Finite Element Stiffness Matrix Research Articles

Multi-criteria parametric optimization of the displacement and weight of a shell of minimal surface on a circular contour consisting of two inclined ellipses under thermal and power loading with consideration of geometric nonlinearity

Thin shells are well suited to optimal design problems, and the finite element method and gradient descent method make it possible to solve inverse problems in structural and applied mechanics. The calculation process takes into account geometric nonlinearity. In modern numerical studies, a formulation taking into account geometric nonlinearity in the finite element method is used, namely, a stepwise loading procedure. It becomes necessary to take into account the relations between displacement vectors and their derivatives and strain increments. These relations help to determine the stiffness matrix of the finite element at each loading step, which leads to a qualitative study of real displacements in the structure. The geometrically nonlinear calculation of a shell of minimum surface on a circular contour consisting of two inclined ellipses in a multi-criteria parametric optimization is performed by the finite element method. The finite element method is a universal variational method that is focused on solving the most complex problems of elasticity theory and applied and structural mechanics using separate calculation complexes. Stiffness matrix - for the whole body is formed on the basis of the finite element stiffness matrix. The system of solving equations of the finite element method is formed using the Lagrange's variational principle, according to which the total potential energy P of a finite element model of a body in a state of stability and equilibrium has a minimum value. As part of the sensitivity analysis, gradients of the design variables of the structure, displacements in the form of partial derivatives of these characteristics along the design variables, and shell thickness are calculated. The sensitivity information serves as the basis for building an optimal design algorithm using the gradient descent method of the objective function. Standard multi-criteria parametric optimization allows for an average of 10% steel savings, with geometric nonlinearity increasing to 20%. At this study site, 24.4% of sheet steel was saved, which is a significant relative saving. This methodology of the authors shows high results for innovative design and production of steel thin shells in Ukraine and around the world, and also makes it possible to apply several types of optimization to one research object.

CALCULATION OF SHELLS BASED ON THE MIXED FEM VARIANT WITH TENSOR-VECTOR APPROXIMATION OF THE DESIRED VALUES

An algorithm for calculating thin shells based on the mixed finite element method (FEM) in a two-dimensional formulation is proposed. To correctly take into account possible displacements of the shell as a rigid body, the algorithm implements the developed tensor-vector form of the interpolation procedure of the desired values, which were chosen as the tensors of deformations and curvature at the point of the middle surface and the displacement vectors of this point. The discretization element was a quadrangular fragment of the middle surface with unknown nodes in the form of location and their first derivatives, as well as components of strain and curvature tensors among the inner surface at the nodes of the finite element. After minimizing the modified mixed functional, a 36?36 finite element stiffness matrix is formed. In order to verify the developed algorithm, a number of test problems were solved for calculating a fragment of an elliptical cylinder with an analytical solution, as well as for calculating a shell with spring supports that allow the ellipsoidal shell to move as an absolutely rigid body. Analysis of the obtained results showed that when calculating a fragment of an elliptical cylinder, the calculation scheme of which makes it possible to obtain an analytical solution, the numerical values of bending moments and normal stresses completely coincide with the values calculated from the condition of static equilibrium. An analysis of the calculation results for an ellipsoidal shell with spring supports led to the conclusion that the developed tensor-vector interpolation procedure in the mixed FEM version allows one to correctly take into account the displacements of shells as solids and obtain an adequate assessment of their stress-strain state.

FEATURES OF THE CONSTRUCTION OF THE STIFFNESS MATRIX OF THE SPATIAL HEXAGONAL FINITE ELEMENT FOR COMPOSITE MATERIAL WITH DISCRETE INCLUSIONS BASED ON THE MOMENT SCHEME

An approach to numerical modeling of the stress-strain state of composite structures with discrete inclusions is presented in the paper. The finite element method is used as the main method, namely its modification – the moment finite-element scheme. The moment scheme, in contrast to the classic scheme of finite elements, allows to avoid such negative properties as not taking in consideration the rigid rotation of the finite element and “false” shear. If both the material of the matrix and the material of the reinforcing inclusions are weakly compressible, then problems arise due to the fact that some elastic constants approach very large values. The Taylor series expansion of the components of the displacement vector, the components of the strain tensor, and the volume change function is used in order to eliminate the mentioned shortcomings, after that, according to the moment scheme, certain sums are removed from these expansions. Homogenization of the material with lamellar inclusions, a small proportion of spherical inclusions, and a large proportion of spherical inclusions is used for modeling the elastic properties of the composite. The chaotic nature of the location of inclusions after homogenization makes it possible to present a non-homogeneous composite material as a homogeneous quasi-isotropic one. The described approaches are used in the construction of the stiffness matrix of the spatial hexagonal finite element. The obtained expressions for the stiffness matrix are done in the software package for calculating structures from composite materials. The calculation of a thick-walled pipe under the action of internal pressure from a composite material with lamellar inclusions, a small proportion of spherical inclusions, and a large proportion of spherical inclusions was carried out using the software package. For different volume fractions of discrete inclusions, the numerical convergence of the results with different finite element meshes has been investigated, which shows great congruence with analytical solutions.

An enriched finite element for the simplified stress analysis of an entire bonded overlap: Continuum macro-element

A methodology for the formulation of the stiffness matrix of an enriched finite element, called continuum macro-element (CME), representing for the full length of a bonded overlap and both the adhesive and the adhesive in only one four-node elements, is presented. Compared to earlier macro-elements modelling the bonded overlap as beams on elastic foundation, the CME supposed the adherends and the adhesive as plane continuum media, for which higher-order displacement fields is supposed for the adhesive. The formulation of the stiffness matrix for the adherend outside of the overlap is presented as well to address the stress analysis of single-lap bonded joint for assessment purpose. The assessment is performed by comparisons with the results from (i) a plane strain finite element and (ii) two recent papers by Nguyen and Le Grognec (2021) and Methfessel and Becker (2022). Good agreements are shown.

Consistency and misconceptions in co-rotational 3D continuum finite elements: A zero-macrospin approach

The seminal work of Crisfield on co-rotational finite elements came to a profound conclusion that the geometric stiffness matrix of 3D continuum finite elements is inherently asymmetric, though it becomes symmetric under equilibrium conditions. Contending that these outcomes are misconceptions, this paper presents a novel zero-macrospin co-rotational approach for 3D continuum elements in which the geometric stiffness matrix is always symmetric for work-conjugate conservative loads, suggesting that Crisfield’s geometric stiffness is erroneously asymmetric and inconsistent. This contention is subsequently upheld by identifying crucial terms omitted in Crisfield’s formulation which are responsible for the erroneous asymmetry, and the notion that symmetry is eventually recovered at equilibrium is shown to be a further misconception. Importantly, this critical appraisal of Crisfield’s formulation, which has been adopted by numerous researchers, and the delineation of its underlying misconceptions exceed academic interest to real practical relevance. Several numerical examples are presented in the paper which highlight the superiority of the proposed approach, lending incontrovertible practical evidence that the consistent geometric stiffness matrix for co-rotational 3D continuum elements is always symmetric for work-conjugate conservative loads.

Vibration analysis of a cracked beam using the finite element method

This study is motivated by the condition monitoring and fault diagnostics of structural beams used in large-scale vibrating screens for the mining industry. For the purpose of developing a reliable model-based approach, a new stiffness matrix of a three-dimensional finite element is proposed for modelling a beam with a breathing crack. Using the obtained stiffness matrix, a finite element model was derived for a cracked beam subjected to a bidirectional base excitation. With the model, a computer simulation was conducted to examine the influence of the crack depth on the natural frequencies of the beam. The numerical results show that the crack influences mainly the natural frequencies of the bending modes in the direction of the crack's growth. The simulation also investigates the responses of the beam subjected to a harmonic base excitation in two directions. The numerical results show that the responses of the cracked beam contain several superharmonic components, and the amplitude of superharmonic components increases with the increase in crack depth. Experiments are conducted to validate the proposed dynamic model using both impact testing and forced testing.

Kinematic Calibration Method for Large-Sized 7-DoF Hybrid Spray-Painting Robots

Large-sized seven-degrees-of-freedom (7-DoF) hybrid spray-painting robots combine ample working space and high flexibility, making them lucrative for the spray painting of aircraft and rocket surfaces. However, their kinematic calibration is hindered by gravitational deformation, which problem is addressed in this study by introducing a rigid-flexible coupling error modeling method. The latter combines the finite element method (FEM) and stiffness matrix method to assess the spatial gravitational deformation of a hybrid robot, which is then introduced into a geometric error model to establish the rigid-flexible coupling error identification model. Given many redundant parameters in the identification model for 7-DoF robots, these parameters are classified and simplified using the nonlinear least-square regularization method for parameter identification. Combining the inverse solution of 7-DoF spray-painting robots with dynamic characteristics considered, an error compensation method for 7-DoF robots is proposed. The kinematic calibration test results strongly indicate that position errors are significantly reduced with gravity compensation taken into consideration, and error convergence speed increases, demonstrating that the kinematic calibration method is feasible and can effectively improve the accuracy of spray-painting robots. The mean errors in the X- and Y-directions are reduced by 20 and 17%, respectively, compared to the conventional method. The proposed method is instrumental in the accurate kinematic calibration of large-sized 7-DoF hybrid robots.

Volumetric element with vector approximation of the desired values for nonlinear calculation of the shell of rotation

The usage of traditional approximating functions directly to the desired displacement vector of the internal point of a finite element to determine it through nodal unknowns in the form of displacement vectors and their derivatives is described. To analyze the stress state of a geometrically non-linearly deformable shell of rotation at the loading step, the developed algorithm for forming the stiffness matrix of a hexagonal finite element with nodal values in the form of displacement increments and their derivatives was used. To obtain the desired approximating expressions, the traditional interpolation theory is used, which, when calculated in a curved coordinate system, is applied to the displacement vector of the internal point of a finite element for its approximation of class C(1) through nodal displacement vectors and their derivatives. For the coordinate transformation, expressions of the bases of nodal points are obtained in terms of the basis vectors of the inner point of the finite element. After the coordinate transformations, approximating expressions of class C(1) are found for the components of the displacement vector of the internal point of the finite element, leading in a curved coordinate system to implicitly account for the displacement of the finite element as a rigid whole. Using calculation examples, the results of the developed method of approximation of the required values of the FEM with significant displacements of the structure as an absolute solid are obtained.

Effect of boundary conditions on values of homogenized fabrics’s elastic parameters and inelastic response

The effect of displacement, periodic and surface traction boundary conditions (BCs) on predicted elastic parameters and inelastic response of a homogenized material equivalent to either a unidirectional or a randomly distributed short fibers reinforced composite has not been extensively studied. A few such studies have focused on their elastic analyses. Here, we analyze the effect of the three types of BCs on both elastic and inelastic deformations of weaves and braids with continuum damage mechanics used in the inelastic analyses. We employ simple unit cells, Hill’s theorem, and the displacement, periodic and uniform traction BCs to deduce from the finite element stiffness matrix K the matrix C of homogenized moduli. For both elastic and inelastic problems, it is seen that predictions using the displacement and the periodic BCs are close to those reported in the literature while those from the traction BCs are not. For unidirectional loading, it is found that the nonlinearity in the axial stress - axial strain curve, the localized deformations and the resin damage occurring in regions unsupported by fibers are significantly more for the traction BCs than that for the other two BCs. Also, there is more damage for the periodic BCs than that for the affine displacement BCs.

ТРЕУГОЛЬНЫЙ КОНЕЧНЫЙ ЭЛЕМЕНТ С ШЕСТЬЮ СТЕПЕНЯМИ СВОБОДЫ В УЗЛЕ

The work is devoted to obtaining the stiffness matrix of a high-precision, flat finite element with 6 degrees of freedom in the node, for solving plane elasticity problems by the finite element method.
 The scientific literature describes higher order elements. However, the theoretical results of these studies are quite far from their practical application. This paper gives a very detailed derivation of the stiffness matrix of a high-precision finite element. Similarly, a stiffness matrix of a tetrahedral finite element with 12-degree-of-freedom node can be obtained. To test the obtained stiffness matrix, a program based on the FEM was written, with the help of which the cantilever beam is calculated. The error in calculating displacements is only 0.22%.
 Conclusion: the stiffness matrix presented in the paper can be used with great success in numerical methods for calculating the stress-strain state.

ЧИСЛЕННЫЙ ПРИМЕР ПОСТЕПЕННОГО ПРЕОБРАЗОВАНИЯ МАТРИЦ ЖЕСТКОСТИ ТРЕУГОЛЬНОГО КЭ БАЛКИ-СТЕНКИ И ЕГО ДКЭ.

The paper considers the numerical example of gradual transformation of the stiffness matrix of triangular finite element of deep beam with the help of stiffness matrix of its additional finite elements. Material FE (concrete) reveals from two (under compression) to four (under tension) physical nonlinear properties before collapse. Initial stiffness matrix changes due to the every one. Therefore two variants of its numerical transformation are considered. Both variants confirm the possibility of realization of the analysis of structures with several nonlinear properties at strength limit stae when the Additional Finite Element Method (AFEM) is used.

Stress state of hydrotechnical facilities resistant to natural disasters

The equations of the theory of the stress-strain state of an elastic body for practical engineering structures of the agro-industrial complex do not have analytical solutions. Therefore, the development of numerical methods for determining the strength parameters of agricultural facilities is an urgent task. Among the numerical methods of strength calculations, the finite element method is currently widely used. In a Cartesian coordinate system, a finite element of a hexahedral shape is used to determine the stress-strain state of an elastic body under bulk loading in two formulations: in the formulation of the method of displacements with nodal unknowns in the form of displacements and their derivatives, and in a mixed formulation with nodal unknowns in the form of displacements and stresses. Approximation of displacements through nodal unknowns in obtaining the finite element stiffness matrix was performed using a form function, the elements of which were Hermite polynomials of the third degree. When obtaining the deformation matrix, displacements and stresses of the internal points of the finite element were approximated in terms of nodal unknowns using bilinear functions. The calculation example shows a significant advantage of using a finite element in a mixed formulation.

Strength calculation algorithm for shell structures based on a four-node discretization element

The article describes a method for calculating shell structures taking into account transverse shear deformations during elastoplastic deformation. The geometric relationships between the increments of deformations in an arbitrary layer of the shell and the increments of the components of the displacement vector, the components of the vector of inclination angles of the normal are obtained on the basis of the hypothesis of S.P. Tymoshenko. The slope of the normal and the increment of the slope are performed from the original position of the normal. The physical relationships between the increments of the stress tensor components and the increments of the components of the deformation tensor at the loading step were obtained on the basis of the hypothesis that they were proportional, based on the deformation theory of plasticity. In the finite element implementation of the developed algorithm, a four-node fragment of the middle surface of the shell structure with nodes located at the vertices of the quadrangular fragment was used. The increments of the displacement vector components, their first derivatives with respect to curvilinear coordinates, as well as the increments of the components of the normal rotation angle vector were selected as the nodal variable parameters of the four-node finite element at the loading step. The size of the stiffness matrix of the four-node finite element was 44x44. The developed algorithm was tested on test problems for calculating a cylindrical shell using a duralumin alloy with a deformation diagram in the form of a two-link broken line as a material.

Approximating inverse FEM matrices on non-uniform meshes with {mathcal{H}}-matrices

We consider the approximation of the inverse of the finite element stiffness matrix in the data sparse {mathcal{H}}-matrix format. For a large class of shape regular but possibly non-uniform meshes including algebraically graded meshes, we prove that the inverse of the stiffness matrix can be approximated in the {mathcal{H}}-matrix format at an exponential rate in the block rank. Since the storage complexity of the hierarchical matrix is logarithmic-linear and only grows linearly in the block-rank, we obtain an efficient approximation that can be used, e.g., as an approximate direct solver or preconditioner for iterative solvers.

Modeling multi-fracturing fibers in fiber networks using elastoplastic Timoshenko beam finite elements with embedded strong discontinuities — Formulation and staggered algorithm

To model fiber failures in random fiber networks, we have developed an elastoplastic Timoshenko beam finite element with embedded discontinuities. The method is based on the theory of strong discontinuities where the generalized displacement field is enhanced by a jump. The continuum mechanics formulation accounts for a fracture process zone and a bulk material while retaining traction continuity across the discontinuity. The additional degrees of freedom that are associated with the discontinuity are represented by a midpoint node, which is statically condensed to enable the implementation in commercial software through the user element interface. We propose a quasi-brittle fracture model, where the failure-related deformation is uncoupled from the plastic deformation in the bulk material. To retain the positive definite finite element stiffness matrix of the bulk material, we neglect the fracture-related softening of the discontinuity and employ a modified Newton iteration in the strain softening domain. Our implementation facilitates the integration into commercial finite element software and examples illustrate the robustness of the method. The FORTRAN source code is freely available to benchmark our model. We show that fiber failures contribute to the nonlinear stress–strain response of paper. Together with fiber–fiber bond failures, they can potentially explain the nonlinear stress–strain response of paper and nanopaper.

Determination of the stress-strain state of structures made of viscoelastic fibrous composite materials based on the stiffness matrix of a spatio-temporal finite element

Abstract. Increase in spheres of composite materials application leads to the need to improve the methods of composite structures calculation, increase their efficiency and adequacy to real processes. The use of fibrous composite materials creates difficulties in determining the stress-strain state of the structure due to the different physical and mechanical properties of the material components. As a rule, a fibrous composite consists of two components: a matrix and a fiber. The matrix ensures the material monolithicity fixing the shape of the product and the relative position of the reinforcing fibers. The fibers absorb the main stresses that occur during operation, and provide rigidity and strength of the composite. The combination of the matrix and fiber properties allows creating special types of composites to be used in various equipment. The use of rubber as a matrix gives the structure viscoelastic properties, which can be seen in the deformation creeping and stress relaxation in the material, which complicates the calculation of the stress-strain state of the whole structure greatly. The paper proposes an approach that is the development of a moment scheme of finite elements in the form of the displacements method based on the Lagrange principle. Using this approach, a special space-time finite element has been developed that takes into account the viscoelastic properties of the material matrix and the fiber elastic properties. The proposed numerical approach allows obtaining a solution to the problems of viscoelasticity of composite materials mechanics in three-dimensional model. Using the applications package created on the basis of this approach, the calculation of the stress-strain state of a single-cavity pneumatic cylinder with a rubber cord shell has been done. The influence of the reinforcement scheme and rubber viscoelastic properties on the parameters of the pneumatic cylinder deformation has been analyzed. In particular, it becomes possible to adjust the structure stiffness properties by changing the orientation, type and concentration of fibers, as well as the type of matrix and other material characteristics. Keywords: fibrous composite material, viscoelasticity, space-time finite element.

Comparative analysis of finite element formulations at plane loading of an elastic body

The aim of the work - comparison of the results of determining the parameters of the stress-strain state of plane-loaded elastic bodies based on the finite element method in the formulation of the displacement method and in the mixed formulation. Methods. Algorithms of the finite element method in various formulations have been developed and applied. Results. In the Cartesian coordinate system, to determine the stress-strain state of an elastic body under plane loading, a finite element of a quadrangular shape is used in two formulations: in the formulation of the method of displacements with nodal unknowns in the form of displacements and their derivatives, and in a mixed formulation with nodal unknowns in the form of displacements and stresses. The approximation of displacements through the nodal unknowns when obtaining the stiffness matrix of the finite element was carried out using the form function, whose elements were adopted Hermite polynomials of the third degree. Upon receipt of the deformation matrix, the displacements and stresses of the internal points of the finite element were approximated through nodal unknowns using bilinear functions. The stiffness matrix of the quadrangular finite element in the formulation of the displacement method is obtained on the basis of a functional based on the difference between the actual workings of external and internal forces under loading of a solid. The matrix of deformation of the finite element was formed on the basis of a mixed functional obtained from the proposed functional by repla-cing the actual work of internal forces with the difference between the total and additional work of internal forces when loading the body. The calculation example shows a significant advantage of using a finite element in a mixed formulation.

On explicit form of the FEM stiffness matrix for the integral fractional Laplacian on non-uniform meshes

We derive exact form of the piecewise-linear finite element stiffness matrix on general non-uniform meshes for the integral fractional Laplacian operator in one dimension, where the derivation is accomplished in the Fourier transformed space. With such an exact formulation at our disposal, we are able to numerically study some intrinsic properties of the fractional stiffness matrix on some commonly used non-uniform meshes (e.g., the graded mesh), in particular, to examine their seamless transition to those of the usual Laplacian.

The prismatic finite element with the interpolation procedure vector form for the engineering structures’ strength calculations

The technique of forming a stiffness matrix of a volume prismatic finite element with a triangular base and with six nodes located at the vertices of the prism is presented. The discretization element is formed on the basis of the interpolation procedure vector form with consideration as the interpolation object of the displacement vector of an arbitrary point of the engineering structure. It is proposed to improve the compatibility of the prismatic discretization element at the boundaries of the docking of the bases by using Lagrange multipliers as additional nodal unknowns, which are introduced in additional nodes located in the middle of the prismatic discretization element bases’ sides. The presented vector form of interpolation allows one to obtain the correct finite element solutions in the problems of determining the stress-strain state of engineering structures using curvilinear coordinate systems.

A three dimension lattice-spring model with rotational degree of freedom and its application in fracture simulation of elastic brittle materials

Applying parameter mapping theory, this paper establishes a new three dimension lattice-spring model with rotational degree of freedom (3D-LSMR). This 3D-LSMR contains nearest neighbor, next nearest neighbor and third nearest neighbor which highly increases the accuracy of system. Compared with the general lattice spring model, 3D-LSMR adds the rotational and tangential degrees of freedom, whose spring parameters are obtained by finite element parameter mapping. The traditional lattice spring model only considers the normal interaction between two points (the Poisson’s ratio is fixed), which limits the application of the lattice spring model in a wider range of materials. Some scholars proposed to add shear spring into the traditional model to solve the problem of fixed Poisson’s ratio, but the rotation invariance could not be maintained. The main reason is that the shear spring can’t distinguish the difference of tangential velocity (or displacement) between the two particles due to the common rotation or shear. Therefore, the rotation of the whole rigid body may cause incorrect generation of additional strain energy inside the shear spring. 3D-LSMR model is able to maintain rotation invariance and reproduces different Poisson’s ratios because of the spring parameters with rotational degrees of freedom and the rotation effect of lattice points. In this paper, the finite element stiffness matrix with rotational freedom is derived by using 3D-LSMR as the basic mechanical model, and the parameter mapping theory of spring stiffness coefficient is introduced. By means of numerical simulation, the large deflection of slender cantilever beam, elastic symmetrical collision and the simulation of dynamic fracture for concrete L-specimen are checked and calculated respectively, and then the correctness of the model is verified.