scholarly journals An offline-online strategy for multiscale problems with random defects

2022 ◽  
Author(s):  
Axel Målqvist ◽  
Barbara Verfürth
Keyword(s):  
Stiffness Matrix ◽  
Numerical Experiments ◽  
Linear Combinations ◽  
Reference Element ◽  
Multiscale Problems ◽  
Stiffness Matrices ◽  
Online Strategy ◽  
Local Defects ◽  
Random Defects ◽  
Selection Of

In this paper, we propose an offline-online strategy based on the Localized Orthogonal Decomposition (LOD) method for elliptic multiscale problems with randomly perturbed diffusion coefficient. We consider a periodic deterministic coefficient with local defects that occur with probability $p$. The offline phase pre-computes entries to global LOD stiffness matrices on a single reference element (exploiting the periodicity) for a selection of defect configurations. Given a sample of the perturbed diffusion the corresponding LOD stiffness matrix is then computed by taking linear combinations of the pre-computed entries, in the online phase. Our computable error estimates show that this yields a good approximation of the solution for small $p$, which is illustrated by extensive numerical experiments.  This makes the proposed technique attractive already for moderate sample sizes in a Monte Carlo simulation.

scholarly journals Diagonal scaling of stiffness matrices in the Galerkin boundary element method

2000 ◽  
Vol 42 (1) ◽  
pp. 141-150 ◽  
Author(s):  
Mark Ainsworth ◽  
Bill McLean ◽  
Thanh Tran
Keyword(s):  
Integral Equation ◽  
Stiffness Matrix ◽  
Numerical Experiments ◽  
Degrees Of Freedom ◽  
Boundary Integral ◽  
Piecewise Constant ◽  
Stiffness Matrices ◽  
Diagonal Scaling ◽  
Mesh Ratio ◽  
Galerkin Boundary Element Method

AbstractA boundary integral equation of the first kind is discretised using Galerkin's method with piecewise-constant trial functions. We show how the condition number of the stiffness matrix depends on the number of degrees of freedom and on the global mesh ratio. We also show that diagonal scaling eliminates the latter dependence. Numerical experiments confirm the theory, and demonstrate that in practical computations involving strong local mesh refinement, diagonal scaling dramatically improves the conditioning of the Galerkin equations.

Physical Realization of Generic-Element Parameters in Model Updating

2002 ◽  
Vol 124 (4) ◽  
pp. 628-633 ◽  
Author(s):  
H. Ahmadian ◽  
J. E. Mottershead ◽  
M. I. Friswell
Keyword(s):  
Model Updating ◽  
Finite Element Models ◽  
Physical Realization ◽  
Governing Equations ◽  
Stiffness Matrices ◽  
Model Predictions ◽  
Physical Effects ◽  
Realization Process ◽  
Physical Phenomena ◽  
Selection Of

The selection of parameters is most important to successful updating of finite element models. When the parameters are chosen on the basis of engineering understanding the model predictions are brought into agreement with experimental observations, and the behavior of the structure, even when differently configured, can be determined with confidence. Physical phenomena may be misrepresented in the original model, or may be absent altogether. In any case the updated model should represent an improved physical understanding of the structure and not simply consist of unrepresentative numbers which happen to cause the results of the model to agree with particular test data. The present paper introduces a systematic approach for the selection and physical realization of updated terms. In the realization process, the discrete equilibrium equation formed by mass, and stiffness matrices is converted to a continuous form at each node. By comparing the resulting differential equation with governing equations known to represent physical phenomena, the updated terms and their physical effects can be recognized. The approach is demonstrated by an experimental example.

A Heuristic Algorithm for Feature Selection Based on Optimization Techniques

2011 ◽  
pp. 13-26 ◽  
Author(s):  
A. M. Bagirov ◽  
A. M. Rubinov ◽  
J. Yearwood
Keyword(s):  
Feature Selection ◽  
Heuristic Algorithm ◽  
Real World ◽  
Numerical Experiments ◽  
Optimization Techniques ◽  
Selection Problem ◽  
Feature Selection Problem ◽  
Selection Of

The feature selection problem involves the selection of a subset of features that will be sufficient for the determination of structures or clusters in a given dataset and in making predictions. This chapter presents an algorithm for feature selection, which is based on the methods of optimization. To verify the effectiveness of the proposed algorithm we applied it to a number of publicly available real-world databases. The results of numerical experiments are presented and discussed. These results demonstrate that the algorithm performs well on the datasets considered.

Elastoplastic Analysis of Layered Metal Matrix Composite Cylinders—Part I: Theory

1996 ◽  
Vol 118 (1) ◽  
pp. 13-20 ◽  
Author(s):  
R. S. Salzar ◽  
M.-J. Pindera ◽  
F. W. Barton
Keyword(s):  
Metal Matrix Composite ◽  
Matrix Composite ◽  
Stiffness Matrix ◽  
Metal Matrix ◽  
Plastic Response ◽  
Solution Strategy ◽  
Composite Tubes ◽  
Elastic Plastic ◽  
Stiffness Matrices ◽  
Local Stiffness

An exact elastic-plastic analytical solution for an arbitrarily laminated metal matrix composite tube subjected to axisymmetric thermo-mechanical and torsional loading is presented. First, exact solutions for transversely isotropic and monoclinic (off-axis) elastoplastic cylindrical shells are developed which are then reformulated in terms of the interfacial displacements as the fundamental unknowns by constructing a local stiffness matrix for the shell. Assembly of the local stiffness matrices into a global stiffness matrix in a particular manner ensures satisfaction of interfacial traction and displacement continuity conditions, as well as the external boundary conditions. Due to the lack of a general macroscopic constitutive theory for the elastic-plastic response of unidirectional metal matrix composites, the micromechanics method of cells model is employed to calculate the effective elastic-plastic properties of the individual layers used in determining the elements of the local and thus global stiffness matrices. The resulting system of equations is then solved using Mendelson’s iterative method of successive elastic solutions developed for elastoplastic boundary-value problems. Part I of the paper outlines the aforementioned solution strategy. In Part II (Salzar et al., 1996) this solution strategy is first validated by comparison with available closed-form solutions as well as with results obtained using the finite-element approach. Subsequently, examples are presented that illustrate the utility of the developed solution methodology in predicting the elastic-plastic response of arbitrarily laminated metal matrix composite tubes. In particular, optimization of the response of composite tubes under internal pressure is considered through the use of functionally graded architectures.

A New Finding on the Dynamic Stiffness Matrices of Asymmetric and Axisymmetric Shafts

2001 ◽  
Author(s):  
Francesco A. Raffa ◽  
Furio Vatta
Keyword(s):  
Stiffness Matrix ◽  
Dynamic Stiffness ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Rotational Inertia ◽  
Dynamic Stiffness Matrix ◽  
Rayleigh Beam ◽  
Principal Moments Of Inertia ◽  
Stiffness Matrices ◽  
New Finding

Abstract In this paper the dynamic stiffness method is developed to analyze a rotating asymmetric shaft, i.e. a shaft whose transverse section is characterized by dissimilar principal moments of inertia. The shaft is modeled according to the Rayleigh beam theory including the effects of both translational and rotational inertia, and gyroscopic moments. The mathematical description is carried out in a reference system rotating at the shaft speed and is based on the exact solution of the governing differential equations of motion. The exact expressions of the shaft displacements are utilized for deriving the 8 × 8 complex dynamic stiffness matrix of the shaft. A new relationship is obtained which links the dynamic stiffness matrix of the asymmetric shaft to the 4 × 4 real dynamic stiffness matrix of the axisymmetric shaft.

scholarly journals Fast Stiffness Matrix Calculation for Nonlinear Finite Element Method

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Emir Gülümser ◽  
Uğur Güdükbay ◽  
Sinan Filiz
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Stiffness Matrix ◽  
Nonlinear Finite Element ◽  
Nonlinear Finite Element Method ◽  
Nonlinear Fem ◽  
Calculation Technique ◽  
Stiffness Matrices ◽  
Matrix Calculation ◽  
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.

scholarly journals Exact and Direct Modeling Technique for Rotor-Bearing Systems with Arbitrary Selected Degrees-of-Freedom

1994 ◽  
Vol 1 (6) ◽  
pp. 497-506 ◽  
Author(s):  
Shilin Chen ◽  
Michel Géradin
Keyword(s):  
Stiffness Matrix ◽  
Degrees Of Freedom ◽  
Dynamic Stiffness ◽  
Modeling Technique ◽  
Dynamic Stiffness Matrix ◽  
Global Dynamic ◽  
Combination Methods ◽  
Stiffness Matrices ◽  
Direct Modeling ◽  
Rotor Bearing

An exact and direct modeling technique is proposed for modeling of rotor-bearing systems with arbitrary selected degrees-of-freedom. This technique is based on the combination of the transfer and dynamic stiffness matrices. The technique differs from the usual combination methods in that the global dynamic stiffness matrix for the system or the subsystem is obtained directly by rearranging the corresponding global transfer matrix. Therefore, the dimension of the global dynamic stiffness matrix is independent of the number of the elements or the substructures. In order to show the simplicity and efficiency of the method, two numerical examples are given.

Reckoning for Element Characteristics Matrix of Space Timoshenko-Beam Based on Energy Variational Principle

2011 ◽  
Vol 66-68 ◽  
pp. 1356-1361
Author(s):  
Wen Jun Pan ◽  
Zhi Wu Wei
Keyword(s):  
Variational Principle ◽  
Timoshenko Beam ◽  
Stiffness Matrix ◽  
Mass Matrix ◽  
Research Work ◽  
Energy Functional ◽  
Displacement Function ◽  
Matrix Stiffness ◽  
Stiffness Matrices

To analyze and calculate the element characteristics matrices of space Timoshenko-beam, research work were carried out on the basis of energy variational principle. Displacement function for the space Timoshenko-beam were put forward, the expressions for element mass matrix, stiffness matrix and load array were deduced by energy functional extremum, and the explicit forms of element mass and stiffness matrices were integrated finally. Results show that the element mass and stiffness matrices computed by this method are consistent with those in related references. It has a good theoretical and practical value in the calculation for characteristics matrices of other elements.

Derivation of New Transcendental Member Stiffness Determinant for Vibrating Frames

2003 ◽  
Vol 03 (02) ◽  
pp. 299-305 ◽  
Author(s):  
F. W. Williams ◽  
D. Kennedy
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Stiffness Matrix ◽  
Infinite Order ◽  
Natural Frequencies ◽  
Dynamic Stiffness ◽  
Structural Stiffness ◽  
Trial And Error ◽  
Stiffness Matrices ◽  
Vibration Problems

Transcendental dynamic member stiffness matrices for vibration problems arise from solving the governing differential equations to avoid the conventional finite element method (FEM) discretization errors. Assembling them into the overall dynamic structural stiffness matrix gives a transcendental eigenproblem, whose eigenvalues (natural frequencies or their squares) are found with certainty using the Wittrick–Williams algorithm. This paper gives equations for the recently discovered transcendental member stiffness determinant, which equals the appropriately normalized FEM dynamic stiffness matrix determinant of a clamped ended member modelled by infinitely many elements. Multiplying the overall transcendental stiffness matrix determinant by the member stiffness determinants removes its poles to improve curve following eigensolution methods. The present paper gives the first ever derivation of the Bernoulli–Euler member stiffness determinant, which was previously found by trial-and-error and then verified. The derivation uses the total equivalence of the transcendental formulation and an infinite order FEM formulation, which incidentally gives insights into conventional FEM results.

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