Assembly Of Stiffness Matrix Research Articles

Homogenized Stiffness Matrix of Two-Node Elements through Experimental Flexibility tests

Homogenization is a numerical technique to obtain the equivalent mediated response of a complex material. However, there are some scenarios in which the complexity of the physical phenomena present in the real structure makes it difficult to generate a faithful Representative Volume Element (RVE). In these cases, it may turn easier and more reliable to perform a homogenization directly by experimental test results. For this reason, we present a procedure to build-up the whole stiffness matrix starting from experiments. This approach is here discussed to get a homogenised two-node beam element. The procedure requires some flexibility measurements, thus allowing fewer measurements if compared to stiffness-based approach. The method is verified with some experiments carried on a frame structure. The comparisons with a Finite Element model build up with stiffness matrix assembly demonstrates the validity and robustness of the proposed procedure.

Distributed Transfer Function Formulations for Vibration Analysis of Cracked Beam and Frame Structures

In this paper, based on distributed transfer function method (DTFM), the closed-form analytical solutions for vibrations of Euler–Bernoulli beam and frame structures with arbitrary number of cracks are studied. First, generalized DTFM is employed to characterize the dynamical model for a single cracked beam and its analytical solutions for eigenvalue problem and frequency response are obtained. Then, a new DTFM cracked element that encapsulates one crack of arbitrary location inside the beam is proposed. Using the DTFM cracked element and global dynamic stiffness matrix assembly technique, damaged frame structures of arbitrary form can be modeled for vibration analysis. Previous analytical methods only addressed low-frequency vibration of simple cracked beam structures, the proposed method can yield analytical solutions in the medium- and high-frequency regions, which is critical for the small crack detection in complex frames. Lastly, three numerical examples are given to illustrate the correctness and effectiveness of the DTFM in analyzing natural frequencies, modal shapes and frequency responses for cracked structures. By comparing with the Finite Element Method (FEM) and benchmarks from literatures, we proved that the DTFM has better performances in terms of accuracy and efficiency.

A Multilevel Extension of the GDSW Overlapping Schwarz Preconditioner in Two Dimensions

Abstract Multilevel extensions of overlapping Schwarz domain decomposition preconditioners of Generalized Dryja–Smith–Widlund (GDSW) type are considered in this paper. The original GDSW preconditioner is a two-level overlapping Schwarz domain decomposition preconditioner, which can be constructed algebraically from the fully assembled stiffness matrix. The FROSch software, which belongs to the ShyLU package of the Trilinos software library, provides parallel implementations of different variants of GDSW preconditioners. The coarse problem can limit the parallel scalability of two-level GDSW preconditioners. As a remedy, in the past, three-level GDSW approaches have been proposed, which can significantly extend the range of scalability. Here, a multilevel extension of the GDSW preconditioner is introduced and analyzed. Finally, parallel results for the implementation in FROSch for up to 40 000 cores of the SuperMUC-NG supercomputer at Leibniz Supercomputing Centre (LRZ) and to 48 000 cores of the JUWELS supercomputer at Jülich Supercomputing Centre (JSC) are presented.

Far-field blast responses of sandwich arbitrary polygonal reinforced plate system

This paper presents a spectral element model for investigating the dynamic response characteristics of the arbitrary polygonal built-up reinforced plate system subjected to far-field blast load. In the researched system, the plate elements are made of sandwich materials with functionally graded carbon nanotube reinforced (FG-CNTRC) panels and three-dimensional graphene foam (3D-GrF) core. The stiffness matrix assembly method is incorporated into the spectral element method (SEM) for establishing the overall dynamic formulas of arbitrary assembled plates. Thereinto, the governing equations of motion of plate elements are derived by the Hamilton's principle based on the simple first-order shear deformation theory (S-FSDT) retaining only four displacement variables, which provides an advantage to maintain interelement continuity conditions completely. After that, the natural frequencies and transient responses of the systems with different built-up forms are solved by introducing a golden section search algorithm and Fourier series expansion of the blast load. The admirable predictive accuracy of the present model is verified by comparing the obtained results and the reference solution from finite element method (FEM). Moreover, the effects of the CNT distribution form, CNT volume fraction, 3D-GrF pore distribution form, 3D-GrF porosity, the thickness ratio of the sandwich panel, TNT mass and explosion radius on the transient vibration properties of the system are revealed through a detailed parametric study.

Locally assembled stiffness matrix: a novel method to obtain global stiffness matrix

Locally assembled stiffness matrix: a novel method to obtain global stiffness matrix

An efficient method for vibration equations with time varying coefficients and nonlinearities

An efficient method for solving vibration equations is the basis of the vibration analysis of a cracked multistage blade–disk–shaft system. However, dynamic equations are usually time varying and nonlinear, and the time required for solving is greatly increased accompanied by an increase in the model order caused by the multistage system and the nonlinearity caused by cracks. In this article, an efficient method for solving the time varying and nonlinear vibration equation is investigated. In the proposed method, the time varying terms are transformed into constant terms, while the local nonlinear matrix of the cracked blade is separated from the assembly stiffness matrix under the constraint of proper orthogonal decomposition (POD) transformation rules. Furthermore, the POD transformations of the constant terms and the linear assembled stiffness matrix can be implemented in the pretreatment steps to achieve a more efficient POD reduction operation. This research provides a method for efficiently performing the comprehensive and rapid analysis of nonlinear vibration characteristics of rotor systems.

Efficient mass and stiffness matrix assembly via weighted Gaussian quadrature rules for B-splines

Calabrò et al. (2017) changed the paradigm of the mass and stiffness computation from the traditional element-wise assembly to a row-wise concept, showing that the latter one offers integration that may be orders of magnitude faster. Considering a B-spline basis function as a non-negative measure, each mass matrix row is integrated by its own quadrature rule with respect to that measure. Each rule is easy to compute as it leads to a linear system of equations, however, the quadrature rules are of the Newton–Cotes type, that is, they require a number of quadrature points that is equal to the dimension of the spline space. In this work, we propose weighted quadrature rules of Gaussian type which require the minimum number of quadrature points while guaranteeing exactness of integration with respect to the weight function. The weighted Gaussian rules arise as solutions of non-linear systems of equations. We derive rules for the mass and stiffness matrices for uniform C1 quadratic and C2 cubic isogeometric discretizations. In each parameter direction, our rules require locally only p+1 quadrature points, p being the polynomial degree. While the nodes cannot be reused for various weight functions as in Calabrò et al. (2017), the computational cost of the mass and stiffness matrix assembly is comparable.

Node-to-Node Realization of Meshless Local Petrov Galerkin (MLPG) Fully in GPU

This paper presents an end-to-end massively parallelized procedure for the solution of boundary value problems on Graphics Processing Units (GPU). The proposal is an integrated strategy that not only entails the calculation of nodal contributions, and the stiffness matrix assembly using the Meshless Local Petrov Galerkin Method (MLPG) but also the iterative solution of the system of algebraic equations in combination with methods from the Conjugate Gradient (CG) family. This end-to-end solution is fully developed using the Compute Unified Device Architecture (CUDA) platform without the need for extra data movement between the device and host after the matrix assembly. This is possible thanks to the parallel nature of the MLPG; each node designates a thread on the device. The introduced solution is wholly executed in the GPU, with minimal auxiliary structures or global synchronization points. The proposed approach was applied to the solution of a simple electromagnetic problem, and a sevenfold speedup was observed.

Algorithm for solving problems related to the natural vibrations of electro-viscoelastic structures with shunt circuits using ANSYS data

ABSTRACTAn algorithm for numerical realisation of a mathematical statement of the natural vibrations problem for electro-viscoelastic bodies with passive external electric circuits (i.e. shunting circuits) with an arbitrary configuration using the finite element method is proposed in the present paper. The proposed algorithm allows considering the viscoelastic properties of materials using the model of linear hereditary viscoelasticity with complex dynamic moduli and is used to solve 3D solid structure problems that are compatible for ANSYS package element types. This technique implies the usage of the global assembled matrices of stiffness and mass, formed in the ANSYS package. The basis of the algorithm is a novel approach that allows performing decomposition of the global assembled stiffness matrix formed in the ANSYS software package into constituents that are needed for calculation of the natural vibration frequencies of the objects under study. These matrix components are used in the program that was written in FORTRAN (Formula Translation) language. This problem could be efficiently applied for analysis of the dynamic processes in smart systems based on piezoelectric materials and could also form a basis for the development of numerical finite element algorithms for optimization of the dissipative characteristics of electromechanical systems with shunted piezoelectric elements.

Three-Dimensional Parametric Finite-Volume Homogenization of Periodic Materials with Multi-Scale Structural Applications

In order to satisfy the increasing computational demands of micromechanics, the Finite-Volume Direct Averaging Micromechanics (FVDAM) theory is developed in three-dimensional (3D) domain to simulate the multiphase heterogeneous materials whose microstructures are distributed periodically in the space. Parametric mapping, which endorses arbitrarily shaped and oriented hexahedral elements in the microstructure discretization, is employed in the unit cell solution. Unlike the finite-element (FE) technique, the expressions for local stiffness matrices are derived explicitly, enabling efficient global stiffness matrix assembly using an easily implementable algorithm. To demonstrate the accuracy and efficiency of the proposed theory, the homogenized moduli and localized stress distributions produced by the FE analyses are given for comparisons, where excellent agreement is always obtained for the 3D microstructures with different geometrical and material properties. Finally, a multi-scale stress analysis of functionally graded composite cylinders is conducted. This extension further increases the FVDAM’s range of applicability and opens new opportunities for pursuing other areas, providing an attractive alternative to the FE-based approaches that may be compared.

Vibration performance evaluation of planar flexible multibody systems with joint clearance

Clearances are necessary in assemblage of mechanisms to allow the relative motion between the members. This clearance is due to machining tolerances, wear, material deformations, and imperfections, and it can worsen mechanism performance such as precision and vibration. As a new study in this topic, the effect of joint stiffness on the variation of instantaneous natural frequencies and mode shapes of a flexible four-bar mechanism with a clearance between coupler and follower is studied in this paper. To model the clearance, the continuous contact approach is used. The Lankarani’s and Nikravesh’s continuous contact force model is used to model the contact force arising from contact between journal and bearing. Finite element method is used to determine the instantaneous natural frequencies and their corresponding mode shapes. The stiffness of the clearance is modeled as a linear spring added to the assembled stiffness matrix. To validate the clearance model in rigid mechanism, the dynamic response is compared with the results in the literature. To show the validity of the formulation which calculates the instantaneous natural frequencies, two methods are used and compared with each other in the case no clearance exists. The results show that taking the joint stiffness into account has a considerable effect on the instantaneous natural frequencies and their corresponding mode shapes of a flexible multibody system.

A consistent implementation for inelastic materials in an ALE formulation for steady state rolling contact

AbstractThe numerical analysis of rolling contact for rubber materials is a challenging task, especially due to the many nonlinearities inherent to the material, large deformations, friction, and energy dissipation, among others. Industrial applications can be found in ball bearings, rollers, and most commonly in tires of vehicles, applications where reliable numerical simulations lead to the improvement of durability, performance and safety. While a transient analysis stands as a practical and powerful tool for the simulation of rotating bodies, the large amount of computational resources required represents its biggest disadvantage. An alternative frequently used lays in a steady state simulation by means of an Arbitrary Lagrangian Eulerian (ALE) formulation, where the rotational velocity and axial loads are assumed to remain constant. Within this framework, the reference configuration is neither attached to the material particles nor fixed in space and special attention should be paid to the history variables of inelastic materials. In this work, a viscoelastic material model is implemented in an in‐house finite element code, based on a generalized Maxwell model. The implementation takes into consideration the contribution of all elements connected in circumferential direction and a consistent linearization is made for each of them, leading to an assembled stiffness matrix with more non‐zero values than a standard one. This approach is combined with smeared reinforcement embedded in base elements. The reinforcing layers are described by a hyperelastic material model, providing additional advantages for the modeling and simulation of reinforced rollers and tires. Numerical results for different examples show the capabilities of this implementation and the efficiency of the numerical algorithms is discussed. Important remarks and an outlook for further research concludes this presentation. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Improved cache utilization and preconditioner efficiency through use of a space-filling curve mesh element- and vertex-reordering technique

Solving partial differential equations using finite element (FE) methods for unstructured meshes that contain billions of elements is computationally a very challenging task. While parallel implementations can deliver a solution in a reasonable amount of time, they suffer from low cache utilization due to unstructured data access patterns. In this work, we reorder the way the mesh vertices and elements are stored in memory using Hilbert space-filling curves to improve cache utilization in FE methods for unstructured meshes. This reordering technique enumerates the mesh elements such that parallel threads access shared vertices at different time intervals, reducing the time wasted waiting to acquire locks guarding atomic regions. Further, when the linear system resulting from the FE analysis is solved using the preconditioned conjugate gradient method, the performance of the block-Jacobi preconditioner also improves, as more nonzeros are present near the stiffness matrix diagonal. Our results show that our reordering reduces the L1 and L2 cache miss-rates in the stiffness matrix assembly step by about 50 and 10 %, respectively, on a single-core processor. We also reduce the number of iterations required to solve the linear system by about 5 %. Overall, our reordering reduces the time to assemble the stiffness matrix and to solve the linear system on a 4-socket, 48-core multi-processor by about 20 %.

PARALLEL COMPUTING TECHNOLOGIES IN THE FINITE ELEMENT METHOD

Nowadays engineers and researchers are faced with solving very complex problems in a mathematical modeling and design. Numerical analysis naturally finds applications in all fields of engineering and the physical sciences. The finite element method is a powerful tool for the numerical simulation of a wide range of problems. Implementation of the finite element method in CAD systems on the basis of modern computers allows researchers to solve large scale problems. The finite element method uses a discretization of a continuum domain into a mesh as the starting point. The discretization of complex domains may give large numbers of elements, thus increasing the requirements of computer memory and speed. Modern parallel computers use multiple processing elements simultaneously to solve a problem. Thus implementation of parallel computing into the finite element method is urgent direction of the research. In the finite element method for the numerical solution of partial differential equations, the stiffness matrix represents the system of linear equations that must be solved in order to ascertain an approximate solution to the differential equation. Therefore the article describes parallel algorithms for assembly of stiffness matrix and for solution of linear equations. Also this article contains two numerical experiments: 1) Dirichlet problem on the complex domain (gear); 2) Elasticity problem of three-layer shell. In the end of the article author compares the speed of a solution of these problems with one, two and four parallel cores and makes discussion about the effectiveness of parallel finite element processing.

Discontinuous cellular automaton method for crack growth analysis without remeshing

A numerical technical of discontinuous cellular automaton method for crack growth analysis without remeshing is developed. In this method, the level set method is employed to track the crack location and its growth path, where the level set functions and calculation grids are independent, so no explicit meshing for crack surface and no remeshing for crack growth are needed. Then, the discontinuous enrichment shape functions which are enriched by the Heaviside function and the exact near-tip asymptotic field functions are constructed to model the discontinuity of cracks. Finally, a discontinuous cellular automaton theory is proposed, which are composed of cell, neighborhood and updating rules for discontinuous case. There is an advantage that the calculation is only applied on local cell, so no assembled stiffness matrix but only cell stiffness is needed, which can overcome the stiffness matrix assembling difficulty caused by unequal degrees of nodal freedom for different cells, and much easier to consider the local properties of cells. Besides, the present method requires much less computer memory than that of XFEM because of it local property.Combined level set method, the discontinuous enrichment shape functions and discontinuous cellular automaton theory, the discontinuous cellular automaton method is proposed, which can conveniently achieve the analysis from continuity to discontinuity. Numerical examples are given to illustrate that the present method is effective, and can be further extended into practical engineering.

A continuous–discontinuous cellular automaton method for regular frictional contact problems

In the present paper, a continuous–discontinuous cellular automaton method is developed to model the discontinuous problems caused by regular frictional contact, in which level set method, discontinuous enriched shape function, discontinuous cellular automaton method and contact friction theory are combined, by which an analysis from continuity to discontinuity can be achieved, and no assembled stiffness matrix but only cell stiffness is needed in the whole calculation, because of the use of discontinuous cellular automaton method. In the present method, level set method is used to track the discontinuous surface, and discontinuous enriched shape function is employed to describe the discontinuity of displacement and stress. Contact friction theory is applied to construct the Coulomb frictional contact model of discontinuous surfaces; besides, combined with discontinuous cellular automaton method, a new mixed iteration method is proposed to obtain the solution of the problem, and no assembled stiffness matrix is constructed. And the frictional contact iterations are done simultaneously with the updating of cellular automaton, in which the contact states and contact areas can be previously obtained in the cellular automaton updating process, and the efficiency can get much higher. Finally, some numerical examples are given to show that the present method is effective and accurate and can be further extended into some practical engineering.

An Overlapping Schwarz Algorithm for Raviart--Thomas Vector Fields with Discontinuous Coefficients

Overlapping Schwarz methods form one of two major families of domain decomposition methods. In this paper, a two-level overlapping Schwarz method for Raviart--Thomas vector fields is developed and analyzed. The global component of the preconditioner is based on local energy-minimizing discrete harmonic extensions of coarse interface values, and the local components use local direct solvers on overlapping subdomains. The algorithm can be implemented algebraically using submatrices of a fully assembled stiffness matrix of the entire finite element problem. Moreover, it gives a condition number bound which depends only on the size of the local subproblems. The condition number of our method is shown to be bounded by $C ( 1 + \log (H/h)) ( 1 + H/\delta)$, where $H$ is the diameter of the subdomain and $h$ and $\delta$ are the sizes of the elements and the overlap between subdomains, respectively. The bound of the condition number is independent of the values and jumps of the coefficients across the interface between the subdomains. Numerical results, which support the theory, are also presented.

Bending Solution of Plates on Winkler Foundation by Element-Free Galerkin Method Based on Mindlin Plate Theory

Element-free Galerkin method (EFGM) is successfully applied to solve the bending problem of plates on a Winkler foundation. The shape function which is characteristic of high-time continuity is formulated by means of weight function. A way to incorpor- ate the self-adaptive influential radius in weight function is proposed. Based on variational principle, this paper derives control equation for the bending of plates on a Winkler foundation from Mindlin-Reissner plate theory. Using penalty function mothed, assembled stiffness matrix which is real symmetry positive definite matrix is deduced. This method can solve the bending problem of plates with different boundary conditions on a Winkler foundation. The corresponding computer programs of EFGM and post-process programs are also developed. Numerical examples show that EFGM solving the bending of plates on a Winkler found is reasonable and feasible. This present study provides a newly effective numerical method for the Winkler foundation bending problem and expands the application field of EFGM.

A posteriori error analysis of the heterogeneous multiscale method for homogenization problems

In this Note we derive a posteriori error estimates for a multiscale method, the so-called heterogeneous multiscale method, applied to elliptic homogenization problems. The multiscale method is based on a macro-to-micro formulation. The macroscopic method discretizes the physical problem in a macroscopic finite element space, while the microscopic method recovers the unknown macroscopic data on the fly during the macroscopic stiffness matrix assembly process. We propose a framework for the analysis allowing to take advantage of standard techniques for a posteriori error estimates at the macroscopic level and to derive residual-based indicators in the macroscopic domain for adaptive mesh refinement. To cite this article: A. Abdulle, A. Nonnenmacher, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

A new finite element to represent prismatic joint constraints in mechanisms

Among existing kinematic analysis methods of mechanisms, the techniques based on finite elements represent a generally applicable alternative which enable a wide variety of problems to be solved, including linear (velocities, accelerations, jerk, …) and non-linear ones (position). To modelize a mechanism via these techniques, the link element may be used to introduce a distance constraint between two points. The stiffness matrix assembly of these link elements enables stiffness matrix construction from the model, from which the kinematic behaviour of the mechanism may be extracted. Normally kinematic link conditions introduced directly into the system stiffness matrix are used to introduce point to line constraints like those originated by prismatic joints. A new finite element is presented in this paper, defined by its stiffness or geometric matrix, capable of alternatively modelizing the constraints imposed by the prismatic joint. This new element offers numerous advantages against the procedure based on anterior link conditions, particularly in the case of non-linear problems.