This paper proposes an analytical solution and isogeometric analysis numerical approach for buckling analysis of size-dependent beams based on a reformulated strain gradient elasticity theory (RSGET). The superiority of this method is that it has only one material parameter for couple stress and another material parameter for strain gradient effects. Using the RSGET and the principle of minimum potential energy, both non-classical Euler–Bernoulli and Timoshenko beam buckling models are developed. Moreover, the obtained governing equations are solved by an exact solution and isogeometric analysis approach, which conforms to the requirements of higher continuity in gradient elasticity theory. Numerical results are compared with exact solutions to reveal the accuracy of the current isogeometric analysis approach. The influences of length–scale parameter, length-to-thickness ratio, beam thickness and boundary conditions are investigated. Moreover, the difference between the buckling responses obtained by the Timoshenko and Euler–Bernoulli theories shows that the Euler–Bernoulli theory is suitable for slender beams.

An effective free vibration optimization procedure in combination with the isogeometric approach (IGA), particle swarm optimization (PSO) and an integrated global and local parameterization is presented. The natural frequency of functionally graded (FG) plates is calculated by the IGA based on the Bézier extraction of non-uniform rational B-splines (NURBS) with the cubic NURBS basis function. The material composition is assumed to vary only in the thickness direction, and the volumetric fraction is described by the NURBS basis function in light of the superior properties of NURBS curves. The volume fractions of the control points are then optimized by the PSO. In most of the previous work, the control points for the volume fraction are usually equally spaced, which is incapable of identifying the optimal location of the graded zones in most cases. To overcome this bottleneck, a novel local refinement strategy is proposed. The reliability and effectiveness of the proposed approach are demonstrated through several numerical examples. It is interesting to observe that the optimal results are sandwich or laminate plates, and few parameters are involved in the integrated global and local parameterization.

For periodic composite microplates, the microstructure dependent effect has great influence on the elastic wave vibration band gaps. In this paper, an effective isogeometric analysis method based on the modified couple stress theory (MCST) is proposed to calculate the vibration band gaps in periodic composite microplates. The MCST is employed to capture the microstructure effect of vibration band gaps in periodic composite microplates. The high order continuity requirement of MCST can be easily satisfied by applying the NURBS based isogeometric analysis approach. The elasto-dynamics and discrete equations of isogeometric analysis are obtained under the Mindlin kinematics assumptions. The present approach is validated by comparison with the analytical solution. And the influence of microstructure effect, volume fraction and unit cell length on the band gaps are investigated in details.

A typical task of an engineer – be it in civil, mechanical, aeronautical engineering and many other fields – is to design and develop new objects, which fulfill certain physical requirements, generally described by partial differential equations, e.g. for structural problems, heat transfer, fluid dynamics, etc. The realization of an optimal design naturally requires an iteration between geometric design, typically carried out by Computer Aided Design and numerical simulations, such as the Finite Element Method. Unfortunately, CAD models are not directly compatible with a numerical simulation, and thus need to be translated into a simulation suitable format, e.g. a mesh. This transition process is considered to be the bottleneck in the design process and has initiated the development of several new simulation techniques, which allow working directly on the CAD model. The most prominent approach is Isogeometric Analysis which relies on the idea of using the same type of functions for the description of the geometry and for the numerical approximation. Whereas IGA is best suited for dimensionally reduced models, such as shells, immersed boundary- or fictitious domain methods fit better to general solids, particularly those which are described by Boundary Representation or by Constructive Solid Geometry.

The Isogeometric analysis (IGA) [1] employs the same computeraided design (CAD) spline basis functions (e.g., NURBS, T-splines) to describe exact geometries