Computation Of Stiffness Matrix Research Articles

Vibration and Stability Analysis of a Bearing–Rotor System with Transverse Breathing Crack and Initial Bending

This paper focuses on the stability and nonlinear response of a bearing-rotor system affected by a transverse crack and initial bending which was thought to be part of an unbalance or had been neglected before. The differences of breathing functions for the transverse breathing crack caused by initial bending is presented here, and the calculation of time-varying finite elements stiffness matrix of the cracked shaft is improved by replacing traditional the approximate crack segment with an exact area. After establishing the dynamic model of the cracked rotor with initial bending, vibrational characteristics such as amplitude-speed diagram, frequency spectrogram and bifurcations are investigated in detail. The eigenvalues of the transition matrix are calculated and analyzed as an indicator of dynamic stability with the growths of crack depth and initial bending. Many differences are found between the two cases of dynamic response of rotor system by numerical simulation. The frequency change with the growth of initial bending is opposite to the change with the growth of crack depth, and the shapes of amplitude-speed also having great different features. Stable regions are reduced and extended laterally by initial bending. All these results obtained in this paper will contribute to identify the bending fault and assess the stability of the bearing-rotor systems.

Determination of the effective 2D material stiffness matrix using ABAQUS

AbstractA cross‐scale homogenization approach for multi‐phase materials is to calculate an effective material stiffness in the smaller scale for the multiple states. These detailed results are used to identify a calculation rule for the effective material stiffness in the larger scale.The calculation of the stiffness matrix using ABAQUS for 3D finite elements was already described by [1]. This can be transferred to the calculation of the 2D stiffness matrix in ℝ3×3 without much effort. However, some 2D elements in ABAQUS require the specification of a matrix in ℝ4×4. In this article, the methods for calculating this type of stiffness matrix in 2D are presented.

Feasibility of a weakly intrusive Generalized Finite Element Method implementation in a commercial code: Application to Ceramic Matrix Composite micro-structures

As the development of new grades of Ceramic Matrix Composites (CMC) for civil aviation grows, different manufacturing processes have been perfected and several of them can be used successively in order to obtain different types of micro-structures and a variable material quality. Consequently a versatile model should be developed in order to compare these materials and create a tool to help engineers to predict the mechanical behavior at the fiber scale. Here the Generalized Finite Element Method (GFEM) is proposed. It consists in enriching the classical Finite Element (FE) approached displacement by numerical functions to deliver an accurate description of the fiber-scale structure while limiting the number of degrees of freedom compared to a classical finite element description. A pattern-based description of the microscale is depicted using an industrial code for an engineering purpose. Four main difficulties are highlighted (i) the choice of the enrichment functions regarding the literature (ii) their stiffness matrix computation in a commercial code (iii) the construction of the pattern-based structure and (iv) the post-processing. Two GFEM strategies are presented and demonstrate the feasibility of an enriched kinematics within a classical finite element modeler. The selection of such modeler is conditioned by the possibility of weakly intrusive automation of the various stages of construction of the enriched patterns with the help of an external scripting language.

An Efficient and Rapid Numerical Quadrature to generate element matrices for quadrilateral and hexahedral elements in Functionally Graded Materials (FGMs)

This research paper introduces a robust, efficient and stabilized midpoint quadrature to generate element stiffness matrices for quadrilateral and hexahedral elements in finite element method (FEM). This new quadrature is established based on Passive Richardson extrapolation which has the inherent ability to stabilize and improve the accuracy of the extrapolated results. The new quadrature includes a simple artificial stabilizing function with Gauss one-point quadrature (1GP) as an anti-hourglass effect. For the first approximation, the 1GP (sampling point to be considered at origin (0,0)) is done to evaluate element matrices (thermal conductivity matrices, capacitance matrices etc.,) in the 4-square element ( [-2, 2] x [-2, 2] ) in the (ξ, η) plane. As a stabilizing function, sampling points are considered to be either at the center of the four element edges or center of the four quadrants in the reference square element with appropriate constant weighting functions. For hexahedral elements also, the element stiffness coefficients are calculated only in the center of the element and center of all the octants of the reference cubic element ([-2, 2] x [-2, 2] x [-2, 2]), the proposed method deals mostly with integer variables and sampling at (0,0,0), whereas Gauss quadrature uses lengthy float variables corresponding to irrational numbers in terms of decimals in the stiffness matrix calculations, thus consuming time. A number of standard example problems are demonstrated to assess the efficiency of the proposed method. The results of all the numerical examples are found to be in good agreement with the exact results and conventional Gauss quadrature results.

Direct image-based micro finite element modelling of bone tissue

In the paper, a method for the direct image-based creation of the finite element model from images is presented. Image information is taken into account during the calculation of the element stiffness matrix. In this case, material heterogeneity can be included directly in the finite element model. For this purpose, the hypothesis about the correlation between pixel values and elastic properties was used. Four nodes plane element was built. The element can be used with the quantitative phase or scanning electron images and computed tomography data. Simulation for bone data performed. The influence of pixel on the error estimate was studied. The method to parallelize the calculation of the stiffness matrix is presented. As an example, a slice of bone was used in the calculation. Results for average stress distribution for the origin and improved mesh are presented.

An Efficient Space–Time Method for Time Fractional Diffusion Equation

A space–time Petrov–Galerkin spectral method for time fractional diffusion equations is developed in this paper. The Petrov–Galerkin method is used to simplify the computation of stiffness matrix but leads to full non-symmetric mass matrix. However, the matrix decomposition method based on eigen-decomposition is numerically unstable for non-symmetric linear systems. A QZ decomposition is adopted instead of eigen-decomposition. The QZ decomposition has essentially the same computational complexity as the eigen-decomposition but is numerically stable. Moreover, the enriched Petrov–Galerkin method is developed to resolve the weak singularity at the initial time. We also carry out the error analysis for the proposed methods and present ample numerical results to validate the accuracy and robustness of our numerical schemes.

A comprehensive study on the speed-varying stiffness of ball bearing under different load conditions

The stiffness as one of the most important dynamic characteristics of ball bearing, has a great influence on the service performance of bearing and rotor systems. To obtain and analyze the stiffness matrix of ball bearing under arbitrary load conditions, this paper proposed a novel mathematical method for ball bearing modeling and stiffness matrix calculation based on a new initial condition assumption. Based on this, a comprehensive and systematical study to investigate the effect of rotating speed on the stiffness for ball bearing under three typical load conditions has been presented (the pure axial load condition, the pure radial load condition and the combined loads condition). And the influences of bearing basis structure parameters including the internal clearance, ball diameter, inner and outer raceway curvature coefficients on the change rule of the stiffness along with speed of ball bearing under three typical load conditions have been given. The results show that the proposed model has a high versatility, and the change rules of the stiffness along with rotating speed are greatly influenced by the load conditions, bearing structural parameters. Therefore, this paper not only presents a complete algorithm for the modeling and stiffness calculation of ball bearing under arbitrary load conditions, but also provides important guidance for the design of bearing and rotor systems.

Reluctance network lumped mechanical & thermal models for the modeling and predesign of concentrated flux synchronous machine

Abstract The aim of this paper is the multi-physical modeling of synchronous permanent magnet machines, dedicated to hybrid electrical vehicles application, using reluctance network lumped mechanical and thermal models. The modeling approaches are presented and validated by comparing the obtained results to those of finite element method with a close look to the airgap modeling and the consideration of soft magnetic materials non-linearity in the electromagnetic modeling. As well as a close look also on the conduction and convection heat coefficients for the machine different regions in the thermal modeling. Finally a focus on the mass, damping and stiffness matrix computation in lumped mechanical modeling taking into account the temperature influence on the materials mechanical properties. In addition a simplified rotating electrical machine is described and multiple coupled analysis were done in order to derive the structure magneto-vibroacoustic performances.

Improvements in Shear Locking and Spurious Zero Energy Modes Using Chebyshev Finite Element Method

In this paper, the authors present Chebyshev finite element (CFE) method for the analysis of Reissner–Mindlin (RM) plates and shells. Chebyshev polynomials are a sequence of orthogonal polynomials that are defined recursively. The values of the polynomials belong to the interval [−1,1] and vanish at the Gauss points (GPs). Therefore, high-order shape functions, which satisfy the interpolation condition at the points, can be performed with Chebyshev polynomials. Full gauss quadrature rule was used for stiffness matrix, mass matrix and load vector calculations. Static and free vibration analyses of thick and thin plates and shells of different shapes subjected to different boundary conditions were conducted. Both regular and irregular meshes were considered. The results showed that by increasing the order of the shape functions, CFE automatically overcomes shear locking without the formation of spurious zero energy modes. Moreover, the results of CFE are in close agreement with the exact solutions even for coarse and irregular meshes.

Investigation of TLP behavior under tendon damage

Exact calculation of stiffness matrix of TLP is important for investigation of its behavior in intact and tendon damage conditions and it is necessary to design a reliable TLP.A serious attention has been concentrated on this problem in API-RP-2T and various aspects of that such as dynamic and static behavior of TLP, tendon tensions after one tether failure. Also it has introduced safety factors in different loads and healthy conditions. This paper deals with effect of damage tendon condition of TLP on behavior and tensions of tendons in wave frequency range. Equilibrium of TLP is studied and stiffness matrix of TLP is derived in tendon damage condition. Wave loads and hydrodynamic coefficients are calculated by boundary element method. Then, motion equations are solved for 48 load and healthy conditions and wave frequency portion of tensions and motions are extracted from time history of them. It is shown that although damage tendon condition has no extreme influence on response amplitude magnification, it has a relatively major effect on tension of tendons which are close to failed tendon. In addition, roll and pitch magnifications have major effect on increase in tension of tendons. Also, it is seen that critical wave heading occurs in direction of diagonal of group of failed tendon.

Fast native-MATLAB stiffness assembly for SIPG linear elasticity

When written in MATLAB the finite element method (FEM) can be implemented quickly and with significantly fewer lines, when compared to compiled code. MATLAB is also an attractive environment for generating bespoke routines for scientific computation as it contains a library of easily accessible inbuilt functions, effective debugging tools and a simple syntax for generating scripts. However, there is a general view that MATLAB is too inefficient for the analysis of large problems. Here this preconception is challenged by detailing a vectorised and blocked algorithm for the global stiffness matrix computation of the symmetric interior penalty discontinuous Galerkin (SIPG) FEM. The major difference between the computation of the global stiffness matrix for SIPG and conventional continuous Galerkin approximations is the requirement to evaluate inter-element face terms, this significantly increases the computational effort. This paper focuses on the face integrals as they dominate the computation time and have not been addressed in the existing literature. Unlike existing optimised finite element algorithms available in the literature the paper makes use of only native MATLAB functionality and is compatible with GNU Octave. The algorithm is primarily described for 2D analysis for meshes with homogeneous element type and polynomial order. The same structure is also applied to, and results presented for, a 3D analysis. For problem sizes of 106 degrees of freedom (DOF), 2D computations of the local stiffness matrices were at least ≈24 times faster, with 13.7 times improvement from vectorisation and a further 1.8 times improvement from blocking. The speed up from blocking and vectorisation is dependent on the computer architecture, with the range of potential improvements shown for two architectures in this paper.

Dam-zoning modulus reconstruction using adjoint method

Mechanical properties of ageing dams and rock foundations can differ significantly from those design values used during construction. It is necessary to conduct the inverse analysis of mechanical parameters of dam and rock foundation. However, it is inconsistent with the actual situation by utilising the conventional inversion method for structural analysis, as the conventional method can only inverse the integrated elastic modulus. Therefore, a new inversion method is proposed in this paper, which utilises adjustment coefficient to inverse initial elastic modulus, and introduces the adjoint method combined with DFP quasi Newton method in order to inverse zoning elastic modulus of dam and rock foundation. The adjoint method calculates the gradients in a much more efficient way, as it can avoid the repeated computation of large stiffness matrix. Therefore, the time consumption is reduced greatly and the computational efficiency increases significantly. Combined with project case, the feasibility and validity of the proposed method are verified.

New Formula for Geometric Stiffness Matrix Calculation

The standard formula for geometric stiffness matrix calculation, which is convenient for most engineering applications, is seen to be unsatisfactory for large strains because of poor accuracy, low convergence rate, and stability. For very large compressions, the tangent stiffness in the direction of the compression can even become negative, which can be regarded as physical nonsense. So in many cases rubber materials exposed to great compression cannot be analyzed, or the analysis could lead to very poor convergence. Problems with the standard geometric stiffness matrix can even occur with a small strain in the case of plastic yielding, which eventuates even greater practical problems. The authors demonstrate that amore precisional approach would not lead to such strange and theoretically unjustified results. An improved formula that would eliminate the disadvantages mentioned above and leads to higher convergence rate and more robust computations is suggested in this paper. The new formula can be derived from the principle of virtual work using a modified Green-Lagrange strain tensor, or from equilibrium conditions where in the choice of a specific strain measure is not needed for the geometric stiffness derivation (which can also be used for derivation of geometric stiffness of a rigid truss member). The new formula has been verified in practice with many calculations and implemented in the RFEM and SCIA Engineer programs. The advantages of the new formula in comparison with the standard formula are shown using several examples.

An effective fracture analysis method based on the virtual crack closure-integral technique implemented in CS-FEM

In this work, the virtual crack closure integral technique (VCCT) is formulated in the framework of cell-based smoothed finite element (CS-FEM) for evaluating stress intensity factors and for modeling the crack propagation in solids. In the present CS-FEM, the strain smoothing technique is operated over the smoothing domains which are constructed based on elements, and each element is further subdivided into several smoothing cells. The smoothed strain is then obtained by a boundary integral along the boundaries of the smoothing cells. Only shape function itself is involved in computing the strains and no derivatives of the shape functions or coordinate transformation is required for the computation of the discretized stiffness matrix, and thus ideal for fracture mechanics problems to evaluate the stress intensity factors, we utilize the one-step-analysis approach of the VCCT based on the assumption that an infinitesimal perturbation of crack-tip location shall not significantly affect the stress/displacement field. The significant feature of the present CS-FEM method equipped with VCCT is that it requires no domain integration in the analysis of fracture mechanics problems. Several numerical examples are presented to validate the effectiveness of the present method. It uses only the information for displacement openings behind the crack-tip and the nodal forces at the crack-tip for stress intensity factor evaluation. The comparison study has shown that is as accuracy as the FEM-Q4 that need domain integrations for both stiffness matrix computation and interaction integral methods for stress intensity factor evaluation. The present method is further used to successfully predict the crack growth trajectory with excellent agreement between numerical results and the experimental observations.

Stiffness identification of four-point-elastic-support rigid plate

As the stiffness of the elastic support varies with the physical-chemical erosion and mechanical friction, model catastrophe of a single degree-of-freedom (DOF) isolation system may occur. A 3-DOF four-point-elastic-support rigid plate (FERP) structure is presented to describe the catastrophic isolation system. Based on the newly-established structure, theoretical derivation for stiffness matrix calculation by free response (SMCbyFR) and the method of stiffness identification by stiffness matrix disassembly (SIbySMD) are proposed. By integrating the SMCbyFR and the SIbySMD and defining the stiffness assurance criterion (SAC), the procedures for stiffness identification of a FERP structure (SIFERP) are summarized. Then, a numerical example is adopted for the SIFERP validation, in which the simulated tested free response data are generated by the numerical methods, and operation for filtering noise is conducted to imitate the practical application. Results in the numerical example demonstrate the feasibility and accuracy of the developed SIFERP for stiffness identification.

P1-Nonconforming shell element and its application to topology optimization

The P1-nonconforming quadrilateral element developed by Park and Sheen [19] is employed for the formulation of a four-node degenerated shell element. The numerical stability of the P1-nonconforming quadrilateral element verified in plane elasticity problems and Stokes flow problems is investigated for the application of mitigating locking phenomena in shell problems. To facilitate the stiffness matrix computation for a non-flat general quadrilateral shell element, a nonparametric reference scheme using both affine transformation and bilinear transformation is adopted. Based on a field-consistency concept, the spurious constraints that cause locking are analyzed and an effective reduced integration scheme is found. The proposed shell element is applied to multi-physics topology optimization problems involving fluid analysis and shell analysis. For fluid analysis, the P1-nonconforming quadrilateral element is also adopted to utilize its volumetric locking-free property for incompressible materials. Vein layouts of leaves are designed by topology optimization and compared with natural vein layouts to verify the effectiveness of the proposed element.

Fast Stiffness Matrix Calculation for Nonlinear Finite Element Method

We propose a fast stiffness matrix calculation technique for nonlinear finite element method (FEM). Nonlinear stiffness matrices are constructed using Green-Lagrange strains, which are derived from infinitesimal strains by adding the nonlinear terms discarded from small deformations. We implemented a linear and a nonlinear finite element method with the same material properties to examine the differences between them. We verified our nonlinear formulation with different applications and achieved considerable speedups in solving the system of equations using our nonlinear FEM compared to a state-of-the-art nonlinear FEM.

GPU accelerated computation of the isogeometric analysis stiffness matrix

Due to high regularity across mesh elements, isogeometric analysis achieves higher accuracy per degree of freedom and improved spectrum properties, among others, compared with finite element analysis. However, this inherent feature of isogeometric analysis increases the density of the stiffness matrix and requires more elaborate numerical integration schemes for its computation. For these reasons, the assembly of the stiffness matrix in isogeometric analysis is a computationally demanding task, which needs special attention in order to be affordable for real-world applications. In this paper we address the computational efficiency of assembling the stiffness matrix using the standard element-wise Gaussian quadrature. A novel approach is proposed for the formulation of the stiffness matrix which exhibits several computational merits, among them its amenability to parallelization and the efficient utilization of the graphics processing units to drastically accelerate computations.

Research on Semi-Rigid Connections Calculation of Laminated Wood Frame Beam-Column

Generally laminated wood frame beam-column uses the additional steel bolts to connect, its performance is a semi-rigid characteristics under the loads. In the small deflection assumption conditions, this paper is based on the theory of semi-rigid connection, derived the semi-rigid rod end stiffness matrix and moment calculation formula of laminated wood structural frame beam-column element under the arbitrary loads. By SAP2000 the type of flexible coefficient is adopted to establish finite element numerical analysis based on 1# wood frame structure-building of science and technology R&D center in Hebei province. The result shows that each component is basically the same with the original design internal forces while the flexible coefficient is range of 0.41 to 0.82, the framework of semi-rigid connections in the actual calculation should be considered. The internal force calculation method is proposed, which provide a basis for wood design calculation that takes into account the semi-rigid connections.

New Numerical Quadrature of Integrand with Singularity of 1/<i>r</i> and its Application

A new quadrature method is proposed for numerical integration of integrands with the singularity of 1/r occurring at the computation of stiffness matrix when a singular physical cover is introduced to the numerical manifold method (NMM) for linear fracture problems. The detailed proof is presented, which shows the Jacobian has a factor of r that can be used to eliminate the singularity. Compared with the Duffy transformation, it proves more simple and easier to implement while owning the same precision. A numerical example in elastic fracture by the NMM is presented to illustrate the performance of the proposed method. The result has a good agreement with the reference solution.