Non-symmetric Stiffness Matrix Research Articles

Rotationally invariant viscoelastic medium with a non-symmetric stiffness matrix

Rotationally invariant viscoelastic medium with a non-symmetric stiffness matrix

Representation theorem for viscoelastic waves with a non-symmetric stiffness matrix

In an elastic medium, it was proved that the stiffness tensor is symmetric with respect to the exchange of the first pair of indices and the second pair of indices, but the proof does not apply to a viscoelastic medium. In this paper, we thus derive the representation theorem for viscoelastic waves in a medium with a non-symmetric stiffness matrix. The representation theorem expresses the wave field at a receiver, situated inside a subset of the definition volume of the viscoelastodynamic equation, in terms of the volume integral over the subset and the surface integral over the boundary of the subset. For the given medium, we define the complementary medium corresponding to the transposed stiffness matrix. We define the frequency-domain complementary Green function as the frequency-domain Green function in the complementary medium. We then derive the provisional representation theorem as the relation between the frequency-domain wave field in the given medium and the frequency-domain complementary Green function. This provisional representation theorem yields the reciprocity relation between the frequency-domain Green function and the frequency-domain complementary Green function. The final version of the representation theorem is then obtained by inserting the reciprocity relation into the provisional representation theorem.

Frequency-domain ray series for viscoelastic waves with a non-symmetric stiffness matrix

In an elastic medium, it was proved that the stiffness tensor is symmetric with respect to the exchange of the first pair of indices and the second pair of indices, but the proof does not apply to a viscoelastic medium. In order to indicate which phenomena could be observed in the wave field if the stiffness matrix were non–symmetric, we propose the frequency–domain ray series for viscoelastic waves with a non–symmetric stiffness tensor in this paper.

A finite element formulation preserving symmetric and banded diffusion stiffness matrix characteristics for fractional differential equations

Due to the nonlocal property of the fractional derivative, the finite element analysis of fractional diffusion equation often leads to a dense and non-symmetric stiffness matrix, in contrast to the...

Asynchronous modes of vibrations in linear conservative systems: an illustrative discussion of plane framed structures

The theme of asynchronous modes of vibration is recast in this paper to address the case of linear conservative discrete systems. In fact, it is known that linear non-conservative discrete systems such as those characterized by a non-symmetric stiffness matrix may undergo asynchronous free vibrations. Yet, non-conservativeness is not mandatory for the occurrence of asynchronous modes, as it is illustrated in this investigation of simple plane frames, provided the system parameters are properly chosen. Firstly, a one-storey-frame model, with consistent-mass and stiffness matrices, is studied and conditions for the occurrence of asynchronous modes are established. Next, a three-storey shear building is modelled, with lumped-mass matrix, and isolated asynchronous modes are not observed. However, non-isolated asynchronous modes are seen to exist, including a case in which the first floor is at rest. In this scenario, the upper floors play the role of vibration controllers, sparing the lower columns from oscillating. This might be welcome from the architectural viewpoint, so that pilotis would not be a priori ruled out in case of earthquake-borne vibration. Usually the participating masses of the asynchronous modes are low (and even null). To maximize the asynchronous-mode participating mass, a final model of the three-storey frame is discussed, this time taking into account the flexural stiffness of both the columns and the beams, although still using lumped-mass matrix. In this case, it is seen that much larger participating masses can be obtained for the asynchronous mode. Since asynchronous modes are associated with the vibration localization phenomenon, they can be potentially explored in the design of vibration controllers or, alternatively, energy harvesting systems.

A numerical study on the symmetrization of tangent stiffness matrix in non-linear analysis using corotational approach

In the corotational kinematics (EIRC), the movement is decomposed in deformational and rigid body components using projection operators. Large rigid body translations and rotations, and infinitesimal deformation tensors are adopted. In this way, a non-symmetric stiffness matrix is obtained for the finite element. Literature points that this matrix may be symmetrized in the usual way and the original non-symmetric matrix tends to be in a symmetric form as equilibrium is approached. This paper performed symmetrization studies using Frobenius norm and Absolute Maximum Coefficient of the anti-symmetric part of stiffness matrix in several numerical analyses with high non-linearity solved with an incremental-iterative scheme. An element independent corotational (EICR) kinematics is used to analyze non-linear spatial frames with 3D Euler–Bernoulli beam elements. The results show that this symmetrization not always occurs and depends significantly on the magnitude of displacements and finite-element mesh refinement. The authors demonstrated that, in some cases, the symmetrization would only take place with very refined meshes when the discretized structure approaches the continuum. In this way, the current literature and general mathematical proof of that symmetrization have to be restated.

The meshfree strong form methods for solving one dimensional inverse Cauchy-Stefan problem

In this paper, we extend the application of meshfree node based schemes for solving one-dimensional inverse Cauchy-Stefan problem. The aim is devoted to recover the initial and boundary conditions from some Cauchy data lying on the admissible curve s(t) as the extra overspecifications. To keep matters simple, the problem has been considered in one dimensional, however the physical domain of the problem is supposed as an irregular bounded domain in $$\mathbb {R}^2$$R2. The methods provide the space-time approximations for the heat temperature derived by expanding the required approximate solutions using collocation scheme based on radial point interpolation method (RPIM). The proposed method makes appropriate shape functions which possess the important Delta function property to satisfy the essential conditions automatically. In addition, to conquer the ill-posedness of the problem, particular optimization technique has been applied for solving the system of equations $$Ax=b$$Ax=b in which A is a nonsymmetric stiffness matrix. As the consequences, reliable approximate solutions are obtained which continuously depend on input data.

A Strategy for Solving the Non Symmetries Arising in Nonlinear Consolidation of Partially Saturated Soils

The main scope of this paper is to present an alternative to tackle the problem of the non symmetries arising in the solution of the nonlinear couple consolidation problem based on a combination of different stress states. Being originally a non symmetric problem, it may be straightforward reduced to a symmetric one, and the conditions in which this reduction may be carried out, are addressed. Non linear saturation-suction and permeability-suction functions were regarded. The geometric model was developed considering an updated lagrangian description with a co-rotated Kirchhoff stress tensor. This description leads to a non-symmetric stiffness matrix and a simple alternative, using a symmetric constitutive matrix, is addressed to overcome this situation. The whole equation system was solved using an open finite element code FECCUND, developed by the authors. In order to validate the model, various examples, for which previous solutions are known, were solved. The use of either a strongly non linear and no symmetric formulation or a simple symmetric formulation with accurate prediction in deformation and pore-pressures is extremely dependent on the soil characteristic curves and on the shear efforts level, as well. A numerical example show the predictive capability of this geometrically non linear fully coupled model for attaining the proposed goal.

Dynamic asymmetry of piezoelectric shell structures

The non-symmetry and asymmetric dynamic characteristics of piezoelectric shell structures are studied analytically and numerically. The basic formulations of piezoelectric structures are firstly given in the tensor form. The differential equations for displacements and electric potential are derived from the formulations, which validate that the description in curvilinear coordinates with the electrical and mechanical coupling induces the non-symmetry in generalized stiffness and then the dynamic asymmetry of piezoelectric structures. Also expanding the structural displacements and electric potential with various spatial characteristics can induce the non-symmetry in generalized stiffness matrix of the converted ordinary differential equations. Then, for the multi-degree-of-freedom system with asymmetric stiffness matrix, the conventional right modes or eigenvectors are proved to have not the auto-orthogonality relations with the mass or stiffness weight. However, the left and right eigenvectors have the cross-orthogonality relations with the mass or stiffness weight, which can be used for uncoupling the asymmetric systems. Furthermore, the non-symmetric stiffness matrix is divided into the corresponding symmetric and anti-symmetric stiffness matrices. The algebraic equations for eigenvalues and singular values of the asymmetric, symmetric and anti-symmetric systems are given. It is obtained that the eigenvalues are equal to the corresponding singular values for the symmetric system, the singular values are equal to the products of the unit imaginary number and corresponding eigenvalues for the anti-symmetric system, and the differences of the eigenvalues and corresponding singular values for the asymmetric system depend on the anti-symmetric stiffness. Also the upper limits of the absolute and relative differences of the singular values of the asymmetric system to the corresponding eigenvalues of the symmetric system, and the upper and lower limits of the singular values of the asymmetric system are obtained, respectively. Finally, the spherically symmetric piezoelectric shell described in spherical coordinates is studied in detail to show the asymmetric dynamic characteristics. The differential equations for the radial displacement and electric potential of the shell structure are obtained to illustrate the non-symmetric generalized stiffness involving elastic and piezoelectric constants. Eliminating the electric potential, expanding the displacement in space, and using the Galerkin method yield the ordinary differential equations, which represent a multi-degree-of-freedom dynamic system with the asymmetric generalized stiffness matrix. Numerical results are given to illustrate the eigenvalues and modes of the asymmetric system different from those of the corresponding symmetric system, the relative differences of the eigenvalues of the asymmetric to symmetric system for different piezoelectric constants and geometric parameters, and the non-orthogonality of the left or right eigenvectors for the asymmetric system. The analytical and numerical results on the non-symmetric dynamics of piezoelectric shell structures are useful for accurate analysis and design.

A new rigid‐viscoplastic model for simulating thermal strain effects in metal‐forming processes

AbstractA new rigid‐viscoplastic model that includes the effect of thermal strains when modelling steady‐state metal‐forming processes was developed. A symmetric approximation to the resulting non‐symmetric stiffness matrix was derived. The thermo‐mechanical flow formulation was implemented using the pseudo‐concentrations technique. The new formulation was numerically tested showing that it provides reliable results. Copyright © 2003 John Wiley & Sons, Ltd.

On Two Variants of an Algebraic Wavelet Preconditioner

A recursive method of constructing preconditioning matrices for the nonsymmetric stiffness matrix in a wavelet basis is proposed for solving a class of integral and differential equations. It is based on a level-by-level application of the wavelet scales decoupling the different wavelet levels in a matrix form just as in the well-known nonstandard form. The result is a powerful iterative method with built-in preconditioning leading to two specific algebraic multilevel iteration algorithms: one with an exact Schur preconditioning and the other with an approximate Schur preconditioning. Numerical examples are presented to illustrate the efficiency of the new algorithms.

Dynamic stability of finite dimensional linearly elastic systems with unilateral contact and Coulomb friction

Necessary and sufficient conditions are established for the occurrence of dynamic instabilities in finite dimensional linearly elastic systems in unilateral frictional contact with a rigid flat surface. These conditions apply, in particular, to the systems that result from the finite element discretization of linearly elastic bodies. From a numerical point of view, these conditions lead to studying eigenproblems relative to a non-symmetric (tangent) stiffness matrix that incorporates the effect of the current state of the contract candidate particles. Illustrative small-sized examples are presented, together with an application to the case of an experimentally tested block of polyurethane, where friction induced instability phenomena were observed.

Finite element algorithm for jointed concrete pavements subjected to moving aircraft

An algorithm based on the finite element method is developed to analyze the dynamic response of multiple, jointed concrete pavements to moving aircraft loads. In the finite element idealization, the pavement-subgrade system is idealized by thin plate finite elements resting on Winkler-type viscoelastic foundation represented by a series of distributed springs and dashpots. The dowel bars at the transverse joints are represented by beam elements. It i assumed that the dowel bar is fully embedded into the pavement thus neglecting dowel-pavement interaction effects. The longitudinal keyed or aggregate interlock joints are modeled by vertical spring elements. The dynamic aircraft-pavement interaction effects are considered in the analysis by modeling the aircraft by masses supported by spring-dashpot systems representing the landing gear of the aircraft. It is assumed that the aircraft travels along a straight line with a specific initial velocity and acceleration. The aircraft-pavement interaction takes the form of two sets of coupled equations which result ina non-symmetric stiffness matrix. An approximate mixed iteration-direct elimination scheme is used to solve for the dynamic equations. The accuracy of the computer code is verified by the available experimental and analytical solutions. A parametric study is conducted to investigate the effects of various parameters on the dynamic response of pavements.

Stress and deformation analysis of plane‐strain rolling process

A finite‐element elastic‐plastic deformation analysis is required in order to predict a complete stress and deformation in a material for metal forming. There have been a few attempts for rolling analysis using the elastic‐plastic code, because the boundary conditions between roll and deforming material become complicated.In this study, an elastic‐plastic plane‐strain rolling analysis with friction has been attempted by the updated Lagrange code. Both slipping and sticking were taken into account for boundary conditions between roll and material. The stress and deformation analysis is performed using a nonsymmetric stiffness matrix solution (for curved and frictional boundaries).The elastic‐plastic analysis, coupled with the proposed frictional boundary model, evidently provides a reliable stress and deformation solution in the metal rolling process.

Nonlinear finite element analysis of elastic systems under nonconservative loading-natural formulation. part I. Quasistatic problems

The paper presents a nonlinear finite element analysis of elastic structures subject to nonconservative forces. Attention is focused on the stability behaviour of such systems. This leads mathematically to non-self-adjoint boundary-value problems which are of great theoretical and practical interest, in particular in connection with alternative modes of instability like divergence of flutter. Only quasistatic effects are however considered in the present part. The methodology of our theory is general, but the specific thrust of the present research is directed towards the analysis of structures acted upon by displacement-dependent nonconservative (follower) forces. In a finite element formulation the analysis of geometrically nonlinear elastic systems subject to such forces gives, in general, rise to a contributory nonsymmetric stiffness matrix known as the load correction matrix. As a result, the total tangent stiffness matrix becomes unsymmetric - an indication of the non-self-adjoint character of the problem. Our theory is based on the natural mode technique [1, 2, 3]and permits i.a. a simple but elegant derivation of the load correction matrix. The application of the general theory as evolved in this paper is demonstrated on the beam element in space. A number of numerical examples are considered including divergence and flutter types of instability, for which exact analytic solutions are known. The problems demonstrate the efficiency of the present finite element formulation. The paper furnishes also a novel and concise formulation of finite rotations in space which may be considered as a conceptual generalization of the theory presented in [2, 3].