Finite Element Stiffness Matrices Research Articles

Application of the Double Approximation Method for Constructing Stiffness Matrices of Volumetric Finite Elements

Introduction. When numerically solving problems of elasticity theory in a three-dimensional formulation by the finite element method, finite elements (FE) in the form of parallelepipeds, prisms and tetrahedra are used. Regularly, the construction of stiffness matrices of volumetric FE is based on the principle of isoparametricity, which involves the Lagrange polynomials to approximate the geometry and displacements. In computational practice, the most widespread FE are the so-called multilinear isoparametric FE with a linear law of approximation of displacements. The main disadvantage of these elements lies in the “locking” effect when modulating bending deformations. Moreover, the error of the numerical solution increases drastically in the case when the structure, in comparison to conventional deformations, undergoes significant displacements as a rigid whole. Long-term experience in solving problems of deformable solid mechanics by the finite element method has shown that existing volumetric FE have slow convergence, specifically, when modeling bending deformations of plates and shells. This study aims at constructing stiffness matrices of multilinear volumetric FE of increased accuracy allowing for rigid displacements based on the double approximation method.Materials and Methods. The mathematical apparatus of the double approximation method based on the principle of a separate representation of the distribution functions of displacements and deformations inside the element, was used to construct the stiffness matrices of volumetric FE. The storage and processing of the resulting system of equations was implemented in algorithmic terms of sparse matrices. Software development and computational experiments were carried out using the Microsoft Visual Studio 2013 64-bit computing platform and the Intel ® Parallel Studio XE 2019 compiler with the integrated Intel ® Visual Fortran Composer XE 2019 text editor. Visualization of the calculation results was performed using the descriptor graphics of the MATLAB computer mathematics package. A large eight-node SOLID185 CE of the ANSYS Mechanical software complex was used as a test sample.Results. Mathematical tool and software were developed to study the stress-strain state of massive structures under various types of external actions. The authorized application software package was verified on test examples with known analytical solutions. It has been shown that the constructed FE accurately satisfy the basic requirements for finite element modeling of spatial problems of elasticity theory.Discussion and Conclusion. The performed testing of the developed mathematical and program toolkit has shown that the finite elements constructed on the basis of the double approximation method can successfully compete with similar SOLID185 volumetric elements of the ANSYS Mechanical software complex. The proposed elements can be integrated into domestic import-substituting software systems that implement the finite element method in the form of the displacement method.

Thermal vibration analysis of cracked nanobeams submerged in elastic foundations by nonlocal continuum mechanics

Understanding the mechanical behavior of nanoscale structures is crucial in the development of advanced nanotechnologies. In this study, a novel approach to investigate the thermal lateral vibration of cracked nanobeams immersed in an elastic matrix is investigated. For this purpose, Reddy’s third-order shear deformation theory (TSDT) is considered. In contrast to Timoshenko beam theory (First-order Shear Deformation Theory, FSDT), TSDT does not depend on a shear correction coefficient. The nano-scale effect is modeled using Eringen’s nonlocal continuum mechanics theory. The nonlocal form of the governing equation is obtained through the application of Hamilton’s principle. The weak form of the finite element global mass and stiffness matrices are obtained using Lagrange linear and Hermitian cubic interpolation. To model the crack in bending vibration, two rotational springs are used for TSDT, unlike the use of a single rotational spring in traditional Bernoulli–Euler (Classical Beam Theory, CBT) and FSDT. The stiffness of the springs is adjusted based on the severity of the crack. The influences of the nonlocal parameter, beam slenderness, position of crack, crack severity, Pasternak and Winkler foundation parameters, thermal effects and boundary conditions on the natural frequencies are investigated. The model’s outcomes are compared with findings from prior publications, demonstrating a strong level of agreement. This study contributes to the growing research on nanostructures by presenting a novel approach to understanding the dynamics of cracked nanobeams using Reddy beam analysis-based solutions.

Model order reduction of time-domain vibro-acoustic finite element simulations with non-locally reacting absorbers

Time-domain vibro-acoustic finite element simulation gained considerable interest in recent years because of the rise of some applications such as auralization and virtual sensing. Efficiency and stability are of great importance for this topic. The number of elements needed per wavelength to reach acceptable accuracy often results in a large model size, which requires lots of computational resources and cannot run efficiently. Model order reduction can significantly alleviate this problem by reducing the size of these models, while maintaining the high-fidelity property. However, many model order reduction techniques fail to preserve the stability for non-locally reacting absorbers, which are often accounted for by using the Helmholtz equation with frequency-dependent density and bulk modulus, known as the equivalent fluid method. When discretized into a finite element model, frequency-dependent mass and stiffness matrices are introduced, which hinder the preservation of stability in the context of model order reduction. This paper presents a method that enables the creation of a stable reduced order model of a vibro-acoustic system with non-locally absorbers. This method reforms the Helmholtz equation and considers it as the interconnection of several passive systems. Stacking the states of these subsystems gives the final descriptor representation of the non-locally reacting absorbers. Furthermore, a possible second-order representation of the descriptor model is chosen such that it satisfies the stability-preserving condition under model order reduction. The resulting second-order model is coupled with a vibro-acoustic system in a velocity potential-displacement formulation, leading to a second-order model satisfying the aforementioned stability-preserving condition. The proposed method is successfully verified by several numerical simulations.

Refined multilayered beam, plate and shell elements based on Jacobi polynomials

In this paper, theories of structures based on hierarchical Jacobi expansions are explored for the static analysis of multilayered beams, plates and shells. They belong to the family of classical orthogonal polynomials. This expansion is employed in the framework of the Carrera Unified Formulation (CUF), which allows to generate finite element stiffness matrices in a straightforward way. CUF allows also to employ both layer-wise and equivalent single layer approaches in order to obtain the desired degree of precision and computational cost. In this work, CUF is exploited for the analysis of one-dimensional beams and two-dimensional plates and shells, and several case studies from the literature are analysed. Displacements, in-plane, transverse and shear stresses are shown. In particular, for some benchmarks, the shear stresses are calculated using the constitutive relations and the stress recovery technique. The obtained results clearly show the convenience of using equivalent single layer models when calculating displacements, in-plane stresses and shear stresses recovered by three-dimensional indefinite equilibrium equations. On the other hand, layer-wise models are able to accurately predict the structural behaviour, even though higher degrees of freedom are needed.

A direct method for the simultaneous updating of finite element mass, damping and stiffness matrices

This paper is mainly concerned with a model updating problem for damped structural systems, which can be described as the following inverse quadratic eigenvalue problem (IQEP) and an optimal approximate problem (OAP). Problem IQEP: Given matrices and with , for , , and both Λ and X being closed under complex conjugation, find real symmetric matrices M, D and K such that . Problem OAP: Given real symmetric matrices and , find such that , where is the solution set of Problem IQEP and is a weighted Frobenius norm. The representation of the general solution to Problem IQEP and the explicit formula for the optimal approximate solution of Problem OAP are given by applying the singular value decomposition. A given numerical example shows that the updated model can be in good agreement with the modal test data.

Application of parallel computing in finite element analysis of constructions

To optimize finite elements computational processes of the “MIRELA +” computing complex, parallel programming algorithms have been developed for creating stiffness matrices of finite elements and calculating the stress-strain. The parallel data processing is implemented using OpenMP library. On the basis of computational methods of experiments, the use of parallel computing algorithms at the beginning of solving problems with traditional finite element algorithms for large-dimensional grids has been established.

An Differential Quadrature Finite Element and the Differential Quadrature Hierarchical Finite Element Methods for the Dynamics Analysis of on Board Shaft

In this paper the dynamic analysis of a shaft rotor whose support is mobile is studied. For the calculation of kinetic energy and stiffness energy, the beam theory of Euler Bernoulli was used, and the matrices of elements and systems are developed using two methods derived from the differential quadrature method (DQM). The first method is the Differential Quadrature Finite Element Method (DQFEM) systematically, as a combination of the Differential Quadrature Method (DQM) and the Standard Finite Element Method (FEM), which has a reduced computational cost for problems in dynamics. The second method is the Differential Quadrature Hierarchical Finite Element Method (DQHFEM) which is used by expressing the matrices of the hierarchical finite element method in a similar form to that of the Differential Quadrature Finite Element Method and introducing an interpolation basis on the element boundary of the hierarchical finite element method. The discretization element used for both methods is a three-dimensional beam element. In the differential quadrature finite element method (DQFEM), the mass, gyroscopic and stiffness matrices are simply calculated using the weighting coefficient matrices given by the differential quadrature (DQ) and Gauss-Lobatto quadrature rules. The sampling points are determined by the Gauss-Lobatto node method. In the Differential Quadrature Hierarchical Finite Element Method (DQHFEM) the same approaches were used, and the cubic Hermite shape functions and the special Legendre polynomial Rodrigues shape polynomial were added. The assembly of the matrices for both methods (DQFEM and DQHFEM) is similar to that of the classical finite element method. The results of the calculation are validated with the h- and hp finite element methods and also with the literature.

Deep learned finite elements

In this paper, we propose a method that employs deep learning, an artificial intelligence technique, to generate stiffness matrices of finite elements. The proposed method is used to develop 4- and 8-node 2D solid finite elements. The deep learned finite elements practically pass the patch tests and the zero energy mode tests. Through various numerical examples, the performance of the developed elements is investigated and compared with those of existing elements. Computation efficiency is also studied. It was confirmed that the deep learned finite elements can potentially outperform existing finite elements. The proposed method can be applied to generate various types of finite elements in the future.

Free vibration analysis of nanotube based sensors including rotary inertia based on the Rayleigh beam and modified couple stress theories

In this study, a finite element formulation is presented to analyze the free vibration of carbon nanotube based sensor in conjunction with modified couple stress and Rayleigh beam theories. Rotary inertia effect and size dependency are considered for vibration problem of the cantilever single walled carbon nanotube. The aim of this paper is to examine the vibrational frequencies of single-walled carbon nanotube with these effects. Therefore, for the finite element solution, stiffness and mass matrices have been obtained that include these effects in the calculations. Numerical results are presented to show the variation of the frequencies with a variety of parameters such as the material length scale parameter, number of the finite elements, length of the nanotube and mode number.

Characterization of the behaviour of welded steel joints through modal components

ABSTRACTThe analysis and design of joints constitutes one of the most important aspects when designing steel structures. It is a common practice to model the joint behaviour by zero‐length rotational springs placed between the beam and corresponding column, or by mechanical models composed of springs and rigid bars. These models have serious limitations when considering the interactions between the shear, bending and axial forces and deformations of the panel zone.In this research, a new approach based on the characterization of modal components is presented. First, the steel joint is modelled as a cruciform finite element of 12 degrees of freedom. Then, the stiffness matrix is decomposed in eigenvectors and eigenvalues. The eigenvectors provide information about the joint deformation modes, while the eigenvalues give information about the stiffness associated with each mode. To identify the main modes and their characteristics, a parametric study has been carried out for welded joints with and without stiffeners. For both cases, all combinations of IPE 120 to IPE 600 profiles for beams and HEA 160 to HEA 1000 profiles for columns have been studied. The only restraint imposed in the combination process is that the beam width be smaller than the column width.First, the cruciform finite element stiffness matrices are obtained. Subsequently, a spectral decomposition is carried out using MATLAB, and then Python scripts are developed to classify the modes and eigenvalues. The statistical analysis of the results shows that after removing the three rigid solid modes the most relevant deformation modes are the shear and the two possible bending modes including their interaction. The remaining six modes correspond to deformation modes with high stiffness values that are not excited under common loading situations.

Analysis of a rotor supported in bearing with gyroscopic effects

Rotors with one or more offset disks and supported on bearings are common in high speed turbomachinery. During start up, these rotors have to cross through several critical speeds before reaching their operating speed. At critical speeds, the spin speed of the rotor matches with one of its natural frequency leading to the condition of resonance and large vibration amplitude. In rotating systems, these natural frequencies depend on the support stiffness. Also, they are a function on the spin speed because of the phenomenon of gyroscopic effect. The gyroscopic effects on critical speeds of a rotor system supported in bearing can be studied by means of a Campbell diagram, which has been used in the design of turbines. In this paper, gyroscopic effects of a rotor with an offset disk and supported on bearing is studied by means of Campbell diagrams. The numerical study is carried out by modelling of the rotor-bearing system using finite element mass, stiffness and gyroscopic matrices. The solution is obtained by solving the assembled equations of motion; following application of geometric boundary conditions and representing the second order differential equations of motion in statespace form. The results of critical speed obtained are compared with those of the results obtained through modal analysis using FE tool (ANSYS). The analysis of the results can be used to extend the study for a multi-disk rotor with different bearing supports.

ИССЛЕДОВАНИЕ ИНВАРИАНТНОСТИ J-ИНТЕГРАЛА ПРИ СЛОЖНОМ СДВИГЕ

The problem of determining the stress-strain state of an elastic-plastic beam with an edge crack under conditions of a complex shear is considered. A quasi-static external loading is given, which leads to complex loading processes at each point of the beam. Deformations are considered small. A numerical algorithm for solving the problem based on the application of the finite element method and the differential-nonlinear version of the plasticity theory taking into account microdeformation (Novozhilov, Kadashevich, Chernyakov) is proposed. The no-uniformity of plastic deformation is approximately taken into account by representing the plastic deformation tensor as a sum of elementary plastic deformations, each of which has its own yield surface and a system of internal micro-elastic forces. Mathematically, the boundary value problem is formulated as a nonlinear boundary value problem and the Cauchy problem with respect to the loading parameter. To solve the Cauchy problem, a step-by-step method that reduces the solution of the original problem to a sequence of solutions on time layers is used. At each step in time, using the iterative method, the nonlinear boundary value problem is reduced to a sequence of quasilinear problems that are solved by the finite element method. The quadrangular isoparametric finite element is chosen as the basis for the finite element method. To calculate the stiffness matrices of finite elements, an algorithm is used that makes it possible to reduce the number of multiple integrals to be calculated in the defining relations of the theory of plasticity taking into account microdeformations, up to the dimensionality of the loading trajectory. To calculate the J-integral, a direct method is used, which is based on its direct calculation on the basis of finite element method. When carrying out the calculations, St45 steel is chosen as the material of the beam, for which, when describing elastic-plastic deformation within the framework of the theory of plasticity taking into account microdeformations, it is permissible to use constants as universal material functions. Under various loading schemes, zones of plasticity in the section of the beam are constructed. The influence of the loading history on form and size of plastic zones and on the J-integral is investigated. It is shown that under the accepted loading schemes, within the admissible accuracy, the J-integral is invariant under both proportional and complex loading, and also that its magnitude depends on the loading history.

Generalised beam theory (GBT) formulation to analyse the vibration behaviour of thin-walled steel frames

This paper reports the results of an investigation on the use of Generalised Beam Theory (GBT) to analyse the vibration behaviour of thin-walled steel frames. After an overview of the main concepts and procedures involved in performing a GBT vibration analysis, the formulation and implementation of a GBT-based beam finite element are presented, namely the determination of the finite element and frame linear stiffness and mass matrices (incorporating the influence frame joint configurations). Finally, in order to illustrate the application and capabilities of the proposed GBT-based finite element formulation, numerical results concerning the local, distortional and global vibration behaviour of (i) a symmetric portal frame and (ii) an “L-shaped” frame exhibiting four commonly used joint configurations are presented and discussed. For validation purposes, some GBT results (frequency values and vibration mode shapes) are compared with values yielded by Ansys shell finite element analyses.

Comparative analysis of efficiency of use of finite elements of different dimensionality in the analysis of the stress-strain state of thin shells

Relevance. To determine the stress-strain state (SSS) of thin-walled shells due to the complexity of obtaining numerical results, the theory of thin shells was developed with the introduction of the direct normal hypothesis to reduce the three-dimensional SSS to the two-dimensional one. With the modern development of digital technology and numerical methods of calculation, in particular the finite element method (FEM), it became possible to obtain numerical results without the use of the direct normal hypothesis, namely on the basis of the theory of elasticity in three-dimensional formulation even for thin shells. Aims. The aim of this work is to compare the efficiency of algorithms for the use of finite element stiffness matrices obtained on the basis of the theory of thin shells with the hypothesis of a straight normal and on the basis of the relations of the three-dimensional theory of elasticity. Methods. The results of comparative analysis of finite element calculations of thin shells using a two-dimensional sampling element in the form of a quadrangular fragment of the middle surface and a three-dimensional element in the form of an eight-node six-face are presented. The components of the displacement vector and their first derivatives were chosen as the nodal variable parameters. The functions of the form for both types of discretization elements were represented by products of Hermite polynomials of the third degree. Results. On the example of calculation of the cylindrical shell clamped at the ends it is shown that the two-dimensional statement in calculations of thin shells is adequate and allows to receive acceptable results at optimum costs of machine time.

Issues with Proportional Damping

Analytical derivation of damping is extremely difficult and in most cases not possible. As a result, damping properties are typically specified on a mode-by-mode basis because of the availability of test data that can be used either directly, or can be used as a guide for similar systems for which data are not yet available. Occasionally there is need to develop from the modal damping properties damping matrices that correspond to the physical coordinate sets in which the finite element model mass and stiffness matrices were developed. Proportional damping, in which combinations of mass and stiffness matrices are used to develop the physical coordinate damping matrix, has been used extensively. It is the purpose of this work to discuss the appropriateness of including a mass proportional term. It is concluded that damping formulations that include a scaled mass term are fundamentally flawed and should not be used.

Generalized B‐bar: unlocking multiple compliance modes of finite elements with stiff‐fiber anisotropy

SummaryElement locking is often seen in homogenized models of elastic fiber‐reinforced materials, and splitting the material compliance into two separate terms isolates troublesome strain modes. Once isolated, the locking modes can be addressed with tailored integration schemes or the opportune introduction of field variables. The canonical application of this approach is seen in the dilatational‐deviatoric split used to treat so‐called ‘volumetric locking’. In the present work, we invoke the spectral decomposition of the material compliance to provide a generalized split. Doing so naturally parses the response into six independent strain modes, with varying propensity for locking. This split can be used to generalize fundamental techniques, such as selective reduced integration and the B‐bar method. This broadened approach works to remedy locking suffered by lower order finite elements used to discretize troublesome materials. Applying these generalized methods to achieve the dilational‐deviatoric split is trivial. However, the compliance spectrum's ability to naturally isolate stiff material response modes makes it a uniquely valuable tool for use on homogenized anisotropic materials. Applying the split, defined by only the first compliance mode, has given rise to the generalized methods, which have proven effective in unlocking finite element models of anisotropic materials. In the present work, the generalization is broadened to treat more than one constrained mode. While treating six modes is equivalent to simple reduced integration techniques, up to five compliance modes are now separated for advantageous treatment. However, some attention must be paid to the stability of the resulting finite element stiffness matrices. We focus here on the treatment of two principal compliance modes. These ‘two‐mode’ applications of the generalized B‐bar method are shown to be a more robust default treatment of linear hexahedral elements than is provided by classical selective reduced integration. This is achieved with a negligible computational overhead. A framework for assessing element stability is delineated, and commonly arising instabilities are analyzed. Copyright © 2016 John Wiley & Sons, Ltd.

Fast MATLAB assembly of FEM matrices in 2D and 3D: Edge elements

We propose an effective and flexible way to assemble finite element stiffness and mass matrices in MATLAB. We apply this for problems discretized by edge finite elements. Typical edge finite elements are Raviart–Thomas elements used in discretizations of H(div) spaces and Nédélec elements in discretizations of H(curl) spaces. We explain vectorization ideas and comment on a freely available MATLAB code which is fast and scalable with respect to time.

A comparative evaluation of the scalar and vector approximations of sought quantities in the finite-element method of arbitrary shells

New versions of formulas that specify the median surfaces of an ellipsoid, elliptical cylinder, and cone, which allow one to perform calculations without features of the definitional domains of the considered shells, are proposed. The algorithm for forming stiffness matrices of quadrangular curvilinear finite elements with eighteen degrees of freedom at a node is given. The designed algorithm is verified based on the example of the calculation of a circular cylinder using scalar and vector approximations of displacements. The efficiency of the vector approximation of displacement fields in calculations of the elliptical cylinder with different fastening conditions and its displacement as a rigid body is shown.

Improved Finite Element Approach for Modeling Three-Dimensional Linear-Elastic Bodies

Objectives: This paper proposes a general formula for computation of the stiffness matrix of three-dimensional linearelastic bodies by the method of Moment Schema of Finite Elements (MSFE). Methods: Numerical methods based on Lagrange variational principle, such as Finite Element Method (FEM) and the method of Moment Schema of Finite Elements (MSFE). The paper focuses on the problem of obtaining the stiffness matrices of Finite Elements (FE), which ensure effectiveness of the developed version of FEM. Variational principle for calculation of energy functional of rectangular three-dimensional finite element is used. Results: Analyses of various problems of mechanics of deformable solids shows slow convergence of traditional FEM, especially for rigid bodies and surfaces of complex curved shapes. This paper proposes a scheme for inference for FEM ratios, which takes into account the basic properties of rigid displacements for isoparametric and curvilinear FE. The presented version of FEM allows us to obtain the stiffness matrix of FE, taking into account the effect of a false shift and low compressibility of elastomers. Conclusion/Application: The general method to find the potential energy of three-dimensional linear-elastic bodies is proposed, which allows us to solve the problems of mechanics of complex structures with higher accuracy.

Auxiliary space multigrid method based on additive Schur complement approximation

SummaryIn this paper, the idea of auxiliary space multigrid methods is introduced. The construction is based on a two‐level block factorization of local (finite element stiffness) matrices associated with a partitioning of the domain into overlapping or non‐overlapping subdomains. The two‐level method utilizes a coarse‐grid operator obtained from additive Schur complement approximation. Its analysis is carried out in the framework of auxiliary space preconditioning and condition number estimates for both the two‐level preconditioner and the additive Schur complement approximation are derived. The two‐level method is recursively extended to define the auxiliary space multigrid algorithm. In particular, so‐called Krylov cycles are considered. The theoretical results are supported by a representative collection of numerical tests that further demonstrate the efficiency of the new algorithm for multiscale problems. Copyright © 2014 John Wiley & Sons, Ltd.