Stanislav Stoykov

Numerical computation of periodic responses of nonlinear large-scale systems by shooting method

Abstract Geometrically nonlinear vibrations of three-dimensional elastic structures, due to harmonic external excitations, are investigated in the frequency domain. The material of the structure is assumed to be linearly elastic. The equation of motion is derived by the conservation of linear momentum in Lagrangian coordinate system and it is discretized into a system of ordinary differential equations by the finite element method. The shooting method is used, to obtain the periodic solutions. A procedure which transforms the initial value problem into a two point boundary value problem, for the periodicity condition, and then it finds the initial conditions which lead to periodic response, is developed and presented, for systems of second order ordinary differential equations. The Elmer software is used for computing the local and global mass and stiffness matrices and the force vector, as well for computing the correction of the initial conditions by the shooting method. Stability of the solutions is studied by the Floquet theory. Sequential continuation method is used to define the prediction for the next point from the frequency response diagram. The main goal of the current work is to investigate and present the potential of the proposed numerical methods for the efficient computation of the frequency response functions of large-scale nonlinear systems, which often result from space discretization of real life engineering applications.

Buckling analysis of geometrically nonlinear curved beams

Abstract The equation of motion of curved beams is derived in polar coordinate system which represents exactly the geometry of the beam. The displacements of the beam in radial and circumferential directions are expressed by assuming Bernoulli–Euler’s theory. The nonlinear strain–displacement relations are obtained from the Green–Lagrange strain tensor written in cylindrical coordinate system, but only the components related with radial and circumferential displacements are used. The equation of motion is derived by the principle of virtual work and it is discretized into a system of ordinary differential equations by Ritz method. Static analysis is performed in parametrical domain, assuming the magnitude of the applied force as parameter, and stability of the solution is determined. The nonlinear system of equations is solved by Newton–Raphson’s method. Prediction for the next point from the force–displacement curve is defined by the arc-length continuation method. Bifurcation points are found and the corresponding secondary branches with the deformed shapes are obtained and presented.

Isogeometric Analysis for Nonlinear Dynamics of Timoshenko Beams

The dynamics of beams that undergo large displacements is analyzed in frequency domain and comparisons between models derived by isogeometric analysis and \(p\)-FEM are presented. The equation of motion is derived by the principle of virtual work, assuming Timoshenko’s theory for bending and geometrical type of nonlinearity.

Nonlinear Vibrations of 3D Laminated Composite Beams

A model for 3D laminated composite beams, that is, beams that can vibrate in space and experience longitudinal and torsional deformations, is derived. The model is based on Timoshenko’s theory for bending and assumes that, under torsion, the cross section rotates as a rigid body but can deform longitudinally due to warping. The warping function, which is essential for correct torsional deformations, is computed preliminarily by the finite element method. Geometrical nonlinearity is taken into account by considering Green’s strain tensor. The equation of motion is derived by the principle of virtual work and discretized by the -version finite element method. The laminates are assumed to be of orthotropic materials. The influence of the angle of orientation of the laminates on the natural frequencies and on the nonlinear modes of vibration is presented. It is shown that, due to asymmetric laminates, there exist bending-longitudinal and bending-torsional coupling in linear analysis. Dynamic responses in time domain are presented and couplings between transverse displacements and torsion are investigated.

Scalable parallel implementation of shooting method for large-scale dynamical systems. Application to bridge components

The mathematical models applied to real-life engineering structures result into large-scale dynamical systems. The efficient computation of their dynamical characteristics requires the usage of advanced numerical methods with parallel algorithms. The shooting method, in combination with a continuation method, presents a powerful tool for analyzing and investigating the dynamical characteristics of such systems. The efficiency and scalability of the shooting method are analyzed in the current paper for linear and nonlinear equations. Its applicability to large-scale systems is demonstrated by structural component of bridge discretized by three-dimensional finite elements.

Efficient sparse matrix-matrix multiplication for computing periodic responses by shooting method on Intel Xeon Phi

Many of the scientific applications involve sparse or dense matrix operations, such as solving linear systems, matrix-matrix products, eigensolvers, etc. In what concerns structural nonlinear dynamics, the computations of periodic responses and the determination of stability of the solution are of primary interest. Shooting method iswidely used for obtaining periodic responses of nonlinear systems. The method involves simultaneously operations with sparse and dense matrices. One of the computationally expensive operations in the method is multiplication of sparse by dense matrices. In the current work, a new algorithm for sparse matrix by dense matrix products is presented. The algorithm takes into account the structure of the sparse matrix, which is obtained by space discretization of the nonlinear Mindlin’s plate equation of motion by the finite element method. The algorithm is developed to use the vector engine of Intel Xeon Phi coprocessors. It is compared with the standard sparse matrix by dense matr...

Numerical methods and parallel algorithms for computation of periodic responses of plates

Numerical methods that compute the nonlinear frequency-response curves of plates are presented. The equation of motion of the plate is derived by assuming first order shear deformation theory and considering geometrical type of nonlinearity. It is discretized by the finite element method using bilinear quadrilateral elements. The frequency-response curve is computed by shooting and continuation methods. Parallel implementation of the shooting method is presented and its scalability is investigated.

Finite Element Method for Nonlinear Vibration Analysis of Plates

Plates are structures which have wide applications among engineering constructions. The knowledge of the dynamical behavior of the plates is important for their design and maintenance. The dynamical response of the plate can change significantly due to the nonlinear terms at the equation of motion which become essential in the presence of large displacements. The current work presents numerical methods for investigating the dynamical behavior of plates with complex geometry. The equation of motion of the plate is derived by the classical plate theory and geometrical nonlinear terms are included. It is discretized by the finite element method and periodic responses are obtained by shooting method. Next point from the frequency-response curve is obtained by the sequential continuation method. The potential of the methods is demonstrated on rectangular plate with hole. The main branch along the fundamental mode is presented and the corresponding time responses and shapes of vibration are shown.

Scalability of Shooting Method for Nonlinear Dynamical Systems

The computation of periodic solutions of nonlinear dynamical systems is essential step for their analysis. The variation of the steady-state periodic responses of elastic structures with the frequency of vibration or with the excitation frequency provides valuable information about the dynamical behavior of the structure. Shooting method computes iteratively the periodic solutions of dynamical systems. In the current paper a parallel implementation of the shooting method is presented. The nonlinear equation of motion of Bernoulli-Euler beam is used as a model equation. Large-scale system of ordinary differential equations is generated by applying the finite element method. The speedup and efficiency of the method are studied and presented.

Space discretization by B-Splines on discontinuous problems in structural mechanics

Elastic structures with material or geometrical discontinuities often appear among engineering applications. The appropriate usage of space discretization functions is essential for deriving mathematical models with sufficient accuracy. In the current work, a beam with stoppers is considered as an example of elastic structure with discontinuities. The p- and h-versions of the finite element method are compared with B-Spline space discretization model. It is shown that B-Splines with multiple knots can be used to efficiently model problems with discontinuities.