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This study aims to present a new hybrid formulation based on non-uniform rational b-splines functions and enrichment strategies applied to free and forced vibration of straight bars and trusses.

Based on the idea of enrichment from generalized finite element method (GFEM)/extended finite element method (XFEM), an extended isogeometric formulation (partition of unity isogeometric analysis [PUIGA]) is conceived. By numerical examples the methods are tested and compared with isogeometric analysis, finite element method and GFEM in terms of convergence, error spectrum, conditioning and adaptivity capacity.

The results show a high convergence rate and accuracy for PUIGA and the advantage of input enrichment functions and material parameters on parametric space.

The enrichment strategies demonstrated considerable improvements in numerical solutions. The applications of computer-aided design mapped enrichments applied to structural dynamics are not known in the literature.

This paper aims to provide designers/engineers, in engineering structural design and analysis, approaches to freely and accurately modify structures (geometric and/or material), and then quickly provide real-time capability to obtain the numerical solutions of the modified structures (designs).

The authors propose an isogeometric independent coefficients (IGA-IC) method for a fast reanalysis of structures with geometric and material modifications. Firstly, the authors seamlessly integrate computer-aided design (CAD) and computer-aided engineering (CAE) by capitalizing upon isogeometric analysis (IGA). Hence, the authors can easily modify the structural geometry only by changing the control point positions without tedious transformations between CAE and CAD models; and modify material characters simply based on knots vectors. Besides, more accurate solutions can be obtained because of the high order degree of the spline functions that are used as interpolation functions. Secondly, the authors advance the proposed independent coefficients method within IGA for fast numerical simulation of the modified designs, thereby significantly reducing the enormous time spent in repeatedly numerical evaluations.

This proposed scheme is efficient and accurate for modifying the structural geometry by simply changing the control point positions, and material characters by knots vectors. The enormous time spent in repeated full numerical simulations for reanalysis is significantly reduced. Hence, enabling quickly modifying structural geometry and material, and analyzing the modified model for practicality in design stages.

The authors herein advance and propose the IGA-IC scheme. Where, it provides designers to fasten and simple designs and modify structures (both geometric and material). It then can quickly in real-time obtain numerical solutions of the modified structures. It is a powerful tool in practical engineering design and analysis process for local modification. While this method is an approximation method designed for local modifications, it generally cannot provide an exact numerical solution and its effectiveness for large modification deserves further study.

This paper aims to develop a new isogeometric boundary element formulation based on non-uniform rational basis splines (NURBS) curves for solving Reissner’s shear-deformable plates.

The generalized displacements and tractions along the problem boundary are approximated as NURBS curves having the same rational B-spline basis functions used to describe the geometrical boundary of the problem. The source points positions are determined over the problem boundary by the well-known Greville abscissae definition. The singular integrals are accurately evaluated using the singularity subtraction technique.

Numerical examples are solved to demonstrate the validity and the accuracy of the developed formulation.

This formulation is considered to preserve the exact geometry of the problem and to reduce or cancel mesh generation time by using NURBS curves employed in computer aided designs as a tool for isogeometric analysis. The present formulation extends such curves to be implemented as a stress analysis tool.

This paper aims to analyze the stability of laminated shells subjected to axial loads or external pressure with considering various geometries and boundary conditions. The main aim of the present study is developing an efficient combined method which uses the advantages of different methods, such as finite element method (FEM) and isogeometric analysis (IGA), to achieve multipurpose targets. Two types of material including laminated composite and sandwich functionally graded material are considered.

A novel type of finite strip method called isogeometric B3-spline finite strip method (IG-SFSM) is used to solve the eigenvalue buckling problem. IG-SFSM uses B3-spline basis functions to interpolate the buckling displacements and mapping operations in the longitudinal direction of the strips, whereas the Lagrangian functions are used in transverse direction. The current presented IG-SFSM is formulated based on the degenerated shell method.

The buckling behavior of laminated shells is discussed by solving several examples corresponding to shells with various geometries, boundary conditions and material properties. The effects of mechanical and geometrical properties on critical loads of shells are investigated using the related results obtained by IG-SFSM.

This paper shows that the proposed IG-SFSM leads to the critical loads with an approved accuracy comparing with the same examples extracted from the literature. Moreover, it leads to a high level of convergence rate and low cost of solving the stability problems in comparison to the FEM.

This paper proposes a novel approach to impose the Neumann boundary condition for isogeometric analysis (IGA) of Euler–Bernoulli beam with 1-D formulation. The proposed method is for only IGA in which it is difficult to handle the Neumann boundary conditions. The control points of B-spline are equivalent to nodes in finite element method. With 1-D formulation, it is not possible to accommodate multiple degrees of freedom in IGA. This case arises in the analysis of beams. The paper aims to propose a way to work around this issue in a simple way.

Neumann boundary conditions, which are even-order derivatives (example: double derivative) of the primary variable, are inherently satisfied in the weak form. Boundary conditions with an odd number of derivatives (example: slope) are imposed with the introduction of a new penalty matrix.

The proposed method can impose a slope boundary condition for IGA of a beam using 1-D formulation.

From the literature, it can be observed that the beam is formulated in 1-D by considering it as either a rotation-free element or a 2-D formulation by considering shear strain along with the normal strain. The work represents 1-D formulation of a beam while considering the slope boundary condition, which is easy and effective to formulate, compared with the slope boundary conditions reported in previous works.

As one of the earliest high-level programming languages, Fortran with easy accessibility and computational efficiency is widely used in the engineering field. The purpose of this paper is to present a Fortran implementation of isogeometric analysis (IGA) for thin plate problems.

IGA based on non-uniform rational B-splines (NURBS) offers exact geometries and is more accurate than finite element analysis (FEA). Unlike the basis functions in FEA, NURBS basis functions are non-interpolated. Hence, the penalty method is used to enforce boundary conditions.

Several thin plate examples based on the Kirchhoff-Love theory were illustrated to demonstrate the accuracy of the implementation in contrast with analytical solutions, and the efficiency was validated in comparison with another open method.

A Fortran implementation of NURBS-based IGA was developed to solve Kirchhoff-Love plate problems. It easily obtained high-continuity basis functions, which are necessary for Kirchhoff formulation. In comparison with theoretical solutions, the numerical examples demonstrated higher accuracy and faster convergence of the Fortran implementation. The Fortran implementation can well solve the time-consuming problem, and it was validated by the time-consumption comparison with the Matlab implementation. Due to the non-interpolation of NURBS, the penalty method was used to impose boundary conditions. A suggestion of the selection of penalty coefficients was given.

This study implements the fourth-order phase field method (PFM) for modeling fracture in brittle materials. The weak form of the fourth-order PFM requires C¹ basis functions for the crack evolution scalar field in a finite element framework. To address this, non-Sibsonian type shape functions that are nonpolynomial types based on distance measures, are used in the context of natural neighbor shape functions. The capability and efficiency of this method are studied for modeling cracks.

The weak form of the fourth-order PFM is derived from two governing equations for finite element modeling. C⁰ non-Sibsonian shape functions are derived using distance measures on a generalized quad element. Then these shape functions are degree elevated with Bernstein-Bezier (BB) patch to get higher-order continuity (C¹) in the shape function. The quad element is divided into several background triangular elements to apply the Gauss-quadrature rule for numerical integration. Both fourth-order and second-order PFMs are implemented in a finite element framework. The efficiency of the interpolation function is studied in terms of convergence and accuracy for capturing crack topology in the fourth-order PFM.

It is observed that fourth-order PFM has higher accuracy and convergence than second-order PFM using non-Sibsonian type interpolants. The former predicts higher failure loads and failure displacements compared to the second-order model due to the addition of higher-order terms in the energy equation. The fracture pattern is realistic when only the tensile part of the strain energy is taken for fracture evolution. The fracture pattern is also observed in the compressive region when both tensile and compressive energy for crack evolution are taken into account, which is unrealistic. Length scale has a certain specific effect on the failure load of the specimen.

Fourth-order PFM is implemented using C¹ non-Sibsonian type of shape functions. The derivation and implementation are carried out for both the second-order and fourth-order PFM. The length scale effect on both models is shown. The better accuracy and convergence rate of the fourth-order PFM over second-order PFM are studied using the current approach. The critical difference between the isotropic phase field and the hybrid phase field approach is also presented to showcase the importance of strain energy decomposition in PFM.

Shell structures are highly efficient and are an elegant way of covering large uninterrupted spaces, but their complex geometry is notoriously difficult to model and analyse. This paper aims to describe a novel free-form shell modelling technique based on structural harmonics.

The method builds on work using weighted eigenmodes for three-dimensional mesh modelling in a computer graphics setting and extends it by specifically adapting the technique to an architectural design context. This not only enables the sculpting of free-form architectural surfaces using only a few control parameters but also takes advantage of the synergies between eigenmodes and structural buckling modes, to provide an efficient means of stiffening a shell against failure by buckling.

The result is a flexible free-form modelling tool that not only enables the creation of arbitrary doubly curved surfaces but also allows simultan. The tool helps to assist in the design of shells at the conceptual stage and encourages an interaction between the architect and engineer. A number of initiatives, including a single degree of freedom design, boundary constraints, visualisation aids and guidelines towards specific spatial configurations have been introduced to satisfactorily adapt the method to an architectural context.

The tool helps to assist in the design of shells at the conceptual stage and encourages an interaction between the architect and engineer. A number of initiatives, including a single degree of freedom design, boundary constraints, visualisation aids and guidelines towards specific spatial configurations have been introduced to satisfactorily adapt the method to an architectural context. This paper includes a full case study of the iconic British Museum Great Court Roof to demonstrate the applicability of the developed framework to real-world problems and the software developed to implement the method is available as an open-source download.

Majority of the existing direct slicing methods have generated precise slicing contours from different surface representations, they do not carry any interior information. Whereas, heterogeneous solids are highly preferable for designing and manufacturing sophisticated models. To directly slice heterogeneous solids for additive manufacturing (AM), this study aims to present an algorithm using octree-based subdivision and trivariate T-splines.

This paper presents a direct slicing algorithm for heterogeneous solids using T-splines, which can be applied to AM based on the fused deposition modeling (FDM) technology. First, trivariate T-splines are constructed using a harmonic field with the gradient direction aligning with the slicing direction. An octree-based subdivision algorithm is then used to directly generate the sliced layers with heterogeneous materials. For FDM-based AM applications, the heterogeneous materials of each sliced layer are discretized into a finite number of partitions. Finally, boundary contours of each separated partition are extracted and paired according to the rules of CuraEngine to generate the scan path for FDM machines equipped with multi-nozzles.

The experimental results demonstrate that the proposed algorithm is effective and reliable, especially for solid objects with multiple materials, which could maintain the model integrity throughout the process from the original representation to the final product in AM.

Directly slicing heterogeneous solid using trivariate T-splines will be a powerful supplement to current technologies in AM.

This paper aims to propose a gradient-based shape optimization framework in which traditional time-consuming conversions between computer-aided design and computer-aided engineering and the mesh update procedure are avoided/eliminated. The scheme is general so that it can be used in all cases as a black box, no matter what the objective and/or design variables are, whilst the efficiency and accuracy are guaranteed.

The authors integrated CAD and CAE by using isogeometric analysis (IGA), enabling the present methodology to be robust and accurate. To overcome the difficulty in evaluating the sensitivities of objective and/or constraint functions by analytic method in some cases, the authors adopt the finite difference method to calculate these sensitivities, thereby providing a universal approach. Moreover, to further eliminate the inefficiency caused by the finite difference method, the authors advance the exact reanalysis method, the indirect factorization updating (IFU), to exactly and efficiently calculate functions and their sensitivities, which guarantees its generality and efficiency at the same time.

The proposed isogeometric gradient-based shape optimization using our IFU approach is reliable and accurate, as well as general and efficient.

The authors proposed a gradient-based shape optimization framework in which they first integrate IGA and the proposed exact reanalysis method for applicability to structural response and sensitivity analysis.