Non-uniform Rational B-spline Basis Research Articles

Design of metamaterial-based heat manipulators using isogeometric level-set topology optimization

We exploit level-set topology optimization to find the optimal material distribution for metamaterial-based heat manipulators. The level-set function, geometry, and solution field are parameterized using the Non-Uniform Rational B-Spline (NURBS) basis functions to take advantage of easy control of smoothness and continuity. In addition, NURBS approximations can produce conic geometries exactly and provide higher efficiency for higher-order elements. The values of the level-set function at the control points (called expansion coefficients) are utilized as design variables. For optimization, we use an advanced mathematical programming technique, Sequential Quadratic Programming. Taking into account a large number of design variables and the small number of constraints associated with our optimization problem, the adjoint method is utilized to calculate the required sensitivities with respect to the design variables. The efficiency and robustness of the proposed method are demonstrated by solving three numerical examples. We have also shown that the current method can handle different geometries and types of objective functions. In addition, regularization techniques such as Tikhonov regularization and volume regularization have been explored to reduce unnecessary complexity and increase the manufacturability of optimized topologies.

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.

Patch-wise quadrature of trimmed surfaces in Isogeometric Analysis

This work presents an efficient quadrature rule for shell analysis fully integrated in CAD by means of Isogeometric Analysis (IGA). General CAD-models may consist of trimmed parts such as holes, intersections, cut-offs etc. Therefore, IGA should be able to deal with these models in order to fulfil its promise of closing the gap between design and analysis. Trimming operations violate the tensor-product structure of the used Non-Uniform Rational B-spline (NURBS) basis functions and of typical quadrature rules. Existing efficient patch-wise quadrature rules consider actual knot vectors and are determined in 1D. They are extended to further dimensions by means of a tensor-product. Therefore, they are not directly applicable to trimmed structures. The herein proposed method extends patch-wise quadrature rules to trimmed surfaces. Thereby, the number of quadrature points can be significantly reduced. Geometrically linear and non-linear benchmarks of plane, plate and shell structures are investigated. The results are compared to a standard trimming procedure and a good performance is observed.

Extended multiscale isogeometric analysis for mechanical simulation of two-dimensional periodic heterogeneous materials

Periodic heterogeneous material (PHM) is widely used in the design of multi-functional composite structures. Embedded particles or void holes with smooth boundaries are common in PHM, and the simulation of such material needs novel multiscale simulation method. Thus, extended multiscale isogeometric analysis (EmsIGA) is developed for the mechanical analysis of planar PHM, particularly for heterogeneous materials with smooth inclusions or voids. The non-uniform rational B-spline basis function is used for geometrical description and also mechanical simulation of the fine-scale unit cells to obtain numerical heterogeneity function. By calculating the numerical heterogeneity function as a static basis function, the equivalent element stiffness matrix of the sub-grids in the coarse scale is obtained and the structural problem can be solved. Downscaling computation is applied to obtain the stress and strain of the structure. Numerical examples validate the efficiency and accuracy of the proposed EmsIGA method in the simulation of heterogeneous structures. Compared with the results of the traditional and multi-scale finite element method, the proposed method can precisely capture the geometry and concentrated stress state of the smooth inclusions and calculate the microscopic stress and strain with improved efficiency.

Stress-related topology optimization with multilevel smoothed isogeometric densities and Bézier elements

Isogeometric analysis (IGA) provides a feasible technique to seamlessly integrate computer-aided design (CAD) into the existing finite element analysis. In this article, a Bézier elements-based IGA method is established to address the stress-related topology optimization of structures, which incorporates geometrical representation, structural analysis and topology optimization into a unified process as well as can naturally handle the curvilinear edge elements for the stress problems. To enhance the smoothness of the structural boundaries, a multilevel isogeometric density field (IDF) defined at control points (CPs) is designed by utilizing the values of the non-uniform rational B-splines (NURBS) basis functions from the previous design iteration. A global stress measure is approximated by a smooth maximum function with an additional error term. The k-means clustering originally from signal processing is used to partition the stress observations with dynamic centroids and indices, which improves the accuracy of the sensitivity results. The sensitivity analyses of the stress-related problems are derived from the multilevel IDF model. Three kinds of numerical results are provided to illustrate the effectiveness of the proposed method to solve stress-related problems via the multilevel IDF model for achieving the optimal solution with smooth boundaries.

Isogeometric analysis based investigation on material filling of coin cavities

The coining technology produces a wide variety of commemorative coins with exquisite patterns. However, it often encounters defects such as insufficient filling, flash lines, light bands, and so on. Process engineers usually perform multiple tryouts to avoid the above-mentioned problems in actual production. This is not only time-consuming and laborious but also ineffective. The virtual tryout of the finite element method (FEM) could assist engineers to avoid the defects in the coining process with a great improvement in product quality. In order to exactly describe complex patterns of commemorative coins, a large number of elements are employed in the classical FEM. Even then, the three dimensional elements, which come in early contact with the reliefs of the punch/die, undergo large deformation and become distorted. Errors of contact judgment between the tools and the workpiece in the FEM occur during the simulation process. Taking into account the advantage of Non-Uniform Rational B-Spline (NURBS) basis functions when accurately describing complex boundaries or surfaces, isogeometric analysis (IGA) is developed for studying the material filling of coin cavities. Six numerical examples involving elastic and plastic analyses with/without contact issues are considered by the presented IGA frameworks and show good performance of the present method in simulating the cavity filling compared with ABAQUS. In addition, numerical findings also indicate that the proposed method exhibits excellent contact detection and strong anti-mesh distortion in large deformation of the coining process. These encouraging observations motivate us to explore the NURBS description of complicated reliefs of coins and the corresponding IGA framework for the coining process.

Three-dimensional bending and free vibration analyses of laminated cylindrical panel with/without elastic foundation using two-dimensional discrete method

In this paper, a novel isogeometric scaled boundary finite element method (IGSBFEM) is developed to provide a comprehensive investigation into bending and free vibration of laminated cylindrical panels resting on Pasternak foundation. The significant novelty of this method lies in that the proposed formulation is derived based on the three-dimensional (3D) theory of elasticity, while only two-dimensional (2D) reference surface associated with the curvilinear coordinate system needs to be discretized by using isogeometric elements constructed with the non-uniform rational B-splines (NURBS) basis functions, which allows to reduce the computation cost and maintain the exact geometry of problem with desired precision flexibly. The SBFEM governing equation of each single lamina is reduced to a matrix form of first-order ordinary differential equations by introducing a vector of dual variables made of nodal displacements and forces, and the solution can be expressed analytically in terms of the radial coordinate. Therefore, shear locking phenomenon can be overcome naturally without any shear correction factors or reduced integration. The global solving equations of laminated cylindrical panels resting on Pasternak foundation are assembled with a layer-wise method. In addition, the high-order continuous features of isogeometric elements are insensitive to distorted mesh. Therefore, it is compatible with the research of complex geometrical structures with holes. The superior performances of the current approach are testified in the analyses of cylindrical panels with or without elastic foundation, solutions obtained by many other researchers are used for comparison purpose, and effects of several parameters including length and thickness, fiber orientation, stacking sequence and foundation stiffnesses coefficient on the bending or free vibration characteristics are also discussed.

New Perspective to Isogeometric Analysis: Solving Isogeometric Analysis Problem by Fitting Load Function

Isogeometric analysis (IGA) is introduced to establish the direct link between computer-aided design and analysis. It is commonly implemented by Galerkin formulations (isogeometric Galerkin, IGA-G) through the use of nonuniform rational B-splines (NURBS) basis functions for geometric design and analysis. Another promising approach, isogeometric collocation (IGA-C), working directly with the strong form of the partial differential equation (PDE) over the physical domain defined by NURBS geometry, calculates the derivatives of the numerical solution at the chosen collocation points. In a typical IGA, the knot vector of the NURBS numerical solution is only determined by the physical domain. A new perspective on the IGA method is proposed in this study to improve the accuracy and convergence of the solution. Solving the PDE with IGA can be regarded as fitting the load function defined on the NURBS geometry (right-hand side) with derivatives of the NURBS numerical solution (left-hand side). Moreover, the design of the knot vector has a close relationship to the NURBS functions to be fitted in the area of data fitting in geometric design. Therefore, the detected feature points of the load function are integrated into the initial knot vector of the physical domain to construct the new knot vector of the numerical solution. Then, they are connected seamlessly with the IGA-C framework for its great potential combining the accuracy and smoothness merits with the computational efficiency, which we call isogeometric collocation by fitting load function (IGA-CL). In numerical experiments, we implement our method to solve 1D, 2D, and 3D PDEs and demonstrate the improvement in accuracy by comparing it with the standard IGA-C method. We also verify the superiority in the accuracy of our knot selection scheme when employed in the IGA-G method, which we call isogeometric Galerkin by fitting load function (IGA-GL).

Sensitivity analysis of flexoelectric materials surrogate model based on the isogeometric finite element method

In this paper proposes a sensitivity analysis method based on a Polynomial Chaos Expansion (PCE) surrogate model for flexoelectric materials. The non-uniform rational B-splines (NURBS) basis functions to discretize the fourth-order partial differential equation for flexoelectricity and obtains a deterministic solution (electric potential). The mathematical expressions of surrogate model for the flexoelectric materials are established by considering uncertain parameters such as independent Young’s modulus, concentrated load and flexoelectric constants. The sensitivity expression is found by derivation the mathematical expression for the surrogate model. Moreover the finite difference method (FDM) are conducted in numerical examples to demonstrate the validity and correctness of the proposed algorithm.

Isogeometric neural networks: A new deep learning approach for solving parameterized partial differential equations

We present a new physics-based machine learning approach for solving parameterized partial differential equations (PDEs) over generalized domains. Central to this method is the coupling of isogeometric representation with deep learning. This approach represents both the physical domain and the solution to a PDE as a linear combination of non-uniform rational B-spline (NURBS) basis functions and employs a deep feed-forward neural network to predict NURBS solution coefficients. Trained networks output the coefficients of NURBS to approximate PDE solutions for an entire range of physical and shape variables. Desirable aspects of the NURBS basis, such as its local support, refinement properties, and its capacity to represent freeform shapes compactly, are synergistically integrated with the ability of neural networks to learn hidden and non-linear mappings. Novel features of this isogeometric neural network (IGN) approach include its ability to perform physics-based learning over changing domains and efficient hierarchical learning. In contrast to classical PDE forward-solving techniques such as finite element analysis and isogeometric analysis, the IGN is able to learn PDE solutions for a range of parametric variables. We show empirically that no labeled data is required to obtain high-fidelity approximations to the solutions of parameterized PDEs. To demonstrate the efficacy of the IGN, 1D and 2D problems of various physics are addressed.

A study on singular boundary integrals and stability of 3D time domain boundary element method

The paper presents a novel numerical scheme to effectively solve the weak, strong, and hyper-singular time-space integrals in the 3D time domain boundary element method (BEM). To deal with the singularities of P- or S-wavefront elements, the element subdivision method is used to identify non-zero valued sub-elements on the wavefronts, and the non-zero valued sub-elements containing source points are solved by the power series expansion method, which achieves the effect of obtaining satisfactory accuracy with fewer Gaussian points. In the present scheme, the non-uniform rational B-Spline (NURBS) basis functions are used to determine the physical coordinates of each sub-element with very high precision. Numerical results show that the proposed scheme can effectively implement a high-precision singular time-space integral solution under a unified framework. In addition, this paper uses the time-weighting method to improve the time integration algorithm to solve the instability problem. The validity and accuracy of the algorithm are verified by dynamics examples of the finite elastic body and infinite elastic body, and the effects of different weight points and integration schemes in the time-weighting method on the stability are discussed. The effectiveness of the proposed algorithm for large-scale elastodynamics problems is also examined using the 27 spherical cavities problem.

Isogeometric Analysis-Based Solidification Simulation and an Improved Way to Apply Supercooling on Droplet Boundary

The classic finite difference method (FDM) has been successfully adopted in the simulation of dendritic solidification, which is based on phase-field theory. Nevertheless, special strategies of boundary integral and projection are required for applying a supercooling rate to a droplet surface. In the present study, isogeometric analysis (IGA) is employed to discretize the phase-field equation due to the two advantages of Non-Uniform Rational B-Splines (NURBS) basis functions, namely an arbitrary order of derivatives and exact description of complex geometry. In addition, an improved, easy way to apply the supercooling rate on a melt droplet surface is proposed to avoid the integral and projection of the cellular boundary required in FDM. Firstly, dendrite growth in a square computational domain is simulated to verify the performance of IGA. Then, the influences of latent heat, anisotropic mode and initial angle on the dendrite shapes are studied by the presented IGA, FDM and finite element method (FEM). Finally, dendritic solidification in a droplet under different cooling rates along irregular boundaries is performed by the proposed IGA.

A dynamic description of material brittle failure using a hybrid phase-field model enhanced by adaptive isogeometric analysis

The objective of this work is to study dynamic crack propagation in brittle materials under time-dependent loading conditions by using the recently developed adaptive isogeometric phase-field approach. The current approach owns several ingredients including the advantages of the phase-field method (PFM), which can be used to model complex crack morphologies without any fracture criterion, and the isogeometric analysis, which possesses some excellent features such as exact geometry and high-order continuity. In addition, the model is further enhanced by employing the locally refined non-uniform rational B-spline (LR NURBS) basis, which is utilized for spatial discretization and phase-field discretization, while the generalized-α or HHT-α method is employed for the temporal discretization. For the coupled elasto-phase field system, a staggered approach is adopted to compute the unknown field variables. Moreover, a hybrid formulation of the PFM is employed to recover the linearity of momentum equation. The structured mesh adaptive refinement is conducted based on the prescribed threshold on the phase-field variable, thus it can effectively alleviate the burden of computational overhead, which is the main shortcoming of the PFM. Three problems involving dynamic crack growth is solved to show the robustness and the applicability of the proposed approach. It is shown that the proposed approach can obtain accurate results at a fraction of degrees-of-freedom without compromising accuracy in comparison with uniformly refined mesh.

A CNN-based surrogate model of isogeometric analysis in nonlocal flexoelectric problems

We proposed a convolutional neural network (CNN)-based surrogate model to predict the nonlocal response for flexoelectric structures with complex topologies. The input, i.e. the binary images, for the CNN is obtained by converting geometries into pixels, while the output comes from simulations of an isogeometric (IGA) flexoelectric model, which in turn exploits the higher-order continuity of the underlying non-uniform rational B-splines (NURBS) basis functions to fast computing of flexoelectric parameters, e.g., electric gradient, mechanical displacement, strain, and strain gradient. To generate the dataset of porous flexoelectric cantilevers, we developed a NURBS trimming technique based on the IGA model. As for CNN construction, the key factors were optimized based on the IGA dataset, including activation functions, dropout layers, and optimizers. Then the cross-validation was conducted to test the CNN’s generalization ability. Last but not least, the potential of the CNN performance has been explored under different model output sizes and the corresponding possible optimal model layout is proposed. The results can be instructive for studies on deep learning of other nonlocal mech-physical simulations.

Bézier extraction based isogeometric topology optimization with a locally-adaptive smoothed density model

This paper presents a Bézier extraction based isogeometric topology optimization framework, in which the pseudo-densities and the weights at control points (CPs) are simultaneously treated as the design variables. A locally-adaptive smoothed density field, that is dynamically updated at each design iteration, is first proposed by utilizing the values of the non-uniform rational B-splines (NURBS) basis functions. The locally-adaptive density model can naturally handle the non-uniform CPs of the curvilinear edge Bézier elements, which enhances the smoothness of structural shape. In the Bézier extraction based topology optimization, the density surface of the optimized structure can be exactly described by NURBS. Then, the optimal shape can be achieved by slicing the density surface with a predefined level-set. The sensitivity analyses of the structural compliance and the compliant mechanism are derived with respect to the design variables and evaluated from the Gaussian quadrature points instead of each element centroid. Some advantages concerning the proposed isogeometric topology optimization are illustrated with several numerical examples that are widely used in recent literature of topology optimization. Meanwhile, the influences of the design parameters on the optimal solution are also discussed to assess the effectiveness of the method.

Multiscale structural analysis of thick sandwich structures using parametric HFGMC micromechanics and isogeometric plate formulation based on refined zigzag theory

This study presents a novel multiscale modeling strategy by combining the parametric high fidelity generalized method of cells (HFGMC) micromechanical model and isogeometric analysis (IGA) based on refined zigzag theory (RZT) for thick sandwich structures. Effective behavior of sandwich face sheets at the macro scale is defined by using constituent (fiber and matrix) properties in the HFGMC model, thereby eliminating the need for time-consuming and costly experiments. At the macro level, the RZT kinematics is utilized to define zigzag distributions of in-plane displacement along thickness coordinate, which can accurately model a thick laminate or a sandwich structure as a single layer regardless of its plies. Moreover, the IGA is leveraged to discretize geometry of the structure effectively with non-uniform rational B-splines (NURBS) basis functions, i.e., free of shear locking phenomena. Various test cases of thin/thick laminates with high heterogeneity and anisotropy are analyzed to demonstrate wide-range applicability of the present approach. Results of the proposed multiscale framework provide excellent distributions of in-plane displacements and stresses along the thickness directions of the laminates as compared to the three-dimensional reference solutions. Hence, computational efficiency, high accuracy, and low cost of the present method are revealed for sandwich structures.

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.

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-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.