transform (CG-FFT) scheme for the numerical solution of the integral equation formulating three-dimensional elastic scattering problems. The formulation is in terms of the stress tensor and particle velocities as the unknown field variables. In contrast with the formulation based on particle displacements, this approach leads to integral representations that do not involve derivatives of the unknown fields, thus resulting in simplified and more stable numerics. The numerical procedure is based on suitable quadrature formulas that provide (second order) accurate approximations while retaining the convolution nature of the relevant integrals that make them amenable to efficient evaluation via FFTs. The scheme is further improved through the introduction of (approximation-based) pre-conditioners that are shown to accelerate the convergence of the CG iterations. Numerical results are presented that demonstrate the accuracy and efficiency of the proposed methodology.

Stochastic modeling of coupled electromechanical interaction for uncertainty quantification in electrostatically actuated MEMS

This work proposes a stochastic framework based on generalized polynomial chaos (GPC), to handle uncertain coupled electromechanical interaction, arising from variations in material properties and geometrical parameters such as gap between the microstructures, applicable to the static analysis of electrostatic MEMS. The proposed framework comprises of two components – a stochastic mechanical analysis, which quantifies the uncertainty associated with the deformation of MEM structures due to the variations in material properties and/or applied traction, and a stochastic electrostatic analysis to quantify the uncertainty in the electrostatic pressure due to variations in geometrical parameters or uncertain deformation of the conductors. The stochastic analysis is based on a stochastic Lagrangian approach, where, in addition to uncertain input parameters and unknown field variables, the random deformed configuration is expanded in terms of GPC basis functions. The spectral modes for the unknown field variables are finally obtained using Galerkin projection in the space spanned by GPC basis functions. The stochastic mechanical and electrostatic analyses are performed in a self-consistent manner to obtain the random deformation of the MEM structures. Various numerical examples are presented to study the effect of uncertain parameters on performance of various MEMS devices. The results obtained using the proposed method are verified using rigorous Monte Carlo simulations. It has been shown that the proposed method accurately predicts the statistics and probability density functions of various relevant parameters.

On the interplay between the solid deformation and fluid flow during the solidification of a metallic alloy

During solidification in casting processes, metallic alloys undergo a continuous transition from liquid to solid by passing a semi-solid or mushy state. In this contribution, we present a continuum-mechanical model describing the deformation and solidification using the thermodynamically consistent theory of porous media (TPM). The model is based on two constituents, namely an elasto-viscoplastic solid skeleton and a viscous pore fluid. The balance equations are formulated on the macroscale and the exchange of mass, species, momentum and energy between the constituents is expressed. Between both constituents a local thermal equilibrium is assumed. The initial boundary value problem is solved by a numerical solution scheme based on mixed Galerkin finite elements. The velocity of the solid skeleton, the pressure of the pore fluid, the common temperature and the chemical compositions are taken as the primary unknown field variables.

The Simulation of Ship Motions Using a B-Spline–Based Panel Method in Time Domain

In this paper, a B-spline-based higher-order method is developed for simulating three-dimensional ship motions with forward speed. The problem is formulated in time domain using a transient free surface Green function. The body geometry is defined by open uniform or nonuniform B-spline basis functions depending on the hull type, whereas the unknown field variables are described by open uniform B-spline basis functions. The collocation method is applied to discretize the integral equation and then solved for the unknown potentials and source strengths. Motion computations in head waves are carried out for three types of ship hulls: a mathematically defined Wigley hull, a typical containership (S175 hull), and a Series 60 hull. Results are obtained for regular and irregular waves and compared with available experimental and computational results. It is found that the results from the present method are in very good agreement with the published results, and in particular with experimental data. Long-duration simulations have also been carried out with an ordinary desktop PC (PIV with 512 MB RAM) to demonstrate the ability of the method to simulate motions over long periods without any visible deterioration using only modest computational resources.

Parallel Meshless EFG Solution for Fluid Flow Problems

ABSTRACT This article deals with the parallel computing solution of fluid flow problems using a meshless element-free Galerkin (EFG) method. The EFG method utilizes moving least-square (MLS) approximants to approximate the unknown field variables. The MLS approximant consists of three components: a weight function, a basis function, and a set of nonconstant coefficients. A new parallel algorithm is proposed for the EFG method. The code has been written in FORTRAN language using MPI message passing library and implemented on a MIMD (multiple-instruction multiple-data)-type PARAM 10000 supercomputer. The code has been validated by solving three model fluid flow problems. A comparison is made among the results (velocities values) obtained by the EFG method with those obtained by the finite-element method (FEM) at a few typical locations. For 8 processors, speedup and efficiency have been obtained as 2.73 and 34.22% for N = 1,552 in 1-D (Example I), 7.20 and 90.00% for N = 1,462 in 2-D (Example II), and 7.30 and 91.27% for N = 1,346 in 2-D (Example III), respectively.

Equivalent single-layer theory for a complete stress field in sandwich panels under arbitrarily distributed loading

In single-layer theory, the displacement components represent the weighted-average through the thickness of the sandwich panel. Although discrete-layer theories are more representative of sandwich construction than the single-layer theories, they suffer from an excessive number of field variables in proportion to the number of layers. In this study, utilizing Reissner’s definitions for kinematics of thick plates, the displacement components at any point on the plate are approximated in terms of weighted-average quantities (displacements and rotations) that are functions of the in-plane coordinates. The equations of equilibrium and boundary conditions of the sandwich panel are derived by employing the principle of virtual displacements. The solution for an arbitrarily distributed load is obtained by employing Fourier series representations of the unknown field variables. This single-layer theory is validated against an analytical solution for simply supported square sandwich panels under pressure over a small region on the face sheet and is also compared with previous single-layer plate theories.

BEM‐computation of antenna near field reactions on conducting materials using curvilinear edge elements

A boundary element method in terms of the field variables is applied to three‐dimensional electromagnetic scattering problems. Especially, the influence of a dipole excited field on low conducting materials situated very close to the antenna will be discussed. We use higher order edge elements of quadilateral shape for the field approximation on curved surfaces. The tangential components of the unknown field variables are interpolated by vector element functions. The Galerkin method is implemented to obtain a set of linear equations. The applicability of the proposed edge element is investigated by the comparison of different BEM‐formulations and FEM‐results.

Application of curvilinear higher order edge elements to scattering problems using the boundary element method

A boundary element method in terms of the field variables is applied to three-dimensional electromagnetic scattering problems. We propose higher order edge elements of quadrilateral shape for the field approximation on curved surfaces. The tangential components of the unknown field variables are interpolated by the edge element. The Galerkin method is implemented to obtain a set of linear equations. The applicability of the proposed edge element is investigated by the comparison of numerical results with analytical solutions.

NON-LINEAR VIBRATION OF A HINGED ORTHOTROPIC CIRCULAR PLATE WITH A CONCENTRIC RIGID MASS

Non-linear free vibration of hinged orthotropic circular plates with a concentric rigid mass at the centre is studied by using the finite element method. Hamilton's principle is applied to derive the basic non-linear partial differential equations and associated boundary conditions for the problem of large amplitude of an orthotropic circular plate. The applications of the finite element method to the dynamic problem rely on the use of a variation principle to derive the necessary element property's equations. The assembled equations for the plate are formed by summing each of the element equations obtained in consideration of a single element. Then, the boundary conditions are imposed on the vector of nodal field variables, so that the appropriate boundary conditions are satisfied. The assembled equations form an eigenvalue problem and are solved for the unknown field variables. The relations between the fundamental frequencies and the amplitudes of non-linear vibrations of the circular orthotropic elastic plate with a rigid core are obtained. The results show that the frequency responses of the plate varies with changes of boundary conditions and the ratio between tangential and radial elastic constant.

A chebyshev collocation method for moving boundaries, heat transfer, and convection during directional solidification

Free and moving boundary problems require the simultaneous solution of unknown field variables and the boundaries of the domains on which these variables are defined. There are many technologically important processes that lead to moving boundary problems associated with fluid surfaces and solid-fluid boundaries. These include crystal growth, metal alloy and glass solidification, melting and name propagation. The directional solidification of semi-conductor crystals by the Bridgman-Stockbarger method is a typical example of such a complex process. A numerical model of this growth method must solve the appropriate heat, mass and momentum transfer equations and determine the location of the melt-solid interface. In this work, a Chebyshev pseudospectra collocation method is adapted to the problem of directional solidification. Implementation involves a solution algorithm that combines domain decomposition, finite-difference preconditioned conjugate minimum residual method and a Picard type iterative scheme.

Finite element analysis of non-linear vibration of a circular plate with a concentric rigid mass

A finite element analysis of non-linear vibration of an elastic circular plate with a concentric rigid mass is examined for different cases of boundary conditions. The non-linear partial differential equations and the associated boundary conditions are derived from an energy method, by using the Hamiltonian principle. The applications of the finite element method to the dynamic problem rely on the use of a variational principle to derive the necessary element properties or equations. The assembled equations for the plate are formed by summing each element equation obtained in consideration of a single element. Then, the boundary conditions are imposed on the vector of nodal field variables, so that the appropriate boundary conditions are satisfied. The assembled equations form an eigenvalue problem and are solved for the eigenvalues as well as the unknown field variables. The relations between the fundamental frequencies and the amplitudes of non-linear vibrations of the circular isotropic elastic plate with a concentric rigid mass are obtained. The results agree well with the solutions obtained by the Kantorovich averaging method. Also, the results yield the upper bounds of the solution to the stated problem.

Concept of general parametric elements and applications

General parametric elements adopt different functions for the expression of unknown field variables and the mapping of geometric configurations. By introducing the concept, paper shows that useful singularity and infinite elements can be produced in a unified manner. In the derivation of these especial elements, a technique is incorporated for imposing respective constraint conditions in the analysis. In the second part of paper, the technique is generalized so that a variety of constraint conditions are implemented in the standard finite element analysis routine at the level of element stiffness formations. It is pointed out that the joint or interface element is also an interesting application of the proposed method.