Isogeometric analysis of 3D straight beam-type structures by Carrera Unified Formulation

This paper applies the isogeometric analysis (IGA) based on unified one-dimensional (1D) models to study static, free vibration and dynamic responses of metallic and laminated composite straight beam structures. By employing the Carrera Unified Formulation (CUF), 3D displacement fields are expanded as 1D generalized displacement unknowns over the cross-section domain. 2D hierarchical Legendre expansions (HLE) are adopted in the local area for the refinement of cross-section kinematics. In contrast, B-spline functions are used to approximate 1D generalized displacement unknowns, satisfying the requirement of interelement high-order continuity. Consequently, IGA-based weak-form governing equations can be derived using the principle of virtual work and written in terms of fundamental nuclei, which are independent of the class and order of beam theory. Several geometrically linear analyses are conducted to address the enhanced capability of the proposed approach, which is prominent in the detection of shear stresses, higher-order modes and stress wave propagation problems. Besides, 3D-like behaviors can be captured by the present IGA-based CUF-HLE method with reduced computational costs compared with 3D finite element method (FEM) and FEM-based CUF-HLE method.     

Algorithms for numerical solution of the modified equal width wave equation using collocation method

Quintic B-spline collocation algorithms for numerical solution of the modified equal width wave (MEW) equation have been proposed. The algorithms are based on Crank–Nicolson formulation for time integration and quintic B-spline functions for space integration. Quintic B-spline collocation method over the finite intervals is also applied to the time split MEW equation and space split MEW equation. Results for the three algorithms are compared by studying the propagation of the solitary wave, interaction of the solitary waves, wave generation and birth of solitons.     

A quadratic b-spline based isogeometric analysis of transient wave propagation problems with implicit time integration method

A uniform quadratic b-spline isogeometric element is exclusively considered for wave propagation problem with the use of desirable implicit time integration scheme. A generalized numerical algorithm is proposed for dispersion analysis of one-dimensional (1-D) and two-dimensional (2-D) wave propagation problems where the quantified influence of the defined CFL number on wave velocity error is analyzed and obtained. Meanwhile, the optimal CFL (Courant–Friedrichs–Lewy) number for the proposed 1-D and 2-D problems is suggested. Four representative numerical simulations confirm the effectiveness of the proposed method and the correctness of dispersion analysis when appropriate spatial element size and time increment are adopted. The desirable computation efficiency of the proposed isogeometric method was confirmed by conducting time cost and calculation accuracy analysis of a 2-D numerical example where the referred FEM was also tested for comparison.     

A parabolic acceleration time integration method for structural dynamics using quartic B-spline functions

In this paper, an explicit time integration method is proposed for structural dynamics using periodic quartic B-spline interpolation polynomial functions. In this way, at first, by use of quartic B-splines, the authors have proceeded to solve the differential equation of motion governing SDOF systems and later the proposed method has been generalized for MDOF systems. In the proposed approach, a straightforward formulation was derived in a fluent manner from the approximation of response of the system with B-spline basis. Because of using a quartic function, the system acceleration is approximated with a parabolic function. For the aforesaid method, a simple step-by-step algorithm was implemented and presented to calculate dynamic response of MDOF systems. The proposed method has appropriate convergence, accuracy and low time consumption. Accuracy and stability analyses have been done perfectly in this paper. The proposed method benefits from an extraordinary accuracy compared to the existing methods such as central difference, Runge–Kutta and even Duhamel integration method. The validity and effectiveness of the proposed method is demonstrated with four examples and the results of this method are compared with those from some of the existent numerical methods. The high accuracy and less time consumption are only two advantages of this method.     

Explicit time integration with lumped mass matrix for enriched finite elements solution of time domain wave problems

We present a partition of unity finite element method for wave propagation problems in the time domain using an explicit time integration scheme. Plane wave enrichment functions are introduced at the finite elements nodes which allows for a coarse mesh at low order polynomial shape functions even at high wavenumbers. The initial condition is formulated as a Galerkin approximation in the enriched function space. We also show the possibility of lumping the mass matrix which is approximated as a block diagonal system. The proposed method, with and without lumping, is validated using three test cases and compared to an implicit time integration approach. The stability of the proposed approach against different factors such as the choice of wavenumber for the enrichment functions, the spatial discretization, the distortions in mesh elements or the timestep size, is tested in the numerical studies. The method performance is measured for the solution accuracy and the CPU processing times. The results show significant advantages for the proposed lumping approach which outperforms other considered approaches in terms of stability. Furthermore, the resulting block diagonal system only requires a fraction of the CPU time needed to solve the full system associated with the non-lumped approaches.     

Free vibration and buckling analysis of plates using B-spline wavelet on the interval Mindlin element

A finite element method (FEM) of B-spline wavelet on the interval (BSWI) is used in this paper to solve the free vibration and buckling problems of plates based on Reissner–Mindlin theory. By aid of the high accuracy of B-spline functions approximation for structural analysis, the proposed method could obtain a fast convergence and a satisfying numerical accuracy with fewer degrees of freedoms (DOF). The numerical examples demonstrate that the present BSWI method achieves the high accuracy compared to the exact solution and others existing approaches in the literatures. The BSWI finite element has potential to be used as a numerical method in analysis and design.     

Quintic B-spline collocation method for numerical solution of the Kuramoto–Sivashinsky equation

In this paper, the quintic B-spline collocation scheme is implemented to find numerical solution of the Kuramoto–Sivashinsky equation. The scheme is based on the Crank–Nicolson formulation for time integration and quintic B-spline functions for space integration. The accuracy of the proposed method is demonstrated by four test problems. The numerical results are found to be in good agreement with the exact solutions. Results are also shown graphically and are compared with results given in the literature.     

Quadratic trigonometric b-spline galerkin methods for the regularized long wave equation

In this study, a numerical solution of the Regularized Long Wave (RLW) equation is obtained using Galerkin finite element method, based on two and three steps Adams Moulton method for the time integration and quadratic trigonometric B-spline functions for the space integration. After two different linearization techniques are applied, the proposed algorithms are tested on the problems of propagation of a solitary wave and interaction of two solitary waves. For the first test problem, the rate of convergence and the running time of the proposed algorithms are computed and the error norm $L_{\infty }$ is used to measure the differences between exact and numerical solutions. The three conservation quantities of the motion are calculated to determine the conservation properties of the proposed algorithms for both of the test problems.     

Geometry + Simulation Modules: Implementing Isogeometric Analysis

Isogeometric analysis (IGA) is a recently developed simulation method that allows integration of finite element analysis (FEA) with conventional computer-aided design (CAD) software [1,3]. This goal requires new software design strategies, in order to enable the use of CAD data in the analysis pipeline. To this end, we have initiated G + SMO (Geometry+Simulation Modules), an open-source, C++ library for IGA. G + SMO is an object-oriented, template library, that implements a generic concept for IGA, based on abstract classes for discretization basis, geometry map, assembler, solver and so on. It makes use of object polymorphism and inheritance techniques to provide a common framework for IGA, for a variety of different basis-types which are available. A highlight of our design is the dimension independent code, realized by means of template meta-programming. Some of the features already available include computing with B-spline, Bernstein, NURBS bases, as well as hierarchical and truncated hierarchical bases of arbitrary polynomial order. These basis functions are used in continuous and discontinuous Galerkin approximation of PDEs over (non-)conforming multi-patch computational (physical) domains. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical solution of the coupled viscous Burgers’ equation

In the present paper, a numerical method is proposed for the numerical solution of a coupled system of viscous Burgers’ equation with appropriate initial and boundary conditions, by using the cubic B-spline collocation scheme on the uniform mesh points. The scheme is based on Crank–Nicolson formulation for time integration and cubic B-spline functions for space integration. The method is shown to be unconditionally stable using von-Neumann method. The accuracy of the proposed method is demonstrated by applying it on three test problems. Computed results are depicted graphically and are compared with those already available in the literature. The obtained numerical solutions indicate that the method is reliable and yields results compatible with the exact solutions.     

On operator split technique for the time integration within finite strain viscoplasticity in explicit FEM

In this contribution, the operator split technique is applied to the time integration within viscoplasticity for explicit FEM. As an example, the finite strain viscoplastic material model of Shutov and Kreißig is analyzed. In the new solution scheme, some evolution equations are solved using an explicit update formula for implicit time stepping. The solution procedure is split into three steps: an elastic predictor and two viscoplastic corrector steps. Aspects of accuracy and stability of the algorithm are discussed. As shown, the proposed method is superior compared to a fully explicit integration of evolution equations. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The construction of plane elastomechanics and Mindlin plate elements of B-spline wavelet on the interval

Based on two-dimensional tensor product B-spline wavelet on the interval (BSWI), a class of C0 type plate elements is constructed to solve plane elastomechanics and moderately thick plate problems. Instead of traditional polynomial interpolation, the scaling functions of two-dimensional tensor product BSWI are employed to form the shape functions and construct BSWI elements. Unlike the process of direct wavelets adding in the previous work, the elemental displacement field represented by the coefficients of wavelets expansions is transformed into edges and internal modes via the constructed transformation matrix in this paper. The method combines the versatility of the conventional finite element method (FEM) with the accuracy of B-spline functions approximation and various basis functions for structural analysis. Some numerical examples are studied to demonstrate the proposed method and the numerical results presented are in good agreement with the closed-form or traditional FEM solutions.     

Optimal balance between mass and smoothed stiffness in simulation of acoustic problems

The Finite Element Method (FEM) is known to behave overly-stiff, which leads to an imbalance between the mass and stiffness matrices within discretized systems. In this work, for the first time, a model is developed that provides optimal balance between discretized mass and smoothed stiffness—the mass-redistributed alpha finite element method (MR-αFEM). This new method improves on the computational efficiency of the FEM and Smoothed Finite Element Methods (S-FEM). The rigorous research conducted ensures that stiffness with the parameter, α, optimally matches the mass with a flexible integration point, q. The optimal balance system significantly reduces the dispersion error of acoustic problems, including those of single and multi-fluids in both time and frequency domains. The excellent properties of the proposed MR-αFEM are validated using theoretical analyses and numerical examples.     

An efficient cubic B-spline and bicubic B-spline collocation method for numerical solutions of multidimensional nonlinear stochastic quadratic integral equations

This paper aims to develop a novel numerical approach on the basis of B-spline collocation method to approximate the solution of one-dimensional and two-dimensional nonlinear stochastic quadratic integral equations. The proposed approach is based on the hybrid of collocation method, cubic B-spline, and bi-cubic B-spline interpolation and Itô approximation. Using this method, the problem solving turns into a nonlinear system solution of equations that is solved by a suitable numerical method. Also, the convergence analysis of this numerical approach has been discussed. In the end, examples are given to test the accuracy and the implementation of the method. The results are compared with the results obtained by other methods to verify that this method is accurate and efficient.     

Numerical solution of second-order one-dimensional hyperbolic equation by exponential B-spline collocation method

In this paper, we propose a method based on collocation of exponential B-splines to obtain numerical solution of a nonlinear second-order one-dimensional hyperbolic equation subject to appropriate initial and Dirichlet boundary conditions. The method is a combination of B-spline collocation method in space and two-stage, second-order strong-stability-preserving Runge–Kutta method in time. The proposed method is shown to be unconditionally stable. The efficiency and accuracy of the method are successfully described by applying the method to a few test problems.     

The construction of numerical integration rules of degree three for product regions

Numerical integration formulas in n-dimensional Euclidean space of degree three are discussed. In this paper, for the product regions a method is presented to construct numerical integration formulas of degree three with 2n real points and positive weights. The presented problem is a little different from those dealt with by other authors. All the corresponding one-dimensional integrals can be different from each other and they are also nonsymmetrical. In this paper an n-dimensional numerical integration problem is turned into n one-dimensional moment problems, which simplifies the construction process. Some explicit numerical formulas are given. Furthermore, a more generalized numerical integration problem is considered, which will shed light on the final solution to the third degree numerical integration problem.     

A numerical technique for solution of the MRLW equation using quartic B-splines

Numerical scheme based on quartic B-spline collocation method is designed for the numerical solution of modified regularized long wave (MRLW) equation. Unconditional stability is proved using Von-Neumann approach. Performance of the method is checked through numerical examples. Using error norms L₂ and L_∞ and conservative properties of mass, momentum and energy, accuracy and efficiency of the new method is established through comparison with the existing techniques.     

Implicitly enriched Galerkin methods for numerical solutions of fourth‐order partial differential equations containing singularities

Highlights are the following:

• For any integer , we construct ‐continuous partition of unity (PU) functions with flat‐top from B‐spline functions to have numerical solutions of fourth‐order equations with singularities. B‐spline functions are modified to satisfy clamped boundary conditions.
• To handle singularity arising in fourth‐order elliptic differential equations, these modified B‐spline functions are enriched either by introducing enrichment basis functions implicitly through particular geometric mappings or by adding singular basis functions explicitly.
• To show the effectiveness of the proposed implicit enrichment methods (mapping method), the accuracy, the number of degrees of freedom (DOF), and matrix condition numbers are computed and compared in the h‐refinement, the p‐refinement, and the k‐refinement of the approximation space of B‐spline basis functions.

Using Partition of unity (PU) functions with flat‐top, B‐spline functions are modified to satisfy boundary conditions of the fourth‐order equations. Since the standard isogeometric analysis (IGA) as well as the conventional FEM have limitations in handling fourth‐order differential equations containing singularities, we consider two enrichment methods (explicit and implicit) in the framework of the p‐, the k, and the h‐refinements of IGA. We demonstrate that both enrichment methods yield good approximate solutions, but explicit enrichment methods give large (almost singular) matrix condition numbers and face integrating singular functions. Because of these limitations of external enrichment methods, we extensively investigate implicit enrichment methods (mapping methods) that virtually convert fourth‐order elliptic problems with singularities to problems with no influence of the singularities. Effectiveness of the proposed mapping method extensively tested to one‐dimensional fourth‐order equation with singularities. The implicit enrichment (mapping) method is extended to the two‐dimensional cases and test it to fourth‐order partial differential equations on cracked domains.     

Lamb wave propagation using Wave Finite Element Method

Some of the available techniques for Lamb wave propagation simulation are the Finite Element Method (FEM), the Boundary Element Method and the Finite Difference Method. The FEM is the best method when complex damage, geometry or boundary is involved. However, high Lamb wave frequency requires very small element size thus high computational cost in FEM analysis. By using the existence of periodicity in plates, an attempt to reduce this computational cost is done using Wave FEM. The applicability of this method to model Lamb wave propagation in plate is first assessed in this paper for the 1-D wave propagation and compared with FEM explicit method. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Implicit stress integration method of Shape Memory material model

The phenomenological SMA material model proposed by Lagoudas [1] is modified and implemented in finite element software PAK [2]. Critical thermodynamic force is derived to match implicit integration method. All variables are derived to depend on effective values of stress, strain and martensitic volume fraction. One scalar equation need to be solved in the integration point using trial stress direction for each time step. The integration in the trial deviatoric stress direction provides possibility to solve large strain problems using the algorithms developed for small strains. Two one-dimensional and one non-proportional large strain example verified accuracy of proposed modifications. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)