An inverse problem for estimation of bending stiffness in Kirchhoff–Love plates

This work deals with the development of a numerical method for solving an inverse problem for bending stiffness estimation in a Kirchhoff–Love plate from overdetermined data. The coefficient is identified using a technique called the Method of Variational Imbedding, where the original inverse problem is replaced by a minimization problem. The Euler–Lagrange equations for minimization comprise higher-order equations for the solution of the displacement and an equation for the bending stiffness. The correctness of the embedded problem is discussed. A difference scheme and a numerical algorithm for solving the parameter identification problem are developed. Numerical results for the obtained values of the bending stiffness as an inverse problem are presented.     

Isogeometric shell analysis: The Reissner–Mindlin shell

A Reissner–Mindlin shell formulation based on a degenerated solid is implemented for NURBS-based isogeometric analysis. The performance of the approach is examined on a set of linear elastic and nonlinear elasto-plastic benchmark examples. The analyses were performed with LS-DYNA, an industrial, general-purpose finite element code, for which a user-defined shell element capability was implemented. This new feature, to be reported on in subsequent work, allows for the use of NURBS and other non-standard discretizations in a sophisticated nonlinear analysis framework.     

A one field full discontinuous Galerkin method for Kirchhoff–Love shells applied to fracture mechanics

In order to model fracture, the cohesive zone method can be coupled in a very efficient way with the finite element method. Nevertheless, there are some drawbacks with the classical insertion of cohesive elements. It is well known that, on one the hand, if these elements are present before fracture there is a modification of the structure stiffness, and that, on the other hand, their insertion during the simulation requires very complex implementation, especially with parallel codes. These drawbacks can be avoided by combining the cohesive method with the use of a discontinuous Galerkin formulation. In such a formulation, all the elements are discontinuous and the continuity is weakly ensured in a stable and consistent way by inserting extra terms on the boundary of elements. The recourse to interface elements allows to substitute them by cohesive elements at the onset of fracture.The purpose of this paper is to develop this formulation for Kirchhoff–Love plates and shells. It is achieved by the establishment of a full DG formulation of shell combined with a cohesive model, which is adapted to the special thickness discretization of the shell formulation. In fact, this cohesive model is applied on resulting reduced stresses which are the basis of thin structures formulations. Finally, numerical examples demonstrate the efficiency of the method.     

A posteriori error estimates for continuous/discontinuous Galerkin approximations of the Kirchhoff–Love plate

We present energy norm a posteriori error estimates for continuous/discontinuous Galerkin (c/dG) approximations of the Kirchhoff–Love plate problem. The method is based on a continuous displacement field inserted into a symmetric discontinuous Galerkin formulation of the fourth order partial differential equation governing the deflection of a thin plate. We also give explicit formulas for the penalty parameter involved in the formulation.     

Application of isogeometric method to free vibration of Reissner–Mindlin plates with non-conforming multi-patch

The isogeometric method is used to study the free vibration of thick plates based on Mindlin theory. The Non-uniform Rational B-Spline (NURBS) basis functions are employed to build the thick plate’s geometry models and serve as the shape functions for solution field approximation in finite element analysis. The Reissner–Mindlin plates built with multiple NURBS patches are investigated, in which several patches of the model have multi-interface and different patches may share a common point. In order to solve the non-conforming interface problems, Nitsche method is employed to glue different NURBS patches and only refers to the coupling conditions in this work. Various plate shapes, different boundary conditions and several kinds of thickness-span ratios are considered to verify the validity of the presented method. The dimensionless frequencies for different cases are obtained by solving the eigenvalue equation problems and compared with the existing reference solutions or the results calculated by ABAQUS software. Several numerical examples exhibit the effectiveness of the isogeometric approach. It shows that the natural frequencies of the Reissner–Mindlin plate can be successfully predicted by the combination of isogeometric analysis and Nitsche method.     

The asymptotic analysis of multiple imaginary characteristic roots for LTI delayed systems based on Puiseux–Newton diagram

The paper proposes a novel procedure for the asymptotic expansions of root loci around multiple imaginary roots of an exponential polynomial, which is necessary for the stability analysis of the LTI systems with commensurate delays. With the LTI delay systems given as exponential polynomials (also called quasi-polynomial), we seek to characterise the asymptotic behaviours of the characteristic roots of such systems in an algebraic way and determine whether the imaginary roots cross from one half-plane into another or only touch the imaginary axis. According to the Weierstrass preparation theorem, the quasi-polynomial equation is equivalent to an algebraic equation in the neighbourhood of a singular point. Furthermore, our result gives an explicit expression of the coefficients of the algebraic equation in infinite power series of delay parameter, and the determinations of such power series coefficients refer to the computation of residues of memorphic functions. Subsequently, the classic Puiseux–Newton diagram algorithm can be used to calculate the algebraic expansions of the reduced equation directly. Thus, the asymptotic behaviours of root loci around singular points of the quasi-polynomial equation are obtained. Some illustrative simulations are given to check the validity of the proposed method on asymptotic analysis with a powerful software.     

Stability analysis of T–S fuzzy models for nonlinear multiple time-delay interconnected systems

In this paper, the Takagi–Sugeno (T–S) fuzzy model representation is extended to the stability analysis for nonlinear interconnected systems with multiple time-delays using linear matrix inequality (LMI) theory. In terms of Lyapunov’s direct method for multiple time-delay fuzzy interconnected systems, a novel LMI-based stability criterion which can be solved numerically is proposed. Then, the common P matrix of the criterion is obtained by LMI optimization algorithms to guarantee the asymptotic stability of nonlinear interconnect systems with multiple time-delay. Finally, the proposed stability conditions are demonstrated with simulations throughout this paper.     

The variational iteration method for solving systems of equations of Emden–Fowler type

In this paper, the variational iteration method (VIM) is used to study systems of linear and nonlinear equations of Emden–Fowler type arising in astrophysics. The VIM overcomes the singularity at the origin and the nonlinearity phenomenon. The Lagrange multipliers for all cases of the parameter α,α>0, are determined. The work is supported by examining specific systems of two or three Emden–Fowler equations where the convergence of the results is emphasized.     

A Rayleigh–Ritz style method for large-scale discriminant analysis

Linear Discriminant Analysis (LDA) is one of the most popular approaches for supervised feature extraction and dimension reduction. However, the computation of LDA involves dense matrices eigendecomposition, which is time-consuming for large-scale problems. In this paper, we present a novel algorithm called Rayleigh–Ritz Discriminant Analysis (RRDA) for efficiently solving LDA. While much of the prior research focus on transforming the generalized eigenvalue problem into a least squares formulation, our method is instead based on the well-established Rayleigh–Ritz framework for general eigenvalue problems and seeks to directly solve the generalized eigenvalue problem of LDA. By exploiting the structures in LDA problems, we are able to design customized and highly efficient subspace expansion and extraction strategy for the Rayleigh–Ritz procedure. To reduce the storage requirement and computational complexity of RRDA for high dimensional, low sample size data, we also establish an equivalent reduced model of RRDA. Practical implementations and the convergence result of our method are also discussed. Our experimental results on several real world data sets indicate the performance of the proposed algorithm.     

Convergence analysis of the Neumann–Neumann waveform relaxation method for time-fractional RC circuits

The classical waveform relaxation (WR) methods rely on decoupling the large-scale ODEs system into small-scale subsystems and then solving these subsystems in a Jacobi or Gauss–Seidel pattern. However, in general it is hard to find a clever partition and for strongly coupled systems the classical WR methods usually converge slowly and non-uniformly. On the contrary, the WR methods of longitudinal type, such as the Robin-WR method and the Neumann–Neumann waveform relaxation (NN-WR) method, possess the advantages of simple partitioning procedure and uniform convergence rate. The Robin-WR method has been extensively studied in the past few years, while the NN-WR method is just proposed very recently and does not get much attention. It was shown in our previous work that the NN-WR method converges much faster than the Robin-WR method, provided the involved parameter, namely β, is chosen properly. In this paper, we perform a convergence analysis of the NN-WR method for time-fractional RC circuits, with special attention to the optimization of the parameter β. For time-fractional PDEs, this work corresponds to the study of the NN-WR method at the semi-discrete level. We present a detailed numerical test of this method, with respect to convergence rate, CPU time and asymptotic dependence on the problem/discretization parameters, in the case of two- and multi-subcircuits.     

The construction of free–free flexibility matrices for multilevel structural analysis

This article considers a generalization of the classical structural flexibility matrix. It expands on previous papers by taking a deeper look at computational considerations at the substructure level. Direct or indirect computation of flexibilities as “influence coefficients” has traditionally required pre-removal of rigid body modes by imposing appropriate support conditions, mimicking experimental arrangements. With the method presented here the flexibility of an individual element or substructure is directly obtained as a particular generalized inverse of the free–free stiffness matrix. This generalized inverse preserves the stiffness spectrum. The definition is element independent and only involves access to the stiffness generated by a standard finite element program and the separate construction of an orthonormal rigid-body mode basis. The free–free flexibility has proven useful in special application areas of finite element structural analysis, notably massively parallel processing, model reduction and damage localization. It can be computed by solving sets of linear equations and does not require processing an eigenproblem or performing a singular value decomposition. If substructures contain thousands of d.o.f., exploitation of the stiffness sparseness is important. For that case this paper presents a computation procedure based on an exact penalty method, and a projected rank-regularized inverse stiffness with diagonal entries inserted by the sparse factorization process. These entries can be physically interpreted as penalty springs. This procedure takes advantage of the stiffness sparseness while forming the full free–free flexibility, or a boundary subset, and is backed by an in-depth null space analysis for robustness.     

The improved Farmer–Loizou method for finding polynomial zeros

An improvement of the Farmer–Loizou method for the simultaneous determination of simple roots of algebraic polynomials is proposed. Using suitable corrections of Newton's type, the convergence of the basic method is increased from 4 to 5 without any additional calculations. In this manner, a higher computational efficiency of the improved method is achieved. We prove a local convergence of the presented method under initial conditions which depend on a geometry of zeros and their initial approximations. Numerical examples are given to demonstrate the convergence behaviour of the proposed method and related methods.     

A study on the free vibration of the joined cylindrical–spherical shell structures

The free vibration characteristics of the joined spherical–cylindrical shell with various boundary conditions are investigated. The boundary conditions considered herein are free–free, simply supported–free and clamped–free for the joined cylindrical–spherical shell structures. The Flügge shell theory and Rayleigh’s energy method are applied in order to analyze the free vibration characteristics of the joined shell structure and individual shell components. In the modal test, the I-DEAS test module is used to calculate mode shapes and natural frequencies of the joined shell structure. The natural frequencies and mode shapes are calculated numerically and they are compared with those of the FEM and modal test to confirm the reliability of the analytical solution. The effects of the shallowness of the spherical shell and length of the cylindrical shell to the free vibrational behavior of joined shell structure are investigated.     

Gauss–Galerkin quadrature rules for quadratic and cubic spline spaces and their application to isogeometric analysis

We introduce Gaussian quadrature rules for spline spaces that are frequently used in Galerkin discretizations to build mass and stiffness matrices. By definition, these spaces are of even degrees. The optimal quadrature rules we recently derived (Bartoň and Calo, 2016) act on spaces of the smallest odd degrees and, therefore, are still slightly sub-optimal. In this work, we derive optimal rules directly for even-degree spaces and therefore further improve our recent result. We use optimal quadrature rules for spaces over two elements as elementary building blocks and use recursively the homotopy continuation concept described in Bartoň and Calo (2016) to derive optimal rules for arbitrary admissible numbers of elements. We demonstrate the proposed methodology on relevant examples, where we derive optimal rules for various even-degree spline spaces. We also discuss convergence of our rules to their asymptotic counterparts, these are the analogues of the midpoint rule of Hughes et al. (2010), that are exact and optimal for infinite domains.     

Homotopy analysis method for solving multi-term linear and nonlinear diffusion–wave equations of fractional order

In this paper we have used the homotopy analysis method (HAM) to obtain solutions of multi-term linear and nonlinear diffusion–wave equations of fractional order. The fractional derivative is described in the Caputo sense. Some illustrative examples have been presented.     

Multigrid Methods for Hellan–Herrmann–Johnson Mixed Method of Kirchhoff Plate Bending Problems

A V-cycle multigrid method for the Hellan–Herrmann–Johnson (HHJ) discretization of the Kirchhoff plate bending problems is developed in this paper. It is shown that the contraction number of the V-cycle multigrid HHJ mixed method is bounded away from one uniformly with respect to the mesh size. The uniform convergence is achieved for the V-cycle multigrid method with only one smoothing step and without full elliptic regularity assumption. The key is a stable decomposition of the kernel space which is derived from an exact sequence of the HHJ mixed method, and the strengthened Cauchy Schwarz inequality. Some numerical experiments are provided to confirm the proposed V-cycle multigrid method. The exact sequences of the HHJ mixed method and the corresponding commutative diagram is of some interest independent of the current context.