Non-uniform Rational B-spline Patches Research Articles

Analyzing free vibration and buckling of heated laminated plate with cutouts: A Nitsche-based isogeometric approach

In this research, a comprehensive Nitsche-based isogeometric analysis framework is presented for studying the free vibration and buckling characteristics of laminated plates with cutouts under uniform and non-uniform thermal loads. The approach utilizes non-overlapping non-uniform rational B-Splines patches to discretize the perforated plate and seamlessly connects them using the Nitsche method. The technique is then applied to investigate the free vibration and buckling behaviors of laminated plates with cutouts under varying thermal conditions. Three tests with different element sizes are used to analyze composite plate under uniform thermal load, demonstrating good convergence and accuracy of the present method. Rectangular laminated plates without cutouts, considering various boundary conditions, fiber orientations, and temperature variations under uniform or non-uniform thermal loads are analyzed. The natural frequencies and critical buckling temperatures are found to be in excellent agreement with existing literature. Additionally, the analysis is expanded for laminated plates with cutouts under diverse boundary conditions, considering variations in fiber orientations, cutout shapes and numbers, and types of thermal loads. Thorough investigations are carried out to understand the influence of these factors on the vibrational and buckling behaviors of the plates. Moreover, numerical study with non-conforming mesh demonstrates strong capability of the proposed method.

Blended isogeometric Kirchhoff–Love and continuum shells

The computational modeling of thin-walled structures based on isogeometric analysis (IGA), non-uniform rational B-splines (NURBS), and Kirchhoff–Love (KL) shell formulations has attracted significant research attention in recent years. While these methods offer numerous benefits over the traditional finite element approach, including exact representation of the geometry, naturally satisfied high-order continuity within each NURBS patch, and computationally efficient rotation-free formulations, they also present a number of challenges in modeling real-world engineering structures of considerable complexity. Specifically, these NURBS-based engineering models are usually comprised of numerous patches, with discontinuous derivatives, non-conforming discretizations, and non-watertight connections at their interfaces. Moreover, the analysis of such structures often requires the full stress and strain tensors (i.e., including the transverse normal and shear components) for subsequent failure analysis and remaining life prediction. Despite the efficiency provided by the KL shell, the formulation cannot accurately predict the response in the transverse directions due to its kinematic assumptions. In this work, a penalty-based formulation for the blended coupling of KL and continuum shells is presented. The proposed approach embraces both the computational efficiency of KL shells and the availability of the full-scale stress/strain tensors of continuum shells where needed by modeling critical structural components using continuum shells and other components using KL shells. The proposed method enforces the displacement and rotational continuities in a variational manner and is applicable to non-conforming and non-smooth interfaces. The efficacy of the developed method is demonstrated through a number of benchmark studies with a variety of analysis configurations, including linear and nonlinear analyses, matching and non-matching discretizations, and isotropic and composite materials. Finally, an aircraft horizontal stabilizer is considered to demonstrate the applicability of the proposed blended shells to real-world engineering structures of significant complexity.

Spatial adaptivity of the SAAF and Weighted Least Squares (WLS) forms of the neutron transport equation using constraint based, locally refined, isogeometric analysis (IGA) with dual weighted residual (DWR) error measures

This paper describes a methodology that enables NURBS (Non-Uniform Rational B-spline) based Isogeometric Analysis (IGA) to be locally refined. The methodology is applied to continuous Bubnov-Galerkin IGA spatial discretisations of second-order forms of the neutron transport equation. In particular this paper focuses on the self-adjoint angular flux (SAAF) and weighted least squares (WLS) equations. Local refinement is achieved by constraining degrees of freedom on interfaces between NURBS patches that have different levels of spatial refinement. In order to effectively utilise constraint based local refinement, adaptive mesh refinement (AMR) algorithms driven by a heuristic error measure or forward error indicator (FEI) and a dual weighted residual (DWR) or goal-based error measure (WEI) are derived. These utilise projection operators between different NURBS meshes to reduce the amount of computational effort required to calculate the error indicators. In order to apply the WEI to the SAAF and WLS second-order forms of the neutron transport equation the adjoint of these equations are required. The physical adjoint formulations are derived and the process of selecting source terms for the adjoint neutron transport equation in order to calculate the error in a given quantity of interest (QoI) is discussed. Several numerical verification benchmark test cases are utilised to investigate how the constraint based local refinement affects the numerical accuracy and the rate of convergence of the NURBS based IGA spatial discretisation. The nuclear reactor physics verification benchmark test cases show that both AMR algorithms are superior to uniform refinement with respect to accuracy per degree of freedom. Furthermore, it is demonstrated that for global QoI the FEI driven AMR and WEI driven AMR produce similar results. However, if local QoI are desired then WEI driven AMR algorithm is more computationally efficient and accurate per degree of freedom.

Isogeometric boundary element for analyzing steady-state heat conduction problems under spatially varying conductivity and internal heat source

This paper proposes an isogeometric boundary element (IGABEM) with the aim to solve the 2D steady-state heat conduction problems under spatially varying conductivity and internal heat source. The IGABEM boundary-domain integral equation is derived underlying the divergence theorem of Gauss and the Laplace equation. The non-uniform rational B-spline (NURBS) basis used in the CAD/CAE industries is applied to approximate both the boundary of problem domain and the unknown physical quantities, and the radial integration method (RIM) is employed to deal with the domain integrals induced by the non-homogeneous thermal conductivity and the internal heat source. The present boundary-domain integral equation is firstly divided into several NURBS patches, upon which the IGABEM can be further defined. The main advantage of the NURBS basis in comparison with the standard piecewise polynomial basis is that there is almost no error in the boundary approximation with IGABEM, which plays a significant role in controlling the accuracy of numerical calculation. Four numerical examples consisting of square regions with cubic and exponential thermal conductivities (with or without internal heat source), annulus region with constant and quadratic conductivities as well as complex region with varying thermal conductivity and complex heat source are provided. The accuracy and convergence of the proposed method are assessed by comparing the present solution with the existing results obtained by several other researchers or ANSYS results, and excellent performance is achieved.

Three-dimensional multi-patch isogeometric analysis of composite laminates with a discontinuous Galerkin approach

In this study, a three-dimensional discontinuous Galerkin isogeometric analysis framework is presented for the analysis of composite laminates. Non-uniform rational B-splines are employed as basis functions for both geometric and computational implementations. From a practical point of view, modeling with multiple non-uniform rational B-spline patches is required in many different applications due to the complexity of computational domains. Then, a special numerical technique is necessary to couple different non-uniform rational B-spline patches to carry out the isogeometric analysis. In this study, therefore, one of the discontinuous Galerkin methods, namely, symmetric interior penalty Galerkin formulation is utilized to deal with multi-patch isogeometric analysis applications. In order to show the applicability of the proposed framework, composite laminates under sinusoidally distributed load with different stacking sequences are studied in the numerical examples. The predicted results are compared with those obtained by the three-dimensional elasticity solutions and various numerical models available in the literature.

Modeling technology of curved surface development for puffer fish

The drag reduction mechanism of puffer epidermis was closely related to its real geometry. In order to solve the modeling problem of epidermal spines on the puffer surface, a modeling method for the expansion of puffer shape was proposed. The three-dimensional scanning and non-uniform rational B-spline surface modeling technology was used to reconstruct the puffer model. According to the curvature characteristics, the surface mathematical equations including exponential, logarithmic, and sinusoidal functions were established based on the multinomial function. The surface was generated by a mathematical equation, and the surface was divided into several non-uniform rational B-spline patches according to curvature. After discretization, the point cloud Gaussian curvature and average value were calculated based on the implicit equation of moving least square surface, and whether the surface is approximately extensible or not was judged. Finally, the puffer surface was divided into 46 curved patches. In this article, the surface expansion algorithm gave priority to ensure the area unchanged, and four feature surfaces were selected according to the epidermal spines arrangement of the puffer surface. The results showed that the technique can simply and efficiently unfold the curved surface of the puffer fish, thus the mapping relationship between the epidermal spines on the surface and the plane was determined, which established a foundation for the accurate arrangement and modeling of the epidermal spines on the surface.

FSDT-Based Isogeometric Analysis for Free Vibration Behavior of Functionally Graded Skew Folded Plates

In this paper, an isogeometric analysis (IGA) approach for free vibration analysis of functionally graded skew folded (FGSF) plates is presented for the first time. This method uses non-uniform rational B-splines (NURBS) as basis functions to provide a flexibility in refinement process. Furthermore, the IGA is able to exactly regenerate the geometry in comparison with the finite-element method (FEM), even at a coarse level of discretization. It is assumed that the mechanical properties of the FG metal–ceramic plates vary continuously through the thickness according to a power-law distribution. Also, the effective mechanical properties are captured by conventional rule of mixture. Regardless of the commonly used plates, the skew folded plates need to be modeled using two NURBS patches. The simple first-order shear deformation theory (S-FSDT) is employed to derive the equations of motion for each patch. After assemblage of the element matrices within each patch, the mass and stiffness matrices are modified and transformed to the global coordinates by employing an appropriate coordinate transformation. Subsequently, the bending strip method is utilized to accomplish the continuity conditions along the interface of the patches. The accuracy and reliability of the proposed IGA model are checked through comparison of the computed results with the available solutions in the open literature. The comparative study exhibits that the presented S-FSDT-based isogeometric formulation is promising to precisely predict the natural frequencies of FGSF plates with a low computational time. Eventually, a parametric study is performed and the variations of the natural frequencies with respect to the effective parameters such as crank angle, skew angle, length-to-thickness ratio, volume fraction and boundary conditions are examined.

A geometry conforming isogeometric method for the self-adjoint angular flux (SAAF) form of the neutron transport equation with a discrete ordinate ([formula omitted]) angular discretisation

This paper presents the application of isogeometric analysis (IGA) to the spatial discretisation of the multi-group, self-adjoint angular flux (SAAF) form of the neutron transport equation with a discrete ordinate (SN) angular discretisation. The IGA spatial discretisation is based upon non-uniform rational B-spline (NURBS) basis functions for both the test and trial functions. In addition a source iteration compatible maximum principle is used to derive the IGA spatially discretised SAAF equation. It is demonstrated that this maximum principle is mathematically equivalent to the weak form of the SAAF equation. The rate of convergence of the IGA spatial discretisation of the SAAF equation is analysed using a method of manufactured solutions (MMS) verification test case. The results of several nuclear reactor physics verification benchmark test cases are analysed. This analysis demonstrates that for higher-order basis functions, and for the same number of degrees of freedom, the FE based spatial discretisation methods are numerically less accurate than IGA methods. The difference in numerical accuracy between the IGA and FE methods is shown to be because of the higher-order continuity of NURBS basis functions within a NURBS patch as well as the preservation of both the volume and surface area throughout the solution domain within the IGA spatial discretisation. Finally, the numerical results of applying the IGA SAAF method to the OECD/NEA, seven-group, two-dimensional C5G7 quarter core nuclear reactor physics verification benchmark test case are presented. The results, from this verification benchmark test case, are shown to be in good agreement with solutions of the first-order form as well as the second-order even-parity form of the neutron transport equation for the same order of discrete ordinate (SN) angular approximation.

Discontinuous Galerkin Based Isogeometric Analysis for Geometric Flows and Applications in Geometric Modeling

We propose a method which combines isogeometric analysis with the discontinuous Galerkin (DG) method for second and fourth order geometric flows to generate fairing surfaces, which are composed of multiple patches. This technique can be used to tackle a challenging problem in geometric modeling---gluing multi-patches together smoothly to create complex models. Non-uniform rational B-splines (NURBS), the most popular representations of geometric models developed in Computer Aided Design, are employed to describe the geometry and represent the numerical solution. Since NURBS basis functions over two different patches are independent, DG methods can be appropriately applied to glue the multiple patches together to obtain smooth solutions. We present semi-discrete DG schemes to solve the problem, and $$\mathcal {L}^{2}$$L2-stability is proved for the proposed schemes. Our method enjoys the following advantages. Firstly, the geometric flexibility of NURBS basis functions, especially the use of multiple patches, enable us to construct surface models with complex geometry and topology. Secondly, the constructed geometry is fair. Thirdly, since only the control points of the NURBS patches evolve in accordance with the geometric flows, and their number (degrees of freedom) is very small, our algorithm is very efficient. Finally, this method can be easily formulated and implemented. We apply the method in mean curvature flows and in quasi surface diffusion flows to solve various geometric modeling problems, such as minimal surface generation, surface blending and hole filling, etc. Examples are provided to illustrate the effectiveness of our method.

Eulerian analysis of bulk metal forming processes based on spline-based meshfree method

An effective Eulerian approach employing Non-Uniform Rational B-Spline (NURBS) curves as a boundary representing technique is presented. This approach is based on the Spline-Based Meshfree Method: analysis domain boundaries are represented by NURBS curves and a NURBS patch is employed as a background mesh. By doing so, clear and neat boundary representations are possible. Moreover, the explicit definition of the analysis domain boundary dispenses with the need to solve convective transport equations. In this work, utilizing the above idea, bulk metal forming processes are treated within Eulerian framework based on the flow formulation of rigid-plastic materials. The implementation issues of the proposed method are discussed in detail. The effectiveness of the present approach is shown by comparing with other numerical methods. It is demonstrated that the method yields reliable results even in the case where severe deformation occurs.

A multi-patch nonsingular isogeometric boundary element method using trimmed elements

One of the major goals of isogeometric analysis is direct design-to-analysis, i.e., using computer-aided design (CAD) files for analysis without the need for mesh generation. One of the primary obstacles to achieving this goal is CAD models are based on surfaces, and not volumes. The boundary element method (BEM) circumvents this difficulty by directly working with the surfaces. The standard basis functions in CAD are trimmed nonuniform rational B-spline (NURBS). NURBS patches are the tensor product of one-dimensional NURBS, making the construction of arbitrary surfaces difficult. Trimmed NURBS use curves to trim away regions of the patch to obtain the desired shape. By coupling trimmed NURBS with a nonsingular BEM, the formulation proposed here comes close achieving the goal of direct design to analysis. Example calculations demonstrate its efficiency and accuracy.

A Nitsche-type formulation and comparison of the most common domain decomposition methods in isogeometric analysis

This paper provides a detailed elaboration and assessment of the most common domain decomposition methods for their application in isogeometric analysis. The methods comprise a penalty approach, Lagrange multiplier methods, and a Nitsche-type method. For the Nitsche method, a new stabilized formulation is developed in the context of isogeometric analysis to guarantee coercivity. All these methods are investigated on problems of linear elasticity and eigenfrequency analysis in 2D. In particular, focus is put on non-uniform rational B-spline patches which join nonconformingly along their common interface. Thus, the application of isogeometric analysis is extended to multi-patches, which can have an arbitrary parametrization on the adjacent edges. Moreover, it has been shown that the unique properties provided by isogeometric analysis, that is, high-order functions and smoothness across the element boundaries, carry over for the analysis of multiple domains. Copyright © 2013 John Wiley & Sons, Ltd.

Spline-based meshfree method

SUMMARY In the present work, the spline-based meshfree method (SBMFM) is presented. The SBMFM uses spline basis functions for field variables and boundary representation. Because of the inherent properties of splines, that is, NURBS (nonuniform rational B-splines), the analysis domain used in the SBMFM has its own parametric domain. Unlike the classical concepts in the Computer-aided design (CAD) community in which the NURBS patches are used to model the body, the NURBS patch is treated as a background mesh in the present work. The real analysis domain is created by trimming operation so that the domain boundaries are described by spline curves only on a single NURBS patch. Building the geometry in this way enables a convenient boundary representation of the domains because direct manipulation of spline curves becomes possible. From a deformation point of view, the mesh distortion problem is effectively avoided even if the analysis domain undergoes a large deformation. In the SBMFM, boundary spline curves freely move with complete independence from the quadrilateral background NURBS patch. The so-called element connectivity is unnecessary for geometry description. In this paper, geometrically nonlinear problems are considered. Implementation strategy is presented and the effectiveness of the present SBMFM is shown through various numerical examples. Copyright © 2012 John Wiley & Sons, Ltd.

Application of the Characteristic Basis Function Method Utilizing a Class of Basis and Testing Functions Defined on NURBS Patches

A novel implementation of the characteristic basis function method (CBFM) is given, in which the high-level basis functions, called characteristic basis functions (CBFs), are represented in terms of curved rooftops generated from nonuniform rational B-splines (NURBS) in the parametric (u,v) domain. The associated macro-testing functions are defined by using curved razor-blade functions corresponding to each rooftop. The underlying objective of the CBFM is the reduction of the number of unknowns that arise from the discretization process when applying the conventional method of moments (MoM). The result is, therefore, an approach which can handle many complex cases via direct solvers, without suffering from convergence problems as many of the iterative techniques are known to do when dealing with ill-conditioned matrices. As a result of the combination of the CBFM with the special class of low-level basis and testing functions directly located over NURBS surfaces, complex and realistic geometries can be efficiently analyzed to yield accurate results while reducing the CPU time as well as required memory resources.

Dynamic maintenance and visualization of molecular surfaces

Molecular surface computations are often necessary in order to perform synthetic drug design. A critical step in this process is the computation and update of an exact boundary representation for the molecular surface (e.g. the Lee–Richards surface). In this paper we introduce efficient techniques for computing a molecular surface boundary representation as a set of non-uniform rational B-splines patches. This representation introduces for molecules the same geometric data structure used in the solid modeling community and enables immediate access to a wide range of modeling operations and techniques. Furthermore, this allows the use of any general solid modeling or visualization system as a molecular modeling interface. However, using such a representation in a molecular modeling environment raises several efficiency and update constraints, especially in a dynamic setting. For example, changes in the probe radius result in both geometric and topological changes to the set of patches. Our techniques provide the option of trading accuracy of the representation for the efficiency of the computation, while still tracking the changes in the set of patches. In particular, we discuss two main classes of dynamic updates: one that keeps the topology of the molecular configuration fixed, and a more complicated case where the topology may be updated continuously. In general the generated output surface is represented in a format that can be loaded into standard solid modeling systems. It can also be directly triangulated or rendered, possibly at different levels of resolution, by a standard graphics library such as OpenGL without any additional effort.

Reverse engineering of composite sculptured surfaces

A reverse engineering technique is proposed in this study for modelling composite sculptured surfaces. The proposed technique includes a surface fitting algorithm and a surface blending algorithm. The former is formulated as the least-squares minimisation problem for which an error expression was minimised which yields the optimised position of control points and parameter values for the non-uniform rational B-spline patch. Cubic spline and bicubic surface algorithms are developed for two-patch and four-patch blending, respectively. Computer simulation results clearly demonstrate that the proposed technique is useful for composite surfaces modelling applications. Factors affecting the convergence speed and surface accuracy in the optimisation process are also discussed.