Non-uniform Rational B-spline Basis Functions Research Articles

Isogeometric proportional topology optimization (IGA-PTO) for multi-material problems

This study addresses the multi-material optimization problem by the effective and robust isogeometric proportional topology optimization (IGA-PTO) approach. Non-uniform rational B-spline basis functions are used for descriptions of geometry, displacement field, and density fields of material phases. The alternating active-phase algorithm is employed to convert the optimization problem with multiple constraints into a series of sub-problems of single constraint. Several examples are exhibited, and intensive comparisons with the gradient-based optimality criteria (OC) algorithm are presented to highlight the superior performance of the proposed PTO algorithm. Finally, the optimized topologies are prototyped using 3D printing technology to evidence the manufacturing feasibility.

Free and forced vibration analysis of piezolaminated plates via an isogeometric layerwise finite element

Isogeometric layerwise finite element (L-IGA) formulation is a recent state-of-the-art approach integrating Non-Uniform Rational B-spline (NURBS) basis functions into the quasi-static solution process of piezolaminated composite plates. This study extends the application of the L-IGA framework to encompass free, forced vibration, and displacement control analyses of laminated composite plates with straight/curvilinear fibers and piezoelectric layers. To this end, the NURBS basis functions, utilized in geometry definition, are employed to solve electromechanically coupled differential equations following Hamilton’s variational principle. The adoption of high-order continuous NURBS shape functions throughout the IGA discretization span both in-plane and through-thickness laminate dimensions. This effectively facilitates precise geometry representation directly from Computer-Aided Design (CAD). Besides, such a discretization accelerates the convergence of displacement and electric potential solution fields toward exact results. Various benchmark problems have been solved to verify the robustness and high accuracy of the proposed dynamic L-IGA method. These include comparative analyses between L-IGA dynamic solutions (i.e. employing the Newmark-Beta method), analytical solutions, and ANSYS-Solid 226 finite element results. All the results are compared across various span-to-thickness ratios, mechanical-potential loading scenarios, and fiber orientation angles. Remarkably, the L-IGA method attains almost excellently accurate time response of various fields (displacement, stress, electric potential) and modal results, with considerably fewer mesh elements than Solid 226 solutions. Overall, such an outcome reveals the high potential and practical merits of the proposed L-IGA formulation as a proficient finite element approach for the dynamic analysis of piezolaminated plates.

Isogeometric boundary element method for axisymmetric steady-state heat transfer

Isogeometric analysis (IGA) utilizes the Non-Uniform Rational B-Splines (NURBS) basis functions, which are commonly employed in Computer Aided Design (CAD), to both construct the problem’s geometry and approximate the field variables. In axisymmetric scenarios, the 3D problem can be simplified to 2D, requiring only the discretization of the 1D curve boundary for the boundary element method (BEM). This study proposes an isogeometric boundary element method (IGABEM) to simulate steady-state heat transfer in axisymmetric domains. Both the precise geometry of the boundary and the physical variables are approximated with the same NURBS basis functions, yielding higher-order continuity, superior accuracy per degree of freedom (DOF), and continuous nodal gradients with fewer DOFs than the traditional BEM. Additionally, a zoning method is introduced to handle heterogeneities. Five cases including a solid cylinder, a revolution hyperboloid with a coaxial cylindrical tube, an ellipsoid with a spherical cavity, a bimaterial ellipsoid and a complex geometry with a toroidal tube validate the accuracy, convergence, computational efficiency of IGABEM, and the applicability in addressing multiply-connected domains. The method outperforms Finite Element Method (FEM) and conventional BEM in accurately handling steady-state axisymmetric heat transfer issues under Dirichlet, Neumann, and Robin boundary conditions.

Hierarchical topology optimization with varying micro-structural aspect ratios

This paper proposes a novel isogeometric topology optimization method to obtain the hierarchical structures, where a special porous microstructures of identical topologies but with different length-width ratios are considered. First, the floating projection technique without penalization is adopted to find the topological configurations of hierachical structures, which make the optimized hierachical structures not easy to fall into the local optimums. Second, the isogeometric analysis based on the non-uniform rational B-spline basis functions, is utilized to calculate the structural compliances and sensitivity information. Third, this paper presents a limited number of microstructures with the changing aspect ratios, to satisfy arbitrary boundary conditions of macrostructures. The macroscopic equivalent properties are evaluated by the isogeometric analysis-based homogenization method. Finally, four numerical examples are provided to illustrate the effectiveness of the proposed method in designing hierarchical structures, and the influence of micro-structural properties on the macro-structural performances.

A non‐uniform rational B‐splines enhanced finite element formulation based on the scaled boundary parameterization for the analysis of heterogeneous solids

AbstractThe contribution is concerned with a finite element formulation for the nonlinear analysis of heterogeneous solids in boundary representation. It results in an element with an arbitrary number of curved boundary edges. The curved edges can be parametrized by, for example, non‐uniform rational B‐splines (NURBS). The presented element formulation is based on the scaling concept, which is adopted from the so‐called scaled boundary finite element method (SBFEM). In contrast to SBFEM, the proposed method uses a numerical approximation for the displacement response in scaling direction. This enables the analysis of geometrically and physically nonlinear problems in solid mechanics. The interpolation at the boundary in circumferential direction is independent of interpolation in scaling direction. Thus, different basis functions can be used for each direction, for example, NURBS basis functions in circumferential and Lagrange basis functions in radial direction. It allows the construction of polygonal elements with an arbitrary number of curved sides, which are described by either whole NURBS curves or NURBS curves' segments. The advantage of the presented element formulation is the flexibility in mesh generation. For example, using Quadtree algorithms, a fast and reliable mesh generation can be achieved. Furthermore, in connection with trimming algorithms, the element formulation allows a precise representation of the geometry even with coarse meshes. Some benchmark tests are presented to evaluate the accuracy of the proposed numerical method against analytical solutions, and a comparison to standard element formulations is given as well.

NURBS-Enhanced polygonal scaled boundary finite element method for heat diffusion in anisotropic media with internal heat sources

We develop polygonal scaled boundary finite element method (SBFEM) with boundaries represented by non-uniform rational B-splines (NURBS) instead of standard Lagrange shape functions. The approach, coined NURBS enhanced SBFEM (NESBFEM), is able to represent the geometry exactly. For elements intersected by the boundary, NURBS basis functions are employed and for other elements, either Lagrange shape functions or NURBS functions are used. The proposed technique inherits the salient features of isogeometric analysis (IGA) and scaled boundary finite element method (SBFEM). Further, well-developed refinement strategies (h−,k−, p-refinements) of IGA are employed to generate higher order polytopal elements. The proposed approach provides a starting point to an efficient framework for practising engineers to solve heat diffusion in anisotropic media with internal heat sources, especially when the domain has internal features, viz., holes/inclusions and to obtain accurate solutions even on coarser mesh with high-order polytopal element.

NLFEM WITH HIGHER ORDER INTERPOLATION FUNCTION FOR EFFICIENT ANALYSIS OF IRREGULAR DOMAIN

This study proposes a new approach to developing a more efficient numerical technique by coupling the non-uniform rational B-spline (NURBS) with the higher-order polynomial basis functions under the framework of the Finite Element Method (FEM). In this technique, denoted as NURBS-Lagrange FEM (NLFEM), the NURBS basis functions are employed to represent the geometry of the problem domain, while the Lagrange interpolation functions are employed for the higher-order polynomial functions to interpolate the field variables. The NURBS is a mathematical model which provides a numerically stable algorithm to exactly represent all conic sections, and the Lagrange interpolation function allows for higher-order basis functions resulting in a faster convergence rate of analysis. By taking advantage of both models, the objective of this study is to propose a new approach, i.e., NLFEM, which can improve the accuracy of the analysis of the irregular domain with more efficient consumption of computer resources. A steady heat transfer formulation for a curved boundary problem is presented to demonstrate the validity and accuracy of the developed technique. The performance is verified against converged solutions obtained using higher-order FEM (FEM/Q9) and NURBS-Augmented FEM (NAFEM). The presented result shows that the NLFEM provides a favorable comparison against other methods. The converged solution is achieved 20% faster than the FEM/Q9 and 80 % faster than the NAFEM. This highlights the potential of the NLFEM as a new approach in numerical techniques for solving problems with irregular boundaries.

An Isogeometric Bézier Finite Element Method for Vibration Optimization of Functionally Graded Plate with Local Refinement

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.

Coupling of NURBS and Meshfree RPIM for plane stress of web with openings

This study presents a novel approach for the coupling of the non-uniform rational B-spline (NURBS) with the radial basis functions (RBF) under the framework of the meshfree radial point interpolation method (RPIM). In this method, denoted as N-RPIM, the NURBS basis functions are employed to model the geometry of the problem domain exactly, whilst the RPIM is used for field variables interpolation. The formulation is applied to the web of a cellular beam under plane stress condition to highlight the advantages. The results show that the method works efficiently, and provides favourable performance against FEM. The solution of the former converges faster, thus highlighting the potential of N-RPIM for future development.

Static and free vibration analyses of functionally graded plates based on an isogeometric scaled boundary finite element method

A novel semi-analytical plate formulation based on the isogeometric analysis (IGA) and scaled boundary element method (SBFEM) is proposed in this paper, and aims to solving the static bending and free vibration problems of functionally graded material (FGM) plates. The modulus of elasticity and density of the material are assumed to vary continuously along the thickness direction with a power-law distribution. The proposed formulation is derived on the basis of 3D theory of elasticity, while only the 2D in-plane dimension of reference surface needed to be discretized with three degrees of freedom (Dofs) per control points, and the non-uniform rational B-spline (NURBS) basis functions are adopted for both the representation of exact geometry and mechanical analysis. The system equations are obtained by using the principle of virtual work. A set of dual variables involving the displacements and nodal forces are introduced to reduce the system equations into first order ordinary differential equations (ODEs), which can be solved by conventional methods. The proposed approach possesses many advantages, such as the problem dimension is successfully reduced by one due to that the physical fields in the thickness direction are expressed analytically as Padé expansion associated with the material property distributed in gradient form, which are able to predict the stress through the thickness more precisely then traditional formulations. The IGSBFEM meshes can maintain exact geometry at any level without further communication with the computer-aided design (CAD) model, which enables the proposed method a flexible method in simulation of FGM plates with complex geometries. The NURBS basis functions can reach to C∞ continuous within a knot span, which means that the discontinuity between adjacent elements on the same patch will vanish naturally. Numerous numerical examples are performed to evaluate the superior accuracy, convergence, locking-free, grid adaptability based on the comparison with other references. Effects of gradient index and several aspect ratios on the static and free vibration responses of FGM plates are investigated.

Isogeometric topology and shape optimization for composite structures using level-sets and adaptive Gauss quadrature

The Level-Set Method (LSM) with the smooth and distinct description of structural boundaries offers more superior benefits for topology optimization. However, the classic Finite Element Method (FEM) and ersatz material method in the LSM cause numerical deficiencies in the optimization, such as iterative instability and the crispness loss of boundaries. This work focuses on developing an Isogeometric Topology and Shape Optimization (ITSO) method for the design of composite structures, in which a Non-Uniform Rational B-splines (NURBS)-based Multi-phase Level-Sets (NM-LS) model is constructed to present the distribution of multiple materials. The NM-LS model consists of an immersed representation using level-sets, NURBS parameterized level-sets and its development. Isogeometric Analysis (IGA) with adaptive Gauss quadrature method are applied to solve unknown structural responses, which can effectively remove several numerical artifacts resulting from the linear interpolation of ersatz material model in both design and analysis. Three models of structural geometry, numerical analysis and topology description can be integrated by the same NURBS basis functions, which can effectively improve numerical precision and iterative stability. Finally, several 2D and 3D numerical examples are addressed to demonstrate the effectiveness and efficiency of the proposed ITSO method on the design of composite structures.

Extended isogeometric analysis of a progressively fracturing fluid‐saturated porous medium

AbstractAn extended isogeometric analysis (XIGA) approach is proposed for modeling fracturing in a fluid‐saturated porous material. XIGA provides a definition of the discontinuity independent of the underlying mesh layout, which obviates the need of knowing the crack extension direction a priori. Unlike Lagrange shape functions used in the standard finite element approach, non‐uniform rational B‐splines (NURBS) provide a higher‐order interelement continuity which leads to a continuous fluid flow also at element boundaries, thereby satisfying the local mass balance. It also leads to an improved estimate of the crack path due to a smoother stress distribution. The NURBS basis functions are cast in finite element data structure using Bézier extraction. To model the discontinuity, the Heaviside sign function is utilized within the displacement and the pressure fields, complemented by the shifting and the blending techniques to enforce compatibility perpendicular and parallel to the crack path, respectively. Different aspects of the approach are assessed through examples comprising straight and curved crack paths for stationary and propagating discontinuities.

Isogeometric-analysis-based stiffness spreading method for truss layout optimization

A novel isogeometric analysis-based stiffness spreading method (IGA-based SSM) is proposed for the truss layout optimization design within a continuum, where the topology, geometry, and cross-sectional areas of the truss elements can be simultaneously optimized. The IGA-based SSM based on energy conservation is employed to obtain the spreading stiffness matrices of the truss elements that are embedded in weak IGA background grids. The IGA background grids are constructed using high-order continuous isogeometric elements that can overcome the limitation of discontinuous sensitivity derived by traditional C0 elements. Because the sensitivity required for the optimization can be obtained analytically, gradient-based optimization algorithms can be easily implemented. To verify the effectiveness of the IGA-based SSM for truss layout optimization design, typical illustrative examples are used, considering the compliance minimization optimization problem. The proposed method can provide a continuous and smooth sensitivity field and overcome the numerical inaccuracy caused by interpolation methods, such as the radial basis function. The different background mesh sizes, initial layouts, and orders p of nonuniform rational B-splines basis functions are investigated in truss layout optimization. The results indicate that the selection of the appropriate background mesh size can avoid the problem of discontinuous truss connections, C1-continuous isogeometric elements can provide better optimization designs with a lower computational cost, and the proposed method can provide a clear layout for different initial layouts. Moreover, the spatial truss layout optimization problem and the min–max displacement optimization problem are also studied to further verify the effectiveness of the proposed method. Finally, the proposed method is extended to solve the concentrated force diffusion problem that widely exists in the interstage adapters of launch vehicles by adding the variance constraint of reaction forces. The results indicate that the proposed method can obtain required concentrated force diffusion results with reasonable stiffness distributions.

A NURBS enhanced polygonal element formulation based on the scaled boundary method for the analysis of solids in boundary representation

AbstractThe contribution is concerned with a numerical element formulation for the nonlinear analysis of solid in boundary representation. Its results in a polygonal element with an arbitrary number of sides. Straight edges are possible as well as curved edges, which are described by e.g. Non‐Uniform Rational B‐Splines (NURBS). The element formulation is based on the so‐called scaled boundary finite element method (SBFEM). The basic idea is to scale the domain's boundary with respect to a scaling center in order to describe the interior domain. In contrast to SBFEM, the proposed method makes use of a numerical aproximation for the displacement response in scaling direction. This enables the analysis of geometrically and physically nonlinear problems in solid mechanics. The interpolation at the boundary in circumferential direction is independent of the interpolation in scaling direction. So, different basis functions can be used for each direction, e.g. NURBS basis functions in circumferential and Lagrange basis functions in radial direction. The advantage of the presented element formulation is the flexibility in mesh generation. The avoidance of so‐called hanging node problems as arbitrary element geometries are possible. For example, using Quadtree algorithms, a fast and reliable mesh generation can be achieved. Furthermore, in connection with trimming algorithms, the element formulation allows a precise representation of the geometry even with coarse meshes. Some benchmark tests are presented to evaluate the accuracy of the proposed numerical method against analytical solutions.

Quasi-3D nonlinear flexural response of isogeometric functionally graded CNT-reinforced plates with various shapes with variable thicknesses

This exploration deals with the nonlinear large deflection behavior of nanocomposite plates reinforced with functionally graded carbon nanotube (FG-CNT) having various shapes with variable thickness. To accomplish this end, a hybrid quasi-3D plate model is constructed by incorporating trigonometric normal and sinusoidal transverse shear functions. The applied plate formulations have the capability to model the nanocomposite plate with only four independent variables which results in a significant reduction in the computational cost of computational calculations. Four different dispersion schemes for the CNT reinforcement are considered, the effective mechanical properties of each one are achieved via an extension of the rule of mixture. In addition, plate thickness is assumed to vary in convex, concave and linear patterns. For solving nonlinear problem, isogeometric technique using non-uniform rational B-spline basis functions is employed to integrate an accurate geometric description. It is revealed that at deeper parts of flexural response, the role of CNT dispersion pattern on the nonlinear flexural stiffness of FG-CNT reinforced composite plates gets lower importance. Moreover, it is demonstrated that by substituting linear thickness variation scheme with convex one, the influence of CNT dispersion pattern reduces, but by changing from the linear type to the concave one, this influence increases.

Three-dimensional crack propagation and inclusion-crack interaction based on IGABEM

The isogeometric boundary element method (IGABEM) is developed to simulate the crack propagation and the inclusion-crack interaction in 3D infinite isotropic medium. The influence of complex shape inclusions on the stress intensity factors (SIFs) along the crack front is studied from the aspects of shape, stiffness, size and position. The non-uniform rational B-spline (NURBS) basis functions can be used to accurately describe the geometric shapes of inclusions and cracks, and the displacement, traction, and discontinuous displacement fields also can be approximated by the same NURBS basis functions. During crack propagation, the normal and tangential vectors of the crack boundary can be uniquely solved. Three examples verify the accuracy and effectiveness of the proposed method. The results show that the SIFs can be calculated accurately even using the single point formula, and the crack propagation process is stable and the path is smooth.

Isogeometric optimization of piezoelectric functionally graded material for energy harvester

Improving the energy harvesting efficiency of piezoelectric energy harvesters (PEHs) in the scientific domain is essential for the application of self-powered microelectromechanical devices. This paper proposes an isogeometric optimization method to optimize the material distribution of piezoelectric functionally graded material (PFGM) energy harvester. Firstly, we establish the isogeometric formulation of the PFGM energy harvester based on the plane strain hypothesis and Hamilton’s principle. Meanwhile, the same non-uniform rational B-splines basis functions are utilized to represent the distribution of the PFGM. Secondly, three optimization models with the specified eigenfrequency and maximum energy conversion efficiency as objective functions are established. Then we derive the analytical sensitivities by the adjoint method and use the method of moving asymptotes to update the design variables. Furthermore, several numerical examples are performed to study the convergence and accuracy of the numerical solutions and analyze the effects of the metallic volume ratio constraint and the initial design domain on the optimized material distribution. More importantly, the dynamic analysis is executed to further test the energy harvesting efficiency based on the optimized designs. The results show that the presented approach can not only realize the specified eigenfrequency but also effectively improve the energy harvesting efficiency.

3D isogeometric boundary element analysis and structural shape optimization for Helmholtz acoustic scattering problems

The capability of boundary element methods (BEM) for solving three-dimensional time harmonic Helmholtz acoustic scattering problems is presented in the framework of the isogeometric analysis (IGA). Both the CAD geometry and the physical boundary variables are approximated using Non-uniform Rational B-splines basis functions (NURBS) in an isogeometric setting. A detailed comparison between two BEM methods: the conventional boundary integral equation (CBIE) and Burton–Miller (BM) is provided including the computational cost. The proposed models are enhanced with a modified collocation scheme with offsets to Greville abscissae to avoid placing collocation points at the corners. Placing collocation points on a smooth surface enables accurate evaluation of normals for BM formulation in addition to straightforward prediction of jump-terms and avoids singularities in O(1∕r) integrals eliminating the need for polar integration. Furthermore, no additional special treatment is required for the hyper-singular integral while collocating on highly distorted elements, such as those containing sphere poles.Acoustic shape optimization in different mediums (air and water) is performed with Particle Swarm Optimization (PSO) and the results are compared with the benchmark solutions from the literature. The reference solutions are obtained with BM which deals with higher order singularities and gradient-based optimization, and requires more complicated sensitivity analysis. The obtained results indicate that, CBIE with PSO is a feasible alternative (except for a small number of fictitious frequencies) which is easier to implement.

Dynamic and static isogeometric analysis for laminated Timoshenko curved microbeams

An effective computational approach that can simultaneously deal with several complexities in geometries, material properties, and size effects is of non-trivial yet important tasks for the design and optimization of device components in micro- and nano-electromechanical systems. This study is devoted to the development of such an effective computational approach, which is formed by integrating the modified couple stress theory into isogeometric analysis framework. The proposed approach is applied to mechanical problems of laminated Timoshenko curved microbeams, i.e., anazlying static bending, natural frequency and transient response. The developed method enables to describe and examine nontrivial features in both material and geometrical properties of laminated curved microbeams. Geometries of curved microbeams and displacement approximation are accurately described by the non-uniform rational B-spline basis functions. For laminated composites, the material coupling is taken into account with the use of equivalent elastic modulus; and the deepness term is properly included in laminate stiffness parameters. In addition, the small-scale size effects are captured with the use of modified couple stress theory. The accuracy and performance of proposed method are demonstrated through several numerical examples, in which the static bending, free vibration and transient response of laminated curved microbeams are considered under effects of some factors such as aspect ratios, lay-ups, boundary conditions and size dependence.

Non-conforming interface coupling and symmetric iterative solution in isogeometric FE–BE​ analysis

In this paper, a three-dimensional (3D) isogeometric finite element (FE) and boundary element (BE) coupling approach based on non-uniform rational B-splines (NURBS) is developed to simulate the deformed body. Based on the geometric exactness and higher smoothness of isogeometric analysis (IGA), NURBS discretization is applied to the approximation of geometric representation, displacement field and traction field simultaneously. An improved power series expansion method (M-PSEM) is used to evaluate the singular integral and is suitable for high order (≥3) NURBS basis functions, which is the limitation of the traditional method. Adaptive integration scheme based on quadtree subdivision technique is adopted to enable the non-singular integration computation more flexible and efficient at optimal computational cost. The relationship between the tractions and nodal external forces on the interface is established by using the virtual work principle. On the other hand, displacement coupling constraints among the non-conforming (or incomplete coincident) surface meshes are constructed by a virtual knot insertion technique. The main advantages of this method are simplicity and robustness, as it is problem-independent and only depends on the NURBS meshes on both sides of the interface. Introducing an enhanced symmetric matrix related to the BE subdomain, the symmetric iterative coupling method (SICM) is extended to solve the final system coupling equation. The final coupling matrix obtained by this iterative method is symmetric, and its dimension is the same as that of the stiffness matrix of IGA finite element method (IGAFEM). Numerical examples are investigated to assess the accuracy and efficiency of the presented method.