Finite Element Matrix Equations Research Articles

Multiple heat sources with multi-parameter inversion of nondestructive infrared detection

This study deals with the case of multiple internal heat source inversion problem of steady-state based on nondestructive infrared detection. We construct homogeneous and heterogeneous steady heat conduction models of different shapes. Neither the number of heat sources, nor their locations, nor their sizes nor their intensities are known. We use the finite element method (FEM) based on numerical algorithm to analyze the two-dimensional model discretely. The internal heat conduction process of model is analyzed. The resultant temperature field can be decomposed into the temperature field caused by the ambient temperature and those given by internal heat sources. We simplify the finite element matrix equation according to decomposition process above. Finally, the problem boils down to solving the highly underdetermined matrix equation of Ax = b. The unknown x item corresponds to internal thermal heat source field. The piecewise polynomial spectral truncated singular value decomposition (PPTSVD) is applied for the first time to the inverse heat source problem. Its regular operator matrix is changed from the original more order differential operator matrix to regional node weighted matrix. After replacement this solution improves the effect of heat source field tending to the boundary. Results of the solution confirms a real heat source field when there are less heat sources or different heat sources are far from each other. But there also exits a serious superimposed effect between neighboring heat sources. We improve the algorithm to study this problem through using the iterative elimination process which complies with the idea of spreading heat source field and then gathering. The iteration tolerance and number of times belonging to one single PPTSVD solving process are reduced. Through iterating the multiple PPTSVD solving process and reconstructing matrixes A and b in each iteration, we obtain the scatter heat source field distribution surrounding real field. Finally, this scattered distribution solution is gathered again. According to the heat source parameter calculated by algorithm, the temperature field of whole model can be reconstructed using FEM. Comsol numerical simulation and real physical experiments are performed to verify the validity and accuracy of the algorithm in different heat conduction models. The results demonstrate that the algorithm can access each parameter of multiple heat sources. Even in the heterogeneous model, it can still obtain accurate results and reconstruct the temperature field of the two-dimensional model. The algorithm can be applied to non-destructive material detection and human infrared medical imaging fields.

On a Wavenumber Optimized Streamline Upwinding Method for Solving Steady Incompressible Navier-Stokes Equations

A stabilized upwind finite-element model is developed to solve the three-dimensional incompressible steady Navier-Stokes equations. The test function is constructed to have a larger weight on the upstream side. This has been achieved by adding a stabilized term to the shape function so as to optimize the numerical wavenumber for convection terms. To avoid Lanczos or pivoting breakdown while solving the resulting unsymmetric and indefinite mixed finite-element matrix equations iteratively, the finite-element equation has been modified by pre-multiplying it with its transpose. The resulting normalized matrix equation becomes symmetric and positive-definite. We can therefore apply a computationally efficient conjugate gradient Krylov iterative solver to get an unconditionally convergent solution. However, the condition number of the new system becomes the square of the original unsymmetric indefinite system. To fully exploit excellent convergence nature of the conjugate gradient iterative solver, an element-by-element strategy is adopted to avoid assembling of all the stiffness matrices obtained at element level. We alleviate the drawback of slower convergence of the conjugate gradient method due to the increased condition number by preconditioning the positive-definite matrix. The resulting preconditioned matrix equation is solved in a matrix-free manner using the preconditioned conjugate gradient iterative solver. The developed finite-element code is first verified by solving a problem amenable to analytical solution. The benchmark lid-driven cavity problem is also solved in a cube for assessing the three chosen iterative solvers.

The Acoustic-Structural Coupling Analysis of Muffler

This paper discusses the acoustic-structural coupling analysis of muffler in which the method of the finite element is used to solve the differential equations. Firstly, according to the symmetry of muffler, by using coordinate transformation the sound field problem is converted from three-dimensional to two-dimensional problem. Secondly, by using Galerkin method derives finite element equations of discrete plane sound field unit. Then the cell boundary conditions and residuals are analyzed which included acoustic-structural coupling border. The unit matrix equations are set to plane acoustic finite element matrix equation. Finally, the design of the finite element method coupled equations of solid sound is designed. A theoretical foundation is established for muffler acoustic-structural coupling characteristic analysis.

Perfectly matched layer-absorbing boundary condition for finite-element time-domain modeling of elastic wave equations

The perfectly matched layer (PML) is a highly efficient absorbing boundary condition used for the numerical modeling of seismic wave equation. The article focuses on the application of this technique to finite-element time-domain numerical modeling of elastic wave equation. However, the finite-element time-domain scheme is based on the second-order wave equation in displacement formulation. Thus, the first-order PML in velocity-stress formulation cannot be directly applied to this scheme. In this article, we derive the finite-element matrix equations of second-order PML in displacement formulation, and accomplish the implementation of PML in finite-element time-domain modeling of elastic wave equation. The PML has an approximate zero reflection coefficients for bulk and surface waves in the finite-element modeling of P-SV and SH wave propagation in the 2D homogeneous elastic media. The numerical experiments using a two-layer model with irregular topography validate the efficiency of PML in the modeling of seismic wave propagation in geological models with complex structures and heterogeneous media.

Full-Wave Real-Life 3-D Package Signal Integrity Analysis Using Nonconformal Domain Decomposition Method

Advances in interconnect technologies, such as the increase of the number of metal layers and 3-D stacking technique, have paved the way for higher functionality and superior performance while reducing size, power, and cost in today's integrated circuit and package products. However, whether or not the package preserves signal integrity (SI) has become a crucial concern for system designers. In this study, a systematic full-wave numerical approach, based on a nonconformal finite-element domain decomposition method (DDM), is proposed for 3-D real-life circuit/package simulations. First, an automatic domain partitioning strategy is utilized to divide the entire model into a number of sub-domains. Each sub-domain is then meshed independently and an <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h</i> -version of adaptive mesh refinement is employed. Next, a nonoverlapping DDM is adopted to efficiently solve the finite-element matrix equation. Afterwards, a model-order reduction technique is exploited to compute the multiport spectral responses. SI effects such as signal delay, coupling, and reflection are simulated on a product-level package benchmark. Finally, numerical results verify the analysis and demonstrate the effectiveness of the proposed method.

U-transformation-finite element method in 3-dimensional analysis of orthotropic laminates

For an orthotropic laminate, an equivalent system with doubly cyclic periodicity is introduced. Then a 3-dimensional finite element model for the equivalent system is transformed into the unitary space, where the large finite element matrix equation is decoupled into some small matrix equations. Such a decoupling very efficiently reduces the computational effort. For an orthotropic laminate with four clamped edges, no exact elasticity solution is available, and the deflection values predicted by different methods have a considerable difference each other for a small length-to-thickness ratio. The present predictions are the largest because the present method is a full 3-dimensional finite element analysis without superfluous constraints. Illustrative numerical examples are presented to observe the distributions of stresses through the thickness of the laminates. (C) 2010 Elsevier Ltd. All rights reserved.

Fractal finite element method based shape sensitivity analysis of multiple crack system

This paper presents fractal finite element based continuum shape sensitivity analysis for a multiple crack system in a homogeneous, isotropic, and two dimensional linear-elastic body subjected to mixed-mode (modes I and II) loading conditions. The salient feature of this method is that the stress intensity factors and their derivatives for the multiple crack system can be obtained efficiently since it only requires an evaluation of the same set of fractal finite element matrix equations with a different fictitious load. Three numerical examples are presented to calculate the first-order derivative of the stress intensity factors or energy release rates.

Simulations of interactions between nonlinear waves and multi or an array of cylinders

Computational methods are developed to simulate interactions of nonlinear waves with multi-structures through the finite element method based on second order and fully nonlinear theories. The three dimensional (3D) mesh with prism elements is generated through an extension of a two dimensional (2D) unstructured grid. The potential and velocity in the fluid field are obtained by solving finite element matrix equations at each time step using the conjugate gradient method with SSOR preconditioner. The combined Sommerfeld-Orlanski radiation condition and the damping zone method is used to minimize wave reflection. The regridding and smoothing techniques are employed to improve the stability of the solution and the accuracy of the result. The method is first used to simulate interactions of waves and an array of cylinders in the time domain based on the second order theory. Numerical simulations show that the influence of mutual interference between cylinders is highly significant. In particular, the first order and the second order results can become quite large when their corresponding wave number is close to the trapped mode. Simulations based on the fully nonlinear theory are also made for the 3D interactions between fixed multi-cylinders and waves generated by a piston type wave maker in a numerical tank with an artificial beach. Extensive results of practical importance have been obtained, which have been overlooked in many previous applications. The developed method is further employed to solve the 3D fully nonlinear radiation problems by multi-cylinders undergoing large amplitude oscillations in the open sea. All these different applications have clearly demonstrated the flexibility of the method. The simulations also show that the developed method is highly efficient and has great potential to be used for large scale calculation in the motions of floating structures in nonlinear waves.

Dynamics of porous media at finite strain

We present a finite element model for the analysis of a mechanical phenomenon involving dynamic expulsion of fluids from a fully saturated porous solid matrix in the regime of large deformation. Momentum and mass conservation laws are written in Lagrangian form by a pull-back from the current configuration to the reference configuration following the solid matrix motion. A complete formulation based on the motion of the solid and fluid phases is first presented; then approximations are made with respect to the material time derivative of the relative flow velocity vector to arrive at a so-called ( v , p)-formulation, which is subsequently implemented in a finite element model. We show how the resulting finite element matrix equations can be consistently linearized, using a compressible neo-Hookean hyperelastic material with a Kelvin solid viscous enhancement for the solid matrix as a test function for the nonlinear constitutive model. Numerical examples are presented demonstrating the significance of large deformation effects on the transient dynamic responses of porous structures, as well as the strong convergence profile exhibited by the iterative algorithm.

A multi‐level frontal algorithm for finite element analysis and its implementation on parallel computation

This paper presents a multi‐level frontal algorithm and its implementation and applications on parallel computation. A multi‐frontal program is given which may be used for unsymmetric finite element matrix equations. The parallel program is developed on a cluster of workstations. The PVM (parallel virtual machine) system is used to handle communications among networked workstations. The method has advantages such as numbering of the finite element mesh in an arbitrary manner, simple programming organisation, smaller core requirements and computation times. An implementation of this parallel method on workstations is discussed, the speedup and efficiency of this method being demonstrated and compared with general domain decomposition method based on band matrix methods by numerical examples.

Application of an element-by-element BiCGSTAB iterative solver to a monotonic finite element model

This paper deals with the advection-diffusion equation in adaptive meshes. The main feature of the present finite element model is the use of Legendre-polynomials to span finite element spaces. The success that this model gives good resolutions to solutions in regions of boundary and interior layers lies in the use of M-matrix theory. In the monotonic range of Peclet numbers, the Petrov-Galerkin method performs well in the sense that oscillatory solutions are not present in the flow. With proper stabilization, finite element matrix equations can be iteratively solved by the Lanczos method, used concurrently with local minimization provided by GMRES(1). The resulting BiCGSTAB iterative solver, supplemented with the Jacobi preconditioner, is implemented in an element-by-element fashion. This gives solutions which are computationally feasible for large-scale flow simulations. The results of two computations are presented in support of the ability of the present finite element model to resolve sharp gradients in the solution. As is apparent from this study is that considerable savings in computer storage and execution time are achieved in adaptive meshes through use of the preconditioned BiCGSTAB iterative solver.

Numerical analysis of thickness shear thin film piezoelectric resonators using a laminated plate theory

Finite element matrix equations are derived from a two-dimensional, piezoelectric high frequency laminated plate theory. Two-layer ZnO/Si and three-layer ZnO/SiO/sub 2//Si thin film resonators vibrating at the fundamental thickness shear mode are investigated using the finite element models. Straight crested waves propagating in a resonator are studied. Resonant frequency, mode shapes and electromechanical coupling coefficient of the fundamental thickness shear mode are calculated and presented with respect to the C-axis orientation of the ZnO layer, and thickness ratios of the layers in a resonator. The results from the laminated plate model are checked against the three dimensional finite element solution and found to be accurate for the fundamental thickness shear and its anharmonic overtones. The plate model does not yield as accurate a frequency spectrum for flexure, face shear and extensional modes. >

Numerical algorithms and results for SC-cut quartz plates vibrating at the third harmonic overtone of thickness shear

Finite element matrix equations, derived from two-dimensional piezoelectric high frequency plate theory are solved to study the vibrational behavior of the third overtone of thickness shear in square and circular SC-cut quartz resonators. The mass-loading and electric effects of electrodes are included. A perturbation method which reduces the memory requirements and computational time significantly is employed to calculate the piezoelectric resonant frequencies. A new storage scheme is introduced which reduces memory requirements for mass matrix by about 90% over that of the envelope storage scheme. Substructure techniques are used in eigenvalue calculation to save storage. Resonant frequency and the mode shapes of the harmonic third overtone thickness shear vibrations for square and circular plates are calculated. A predominant third overtone thickness shear displacement, coupled with the third overtone of thickness stretch and thickness twist, is observed. Weak coupling between the third order thickness shear displacement and the zeroth-, first-, and second-order displacements is noted. The magnitudes of the lower order displacements are found to be about two orders smaller than that of the third overtone thickness shear displacement.

A perturbation method for finite element modeling of piezoelectric vibrations in quartz plate resonators

When the piezoelectric stiffening matrix is added to the mechanical stiffness matrix of a finite element model, its sparse matrix structure is destroyed. A direct consequence of this loss in sparseness is a significant rise in memory and computational time requirements for the model. For weakly coupled piezoelectric materials, the matrix sparseness can be retained by a perturbation method which separates the mechanical eigenvalue solution from its piezoelectric effects. Using this approach, a perturbation and finite element scheme for weakly coupled piezoelectric vibrations in quartz plate resonators has been developed. Finite-element matrix equations were derived specifically for third-overtone thickness-shear, SC-cut quartz plate resonators with electrode platings. High-frequency piezoelectric plate equations were employed in the formulation of the finite element matrix equation. Results from the perturbation method compared well with the direct solution of the piezoelectric finite element equations. This method will result in significant savings in the computer memory and computational time. Resonance and antiresonance frequencies of a certain mode could be calculated easily by using the same eigenpair from the purely mechanical stiffness matrix. Numerical results for straight crested waves in a third overtone SC-cut quartz strip with and without electrodes are presented. The steady-state response to an electrical excitation is calculated.

A fixed grid finite element technique for modelling phase change in steady‐state conduction–advection problems

AbstractA new fixed grid finite element technique for modelling phase‐change effects in steady‐state conduction–advection problems is described. The method is appropriate for materials that transform at a fixed critical temperature and is based on the assumption that the enthalpy experiences a jump discontinuity at this temperature, yielding a singular specific heat. This singularity is described using a Dirac delta function and the appropriate finite element matrix equations are developed for isoparametric elements. A simple first order upwinding scheme is used to avoid numerical oscillations in advection dominated problems. The utility of the method is illustrated through the solution of a one‐dimensional test problem using eight‐node hexahedral isoparametric finite elements. The technique may be applied to a wide range of finite element types and offers a number of advantages over other schemes for modelling phase‐change effects.

Development of a new finite element formulation for the elasto-plastic analysis of fiber-reinforced composites

An analysis model is presented for the elasto-plastic analysis of continuous fiber-reinforced composite structures. The model uses the material properties of each constituent separately instead of the smeared global properties obtained from the lamination theory. Both fiber and matrix stresses are computed using the analysis model. Thus, the elasto-plastic deformation of a metal-matrix composite, composed of elastic fibers and elasto-plastic matrix, can be analyzed using the yield function and the flow rule suitable for each constituent. If each constituent is homogeneous and isotropic, then the usual von Mises yield function and the Prandtl-Reuss flow rule can be used. There is no need to obtain the global elasto-plastic material properties using the rule of mixtures. The formulation for the finite element matrix equation is given for the analysis model. Some numerical examples are solved using the method.

Characteristics of a Lagrangian, high-frequency plate element for the static temperature behavior of low-frequency quartz resonators

Finite-element matrix equations based on the Lagrangian, first-order, incremental plate equations of motion superposed on homogeneous thermal strains were formulated using virtual work principles. A program for an isoparametric, four-node quadrilateral element was written and applied to the study of the frequency-temperature (FT) behavior of flexure-mode quartz resonators. The lumped-mass and consistent-mass matrices were found to yield practically the same FT curves. For simple prismatic resonators, two schemes, reduced/selective integration and incompatible modes, produced relatively similar FT curves. The incompatible modes scheme yielded better results for resonators of more complex shapes, such as the tuning fork. It is concluded that the six-degree-of-freedom per node element is needed for the analysis of the FT behavior of a fully anisotropic flexure-mode resonator.

A block preconditioned conjugate gradient method for solving high-order finite element matrix equations

The Preconditioned Conjugate Gradient algorithms (PCG) are used for solving the matrix equations arising from the Finite Element Method (FEM) with high-order element functions. A vectorizable and a non-vectorizable block preconditioner for use with the conjugate gradient method is presented. The algorithms are tested on the Poisson equation in prolate spheriodal coordinates using the FEM with rectangular elements, and with element functions up to sixth order Lagrange interpolation polynomials.

Relationships between the average offset and the stress drop for two- and three-dimensional faults

Abstract Finite-element results for C , the nondimensional shape factor which relates stress drop to fault offset, are presented for rectangular strike-slip and two dimensional strike-slip and dip-slip faults at various depths of burial. All faults were oriented vertically except for a two-dimensional fault which was dipped at 10°. The finite-element meshes were constructed to limit the error in C to 1 per cent, as demonstrated by comparison with the analytical solution for a circular fault in an infinite region. Attainment of such accuracy results in large, finite-element matrix equations which were solved by a fast iterative technique: the multi-grid method. For the rectangular faults, significant differences were seen in values of C between the finite-element results and some previously reported solutions which were obtained by a different numerical technique.

An algorithm for multipoint constraints in finite element analysis

AbstractA method of introducing general constraint equations into finite element matrix equations is described. The characteristics of the method are that it requires no reordering or condensation of the equations, no large matrix operations, and no increase in the number of unknowns. The method is suitable for application in minicomputer implementations of finite element analysis unless a large number of constraints is to be applied.