Conventional Finite Element Formulation Research Articles

A mixed finite element method for nonlinear radiation–conduction equations in optically thick anisotropic media

We propose a new mixed finite element formulation for solving radiation–conduction heat transfer in optically thick anisotropic media. At this optical regime, the integro-differential equations for radiative transfer can be replaced by the simplified PN approximations using an asymptotic analysis. The conductivity is assumed to be nonlinear depending on the temperature along with anisotropic absorption and scattering depending on both the direction and location variables. The simplified PN approximations are enhanced by considering a diffusion tensor capable of describing anisotropic radiative heat transfer. In the present study, we investigate the performance of the unified and mixed formulations combining cubic P3, quadratic P2, and linear P1 finite elements to approximate the temperature in the simplified P3 model. To demonstrate the performance of the proposed methodology, three-dimensional examples of nonlinear radiation–conduction equations in optically thick anisotropic media are presented. The obtained numerical results demonstrate the accuracy and efficiency of the proposed mixed finite element formulation over the conventional unified finite element formulation to accurately solve the simplified P3 equations in anisotropic media.

A mixed-mode energy-based elastoplastic fatigue induced damage model for the peridynamic theory

In recent years a considerable effort has been dedicated to the development of analysis tools that enable the design of damage tolerant structures, particularly in aerospace applications where weight reduction is a crucial requirement. One of these tools developed in recent years is the peridynamic theory, employed to solve numerically complex elastodynamics problems. One of its advantages reported in the literature is the natural ability to simulate the initiation and crack growth without the need for additional numerical procedures commonly employed in other numerical approaches, like in the conventional finite element formulation. Within this context, this paper presents a novel elastoplastic fatigue-induced damage model whose formulation is based on a strain energy framework combined with a smeared crack approach to simulate the damage process without the need of knowing the location of the crack tip and its length within the domain. The proposed model also can predict the mixed-mode damage propagation in ductile materials without knowing a priori the mixity mode ratio. This approach incorporates an analytical methodology based on the material properties that correlate the strain energy calculated away from the crack tip to the expected propagation rate predicted by the Paris Law. The accuracy of the proposed model is verified by comparing the results obtained using an in-house peridynamic FORTRAN code with the experimental results available in the open literature. Some improvements for the peridynamic material parameters are also presented in this paper, which is also verified by using the in-house code.

New non-intrusive stochastic finite element method for plate structures

Currently, several intrusive stochastic finite element methods (SFEMs) such as perturbation method and spectral SFEM are widely applied for stochastic response analysis of continuous structures. However, the intrusive SFEMs need to modify conventional finite element formulations to establish the stochastic stiffness matrix, and cannot calculate the probability density function of structural response and the reliability straightforwardly. This paper proposes a new non-intrusive SFEM for efficiently computing stochastic responses and reliabilities of plates in a unified way. Firstly, the direct probability integral method (DPIM) is developed to obtain the probability density function of stochastic response by solving probability density integral equation (PDIE). Secondly, the non-intrusive SFEM based on DPIM decouples the deterministic finite element analysis and PDIE to calculate the stochastic responses and reliabilities of uncertain plate structures, and the discretization and quantification of random fields of elastic modulus and thickness are implemented through Karhunen-Loève expansion. Finally, several examples of uncertain Kirchhoff and Mindlin plates demonstrate the efficiency and versatility of the proposed non-intrusive method by comparing with the results from Monte Carlo simulation and literature. The effects of correlation length, mean and variability of random field on the probability distribution of responses and the reliabilities of plates are revealed. For Gaussian random thickness, the linearly elastic plate yields non-Gaussian distributed responses. Increasing the correlation length and variability of random field reduces the reliabilities of plates.

An elastoplastic constitutive damage model based on peridynamics formulation

Peridynamic Theory based models allow simulating the initiation and growth of cracks in solid materials, without the aid of additional methods commonly employed in the conventional finite element formulation. Within this context, a new elastoplastic damage model is proposed to use with the Peridynamic Theory. This proposed damage model combines Von Mises plasticity-based theory with a smeared cracking approach enabling damage prediction within an energy-based framework. The formulation incorporates a mixed-mode propagation criterion to account for the effect of both axial and shear stresses in the simulation, which in turn allows prediction of damage progression in ductile materials under multiaxial loading without knowing a priori the mode mixity ratio. This proposed damage modeling approach can be used within any constitutive peridynamic model, by relying on the displacement field obtained in the simulation.

Analysis of Concrete Tensile Failure using Dynamic Particle Difference Method under High Loading Rates

This study presents the use of the dynamic particle difference method (PDM) to analyze tensile failure in concrete subjected to high loading rates. In general, strong form-based meshfree methods suffer from limitations pertaining to the material modeling of concrete because concrete exhibits both softening and damage behaviors that initiate crack growth under an impact load. These methods are generally based on the direct discretization of the governing equations, such as Navier's equation, which involves second-order differentiation. However, conventional material models are based on the first-order derivatives of displacement. The newly developed dynamic PDM can effectively address the limitations of material modeling using a combination of first-order derivative approximations. This circumvents the requirement for high-order derivative approximations, which are essential in strong formulations, such as the finite difference method and point collocation method. The strain rate effect caused by an extremely high loading speed was successfully modeled by accurately reflecting the energy dissipations that result from the cohesive property of the concrete and brittle cracking. Although the developed method incorporates the elastic constitutive relation, it enables the effective modeling of these nonlinear effects. In addition, it can reduce the computational effort. The proportional damping algorithm simulates the effect of velocity in the equation of motion and the cohesion effect in the concrete material. The damage model and the visibility criterion adequately handle crack initiation and propagation in the concrete member. Furthermore, it is noteworthy that the final discrete forms of the dynamic PDM are similar to the integrands of the weak form in the conventional finite element formulation. We ascertained that the stiffness and mass proportional damping effects are related to the inertia and strain of the material, respectively. It was confirmed that the location and direction of crack propagation in concrete varied with the strain rate. Hence, the accuracy and robustness of the proposed method were successfully verified by simulation, and the strain-rate dependency of concrete fracture was efficiently simulated using the proposed method.

Optimization of the electromagnetic scattering problem based on the topological derivative method.

A new optimization method based on the topological derivative concept is developed for the electromagnetic design problem. Essentially, the purpose of the topological derivative method is to measure the sensitivity of a given shape functional with respect to a singular domain perturbation, so that it has applications in many relevant fields such as shape and topology optimization for imaging processing, inverse problems, and design of metamaterials. The topological derivative is rigorously derived for the electromagnetic scattering problem and used as gradient descent direction to find local optima for the design of electromagnetic devices. We demonstrate that the resulting topology design algorithm is remarkably simple and efficient and naturally leads to binary designs, while depending only on the solution of the conventional finite element formulation for electrodynamics. Finally, several numerical experiments in two and three spatial dimensions are presented to illustrate the performance of the proposed formulation.

Modeling and analysis of the nonlinear indentation problems of functionally graded elastic layered solids

In tribological and thermo-electro-mechanical applications with an appropriate gradation of properties, functionally graded materials provide superior performance for damage and wear resistance of contact systems and thermal and mechanical responses for other mechanical devices. This paper presents a nonlinear mixed variational-based computational model for the analysis of nonlinear plane indentation problems of elastic functionally graded layered elastic solids. The functionally graded elastic layer is perfectly bonded to the elastic substrate and is normally loaded by a cylindrical rigid punch. In addition to nonlinearity inherent to the contact problem, the developed model accounts for the geometric nonlinearity in the sense of larger displacements and rotations, but smaller strains. The modulus of elasticity as well as Poisson's ratio of the graded layer are varying along the thickness of the layer according to both exponential and power laws. On the contrary, in the conventional homogeneous finite element formulation the isoparametric graded finite element formulation is adopted on the level of Gaussian integration points to realize the gradation in material properties. The friction at the contact interface is modeled using Coulomb's friction law. Moreover, the inequality contact constraints are exactly satisfied throughout the contact interface by employing the Lagrange multiplier method, where the indentation force and the displacement fields are treated as independent variables. The obtained results of contact pressure, tangential contact stress, stress distribution, and indentation force show a significant influence of the material distribution, gradation law, gradient index, and thickness of the functionally graded layer.

Numerical study on the thermal buckling analysis of CNT-reinforced composite plates with different shapes based on the higher-order shear deformation theory

In the present study, based on the higher-order shear deformation plate theory, the unified numerical formulation is developed in variational framework to investigate the thermal buckling of different shapes of functionally graded carbon nanotube reinforced composite (FG-CNTRC) plates. Since the thermal environment has considerable effects on the material properties of carbon nanotubes (CNTs), the temperature-dependent (TD) thermo-mechanical material properties are taken into account. In order to present the governing equations, the quadratic form of the energy functional of the plate structure is derived and its discretized counterparts are presented employing the variational differential quadrature (VDQ) approach. The discretized equations of motion are finally obtained based on Hamilton's principle. In order to convenient application of differential quadrature numerical operators in irregular physical domain, the mapping procedure is considered in accordance to the conventional finite element formulation. Some comparison and convergence studies are performed to show validity and efficiency of the proposed approach. A wide range of numerical results are also reported to analyze the thermal buckling behavior of different shaped FG-CNTRC plates.

Investigation of a hybrid polygonal finite element formulation for confined and unconfined seepage

SummaryThis study presents a formulation for field problems using hybrid polygonal finite elements, taking steady state seepage through a porous material as the focus. We make comparisons with a conventional finite element formulation based on a single primary variable, focussing on the advantages of the hybrid formulation in terms of flux field accuracy and extension to convex polygonal shaped elements. For the unconfined case, we adopt a head dependent hydraulic conductivity that does not require remeshing. The performance of the hybrid polygonal element formulation is demonstrated through a series of numerical examples. The results show a sensitivity of the location of the free surface in unconfined seepage to mesh configuration for hybrid quadrilateral meshes with various aspect ratios, but not for hybrid polygonal meshes with various orientations and irregularity. Examination of the free surface location results for several conforming shape function options shows an insensitivity to choice of interpolation function, provided that it conforms with the assumptions in the formulation. Copyright © 2016 John Wiley & Sons, Ltd.

Dynamic modeling of multi-flexible-link planar manipulators using curvature–based finite element method

In conventional displacement-based finite element formulation, the nodal transverse deflections and slopes of a beam element are used to approximate its transverse deflection distribution. In this paper, a curvature-based finite element formulation is presented for dynamic modeling of planar multi-link flexible manipulators. This formulation approximates the curvature distribution of the Euler-Bernoulli beam element by using its nodal curvatures. Therefore, in the curvature-based finite element formulation, fewer numbers of total elastic degrees of freedom for the system are needed. In addition, while the displacement-based finite element formulation yields discontinuous bending stresses at the inter-element nodes, this modeling provides continuous bending stress at those nodes. The proposed modeling will be verified and the dynamic behavior of the flexible planar manipulators will be studied.

A large deformation, rotation-free, isogeometric shell

Conventional finite shell element formulations use rotational degrees of freedom to describe the motion of the fiber in the Reissner–Mindlin shear deformable shell theory, resulting in an element with five or six degrees of freedom per node. These additional degrees of freedom are frequently the source of convergence difficulties in implicit structural analyses, and, unless the rotational inertias are scaled, control the time step size in explicit analyses. Structural formulations that are based on only the translational degrees of freedom are therefore attractive. Although rotation-free formulations using C 0 basis functions are possible, they are complicated in comparison to their C 1 counterparts. A C k -continuous, k ⩾ 1, NURBS-based isogeometric shell for large deformations formulated without rotational degrees of freedom is presented here. The effect of different choices for defining the shell normal vector is demonstrated using a simple eigenvalue problem, and a simple lifting operator is shown to provide the most accurate solution. Higher order elements are commonly regarded as inefficient for large deformation analyses, but a traditional shell benchmark problem demonstrates the contrary for isogeometric analysis. The rapid convergence of the quadratic element is demonstrated for the NUMISHEET S-rail benchmark metal stamping problem.

Two Simple Triangular Plate Elements Based on the Absolute Nodal Coordinate Formulation

In this paper, two triangular plate elements based on the absolute nodal coordinate formulation (ANCF) are introduced. Triangular elements employ the Kirchhoff plate theory and can, accordingly, be used in thin plate problems. As usual in ANCF, the introduced elements can exactly describe arbitrary rigid body motion when their mass matrices are constant. Previous plate developments in the absolute nodal coordinate formulation have focused on rectangular elements that are difficult to use when arbitrary meshes need to be described. The elements introduced in this study have overcome this problem and represent an important addition to the absolute nodal coordinate formulation. The two elements introduced are based on Specht’s and Morley’s shape functions, previously used in conventional finite element formulations. The numerical solutions of these elements are compared with previously proposed rectangular finite element and analytical results.

Wavelet spectral element for wave propagation studies in pressure loaded axisymmetric cylinders

We study transient wave propagation in a pressure loaded isotropic cylinder under axisymmetric conditions. A 2-D wavelet based spectral finite element (WSFE) is developed to model the cylinder with radial and axial displacements. The method involves a Daubechies compactly supported scaling function approximation in the temporal dimension and one spatial (axial direction) dimension. This reduces the governing partial differential wave equation into a set of variable coefficient ODEs, which are then solved using Bessel’s function approximation. This spectral method captures the exact inertial distribution and thus results in large computational savings compared to the conventional finite element (FE) formulation. In addition, the use of localized basis functions in the present formulation circumvents several serious limitations of the previous FFT based techniques. Here, the proposed method is used to study radial and axial wave propagation in cylinders with different configurations. The analysis is performed in both time and frequency domains. The time domain responses are validated with 2-D FE results.

On the equivalence of various hybrid finite elements and a new orthogonalization method for explicit element stiffness formulation

Hybrid finite element methods are superior to conventional displacement-based finite element formulations in many aspects and to date many types of hybrid elements have been proposed based on different assumed stress fields. However, some of them are essentially equivalent to each other and there is no general approach available yet to determine their corresponding relationships. In addition, even though the hybrid techniques have many advantages compared to the displacement-based elements, there is an obvious disadvantage, the additional computational cost of the matrix inversions required in constructing element stiffness matrix. In this paper, we show that a linearly independent transformation of the assumed stress field for a given element leads to an equivalent hybrid finite element. This is verified through the proof of equivalence between various existing hybrid elements proposed by different researchers. To improve the computational efficiency, a new inner product including the compliant matrix of material is introduced. With this new definition of inner product, a simplified orthogonalization procedure for the assumed stress modes is then proposed. Based on the orthonorm stress modes obtained using the proposed orthogonalization, an explicit formulation of the hybrid element stiffness matrix is presented. The effectiveness of the present method is validated by numerical examples.

Adaptive force density method for form-finding problem of tensegrity structures

A numerical method is presented for form-finding of tensegrity structures. Eigenvalue analysis and spectral decomposition are carried out iteratively to find the feasible set of force densities that satisfies the requirement on rank deficiency of the equilibrium matrix with respect to the nodal coordinates. The equilibrium matrix is shown to correspond to the geometrical stiffness matrix in the conventional finite element formulation. A unique and non-degenerate configuration of the structure can then be obtained by specifying an independent set of nodal coordinates. A simple explanation is given for the required rank deficiency of the equilibrium matrix that leads to a non-degenerate structure. Several numerical examples are presented to illustrate the robustness as well as the strong ability of searching new configurations of the proposed method.

Vibration analysis of tapered composite beams using a higher-order finite element. Part I: Formulation

Tapered composite beams are increasingly being used in various engineering applications such as helicopter yoke, robot arms and turbine blades. The objective of the present work is to conduct an investigation of the free undamped vibration response of such tapered composite beams, using the finite element method. Conventional cubic Hermitian finite element formulation requires a large number of elements to obtain reasonably accurate results in the analysis of tapered laminated beams. Since the continuity of curvature at element interfaces cannot be guaranteed with the use of conventional formulation, the stress distribution across the thickness is not continuous at element interfaces. The material and geometric discontinuities at ply drop-off locations leads to additional discontinuities in stress distributions. As a result, efficient and accurate calculation of natural frequencies becomes very difficult. In order to overcome these limitations, a higher-order finite element formulation is developed in Part I of the present work. The stiffness coefficients of the tapered laminated beam are determined based on the stress and strain transformations and classical laminate theory. In Part II of the present work, the developed formulation is used for the free undamped vibration analysis of various types of tapered composite beams and a parametric study is conducted.

Finite element analysis of the geometric stiffening effect. Part 1: A correction in the floating frame of reference formulation

The fact that incorrect unstable solutions are obtained for linearly elastic models motivates the analytical study presented in this paper. The increase in the number of finite elements only leads to an increase in the critical speed. Crucial in the analysis presented in this paper is the fact that the mass matrix and the form of the elastic forces obtained using the absolute nodal coordinate formulation remain the same under orthogonal coordinate transformation. The absolute nodal coordinate formulation, in contrast to conventional finite element formulations, does account for the effect of the coupling between bending and extension. Based on the analytical results obtained using the absolute nodal coordinate formulation, a new correction is proposed for the finite element floating frame of reference formulation in order to introduce coupling between the axial and bending displacements. In this two-part paper, two- and three-dimensional finite element models are used to study the problem of rotating beams. The models are developed using the absolute nodal coordinate formulation that allows for accurate representation of the axial strain, thereby avoiding the ill-conditioning problem that arises when classical displacement-based finite element formulations are used. In the first part of the paper, the case of linear elasticity is considered and assumptions used in the finite element floating frame of reference formulation are investigated. In the second part of the paper, non-linear elasticity is considered. A rotating helicopter blade is simulated, and the complexity of the motion suggests the inclusion of rotary inertia, shear deformation, and non-linear elastic forces in order to obtain an accurate solution that does not suffer from the instability problem regardless of the number of finite elements used.

Integration of geometric design and mechanical analysis using B‐spline functions on surface

AbstractB‐spline finite element method which integrates geometric design and mechanical analysis of shell structures is presented. To link geometric design and analysis modules completely, the non‐periodic cubic B‐spline functions are used for the description of geometry and for the displacement interpolation function in the formulation of an isoparametric B‐spline finite element. Non‐periodic B‐spline functions satisfy Kronecker delta properties at the boundaries of domain intervals and allow the handling of the boundary conditions in a conventional finite element formulation. In addition, in this interpolation, interior supports such as nodes can be introduced in a conventional finite element formulation. In the formulation of the mechanical analysis of shells, a general tensor‐based shell element with geometrically exact surface representation is employed. In addition, assumed natural strain fields are proposed to alleviate the locking problems. Various numerical examples are provided to assess the performance of the present B‐spline finite element. Copyright © 2005 John Wiley & Sons, Ltd.

Assumed strain finite strip method using the non-periodic B-spline

An assumed strain finite strip method(FSM) using the non-periodic B-spline for a shell is presented. In the present method, the shape function based on the non-periodic B-splines satisfies the Kronecker delta properties at the boundaries and allows to introduce interior supports in much the same way as in a conventional finite element formulation. In the formulation for a shell, the geometry of the shell is defined by non-periodic B3-splines without any tangential vectors at the ends and the penalty function method is used to incorporate the drilling degrees of freedom. In this study, new assumed strain fields using the non-periodic B-spline function are proposed to overcome the locking problems. The strip formulated in this way does not posses any spurious zero energy modes. The versatility and accuracy of the new approach are demonstrated through a series of numerical examples.

A DISCONTINUOUS FINITE-ELEMENT FORMULATION FOR MULTIDIMENSIONAL RADIATIVE TRANSFER IN ABSORBING, EMITTING, AND SCATTERING MEDIA

This article presents a discontinuous finite-element formulation for the numerical solution of internal thermal radiation problems in multidimensional domains. In contrast with the conventional finite-element formulation, the discontinuous finite element is based on the idea of allowing the discontinuity of field variables across the internal interelement boundaries and thus is particularly useful for the integral-differential equations governing the thermal radiative transfer phenomena in emitting, absorbing, and scattering media. Mathematical formulation and numerical details using the discontinuous finite element method for internal radiation heat transfer calculations are given. Computational procedures are presented. A parallel computing algorithm exploiting the advantages of the localized discontinuous finite-element formulation and space/solid-angle discretization characteristics is presented. The computed results are given and compared with analytical solutions whenever available as reported in references. Examples include both nonscattering and scattering cases. Parallel computing performance is also evaluated for both 2-D and 3-D cases and results show that the parallel algorithm can be easily implemented within the framework of the discontinuous finite-element method. For the cases tested, the parallel scheme based on the solid-angle discretization gives a reduction in CPU time approaching to almost an idealized parallel CPU scenario.