B-spline Patches Research Articles

Shape optimization of non-matching isogeometric shells with moving intersections

While shape optimization using isogeometric shells exhibits appealing features by integrating design geometries and analysis models, challenges arise when addressing computer-aided design (CAD) geometries comprised of multiple non-uniform rational B-splines (NURBS) patches, which are common in practice. The intractability stems from surface intersections within these CAD models. In this paper, we develop an approach for shape optimization of non-matching isogeometric shells incorporating intersection movement. Separately parametrized NURBS surfaces are modeled using Kirchhoff–Love shell theory and coupled using a penalty-based formulation. The optimization scheme allows shell patches to move without preserving relative location with other members during the shape optimization. This flexibility is achieved through an implicit state function, and analytical sensitivities are derived for the relative movement of shell patches. The introduction of differentiable intersections expands the design space and overcomes challenges associated with large mesh distortion, particularly when optimal shapes involve significant movement of patch intersections in physical space. Throughout optimization iterations, all members within the shell structures maintain the NURBS geometry representation, enabling efficient integration of analysis and design models. The optimization approach leverages the multilevel design concept by selecting a refined model for accurate analysis from a coarse design model while maintaining the same geometry. We adopt several example problems to verify the effectiveness of the proposed scheme and demonstrate its applicability to the optimization of the internal stiffeners of an aircraft wing.

Nitsche's method enhanced isogeometric homogenization of unidirectional composites with cylindrically orthotropic carbon/graphite fibers

An isogeometric homogenization (IGH) technique is constructed for the homogenization and localization of unidirectional composites with radially or circumferentially orthotropic carbon/graphite fibers. The proposed theory employs multiple non-conforming Non-Uniform Rational B-Splines (NURBS) patches to depict repeating unit cells (RUCs) representative of composite microstructures. Displacements are formulated using a two-scale expansion that integrates macroscopic and microscopic contributions, with the latter addressed through the isogeometric analysis technique. Nitsche's method is utilized to apply the interfacial traction and displacement continuity and periodicity conditions. The capability and accuracy of the IGH theory were validated upon comparison with the elasticity solutions that take into account explicitly fiber morphologies, along with classical micromechanics solutions based on equivalent fiber moduli. A comparative analysis with conventional finite-element techniques showcases the developed theory's ability to accurately replicate the singular stress field at the fiber center and to capture smooth stress distributions where significant stress gradients are encountered.

Locking-free isogeometric discretizations of linear plane Timoshenko rods: LAS elements

We make use of linear plane curved Timoshenko rods as a model problem to study how to overcome shear and membrane locking in NURBS-based discretizations. In this work, we propose lumped-assumed-strain (LAS) elements, a projection-based assumed-strain treatment that removes shear and membrane locking for a very broad range of slenderness ratios. For a NURBS patch with basis functions of degree p and Cp−1 continuity across element boundaries, the space of the assumed strains is defined on a B-spline patch with the same elements, but using basis functions of degree p−1 and Cp−2 continuity across element boundaries. The assumed strains are obtained by performing a L2 projection of the compatible strains at the patch level in which the consistent mass matrix is substituted with the lumped mass matrix. Our numerical investigations suggest that LAS elements need either the same or slightly finer mesh resolutions than B̄ elements in order for the relative errors in L2 norm of the unknowns and the stress resultants to be all below 1%. However, LAS elements are significantly more computationally efficient than B̄ elements for a given mesh. The use of C1-continuous quadratic LAS elements with 2 Gauss–Legendre quadrature points per element exhibit superconvergence of the stress resultants for most slenderness ratios, making it a particularly cost-effective choice of obtaining accurate results. When compared with linear Lagrange B̄ elements with 2 Gauss–Legendre quadrature points per element, C1-continuous quadratic LAS elements with 2 Gauss–Legendre quadrature points per element require far fewer elements in order to obtain relative errors in L2 norm of the unknowns and the stress resultants smaller than 1%.

An adaptive multi-patch isogeometric phase-field model for fatigue fracture

Fatigue fracture is one of the main causes of structural failure under cyclic loading, so accurate evaluation of fatigue fracture is very important for the design of structures subjected to cyclic loading. In this work, we propose an adaptive fatigue phase-field model within the framework of multi-patch isogeometric analysis to simulate crack nucleation and propagation in brittle materials under cyclic loading. Multiple non-uniform rational B-splines (NURBS) patches are used to exactly represent complex structures, and the Nitsche’s method is adopted to enforce the compatibility of the displacements, stresses and phase-field variables between two patches. The coupled governing equations of fatigue phase-field model are established based on the Nitsche’s method, and the selection of Nitsche’s parameters are determined from numerical experiments. A refinement-correction adaptive approach based on locally refined non-uniform rational B-splines (LR NURBS) structured mesh refinement strategy is developed to alleviate the computational burden caused by the requirement of small element size around crack surface and the large number of fatigue loading. The proposed multi-patch isogeometric phase-field model can deal with the coupling between equal/unequal length interfaces, thus the fatigue crack propagation simulation in arbitrary complex structure can be implemented. Several numerical examples are investigated to verify the practicality of the multi-patch isogeometric phase-field model and the effectiveness of the adaptive scheme.

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.

A corotational isogeometric assumed natural strain shell element in updated lagrangian formulation for general geometric nonlinear analysis of thin-walled structures

A new general geometric nonlinear curved quadrilateral shell element, which can handle large displacements and large rotations, is presented in this paper. It is formulated based on the incremental virtual work equation in updated Lagrangian formulation for general large displacement analysis. The kinematics of shell element is described using non-uniform B-spline patch, which results in the element displacement field with higher order continuity and the element performance more resistive to shear locking and membrane locking. In order to perfectly avoid such membrane locking and shear locking behaviors, the assumed in-plane membrane strain field and the assumed transverse shear strain field for the incremental covariant Green-Lagrange strain are constructed as linear combinations of non-uniform B-spline basis functions with degrees lower than those adopted by the original displacement field interpolation. In addition, based on Euler parameters, the corotational procedure available for high order isogeometric shell element is developed to extract the strain-producing deformational displacements and rotations and to update the element stresses and realistic internal force vectors. The accuracy and robustness of proposed shell element are demonstrated through the solutions of various benchmark problems.

Multi-material isogeometric topology optimization in multiple NURBS patches

In design optimization of structures, design domains are often described by multiple non-uniform rational B-spline (NURBS) patches. To perform multi-material topological design in such design domains, an effective Nitsche-based multi-material isogeometric topology optimization (ITO) method is proposed in this paper. Specifically, the Nitsche's method is utilized to effectively couple the interfaces between adjacent NURBS patches. In order to preserve the coercivity of the bilinear form, the stabilization parameters are estimated locally according to the element size. Meanwhile, the discrete material optimization (DMO) approach is employed to describe the mixing of candidate materials. Analytical computation of sensitivities for coupled elements is developed as well. The effectiveness of the proposed method is demonstrated via several 2D and 3D numerical examples, where the design domains are constituted by conforming and non-conforming NURBS patches, respectively.

Bivariate non-uniform subdivision schemes based on L-systems

L–systems have been used to describe non-uniform, univariate, subdivision schemes, which offer more flexible refinement processes than the uniform schemes, while at the same time are easier to analyse than the geometry driven non-uniform schemes. In this paper, we extend L–system based non-uniform subdivision to the bivariate setting. We study the properties that an L–system should have to be the suitable descriptor of a subdivision refinement process. We derive subdivision masks to construct the regular parts of the subdivision surface as cubic B-spline patches. Finally, we describe stencils for the extraordinary vertices, which after a few steps become stationary, so that the scheme can be studied through simple eigenanalysis. The proposed method is illustrated through two new subdivision schemes, a Binary-Ternary, and a Fibonacci scheme with average refinement rate below two.

Multiscale topology optimization of cellular structures using Nitsche-type isogeometric analysis

This paper proposes a multiscale topology optimization method using Nitsche-type isogeometric analysis (IGA) for design of structures described by multiple non-uniform rational B-spline (NURBS) patches. In this method, at macroscale, unknown structural responses are calculated by Nitsche-type IGA, which not only ensures the consistency between geometric and analytical models, but also offers an effective way to enforce kinematic compatibility and mechanical equilibrium on interfaces between adjacent NURBS patches. At microscale, mechanical properties of microstructures are predicted by a Kriging metamodel at a low computational cost, which is constructed via some sample lattice unit cells (LUCs). In addition, analytical computation of sensitivities for coupled elements is developed. Finally, the structures are infilled by graded LUCs according to the density distribution within the design space. Some 2D and 3D numerical examples with conforming and non-conforming NURBS patches are presented to test the proposed method. Results indicate that the proposed method is effective for design of structures described by multiple NURBS patches.

New hybrid quadrature schemes for weakly singular kernels applied to isogeometric boundary elements for 3D Stokes flow

This work proposes four novel hybrid quadrature schemes for the efficient and accurate evaluation of weakly singular boundary integrals (1/r kernel) on arbitrary smooth surfaces. Such integrals appear in boundary element analysis for several partial differential equations including the Stokes equation for viscous flow and the Helmholtz equation for acoustics. The proposed quadrature schemes apply a Duffy transform-based quadrature rule (Duffy, 1982) to surface elements containing the singularity and classical Gaussian quadrature to the remaining elements. Two of the four schemes additionally consider a special treatment for elements near to the singularity, where refined Gaussian quadrature and a new moment-fitting quadrature rule are used.The hybrid quadrature schemes are systematically studied on flat B-spline patches and on NURBS spheres considering two different sphere discretizations: An exact single-patch sphere with degenerate control points at the poles and an approximate discretization that consist of six patches with regular elements. The efficiency of the quadrature schemes is further demonstrated in boundary element analysis for Stokes flow, where steady problems with rotating and translating curved objects are investigated in convergence studies for both, mesh and quadrature refinement. Much higher convergence rates are observed for the proposed new schemes in comparison to classical schemes.

An approach for multi-patch surface modification with a curve constraint satisfying convergentG1 continuity

AbstractConceptual shape design plays a critical role in determining the appearance and functionality of a product. Currently, computer-aided design systems can represent a complex shape through multiple surface patches; however, the methods used for multi-patch surface modification are still limited and counterintuitive. There is an ongoing need for more intuitive and efficient surface modification tools that allow designers to directly control shape changes through specific constraints while maintaining surface quality. In this paper, we propose an intuitive modification approach based on the designer’s activity to interactively manipulate a multi-patch surface with an arbitrary curve constraint with tangent continuity (G1) across connected B-spline patches. The advantages of the curve and surface superposition were used to change the shape of the surface. The first step was a shape-control mechanism that used shape functions to control the deformation of a superposed surface. A fine-tuning step was followed to ensure the aesthetic requirements and quality of the surface by achieving convergent G1 continuity in a linear manner. The efficiency and practicality of the proposed approach have been verified using application examples. The proposed approach can be applied to modify surfaces used in the design of a complex geometric model.

Dynamic projection mapping on deformable stretchable materials using boundary tracking

We present the first method for dynamic projection mapping on rectangular, deformable and stretchable elastic materials using consumer-grade time of flight (ToF) depth cameras. We use a B-Spline patch to model the projection surface without explicitly modeling the deformation. Leveraging the nature of deformable and stretchable materials, we propose an efficient tracking method that can track the boundary of the surface material. This achieves realistic mapping even in the interior of the surface such that the projection appears to be printed on the material. The surface representation is updated in real-time using GPU based computations. Further, we also show that the speed of these updates is limited by the speed of the camera and therefore can be adopted for higher speed cameras as well.

IGA Using Offset-based Overlapping Domain Parameterizations

Isogeometric analysis (IGA) is a numerical method, proposed in Hughes et al. (2005), that connects computer-aided design (CAD) with finite element analysis (FEA). In CAD the computational domain is usually represented by B-spline or NURBS patches. Given a B-spline or NURBS parameterization of the domain, an isogeometric discretization is defined on the domain using the same B-spline or NURBS basis as for the domain parameterization. Ideally, such an isogeometric discretization allows an exact representation of the underlying CAD model.CAD models usually represent only the boundary of the object. For planar domains, the CAD model is given as a collection of curves representing the boundary. Finding a suitable parameterization of the interior is one of the major issues for IGA, similar to the mesh generation process in the FEA setting. The objective of this isogeometric parameterization problem is to obtain a set of patches, which exactly represent the boundary of the domain and which are parameterized regularly and without self-intersections. This can be achieved by segmenting the domain into patches which are matching along interfaces, or by covering the domain with overlapping patches. In this paper we follow the second approach.To construct a planar parameterization suitable for IGA from a given boundary curve, we propose an offset-based domain parameterization algorithm. Given a boundary curve, we obtain an inner curve by generalized offsetting. The inner curve, together with the boundary curve, naturally defines a ring-shaped patch with an associated parameterization. By definition, the ring-shaped patch has a hole, which can be covered by a multi-cell domain. Consequently, the domain is represented as a union of two overlapping subdomains which are regularly parameterized. On such a configuration, one can employ the overlapping multi-patch (OMP) method, as introduced in Kargaran et al. (2019), to solve PDEs on the given domain. The performance of the proposed method is reported in several numerical examples, considering different shape properties of the given boundary curve.

Multi-sided B-spline surfaces over curved, multi-connected domains

We propose a new surface representation, the Generalized B-spline (GBS) patch, that combines ribbon interpolants given in B-spline form. A GBS patch can connect to tensor-product B-spline surfaces with arbitrary Gm continuity. It supports ribbons not only along the perimeter loop, but also around holes in the interior of the patches.This is a follow-up paper of a recent publication (Várady et al., 2020) that described multi-sided Bézier surfaces over curved multi-sided domains. While the fundamental concept is retained, several new details have been elaborated. The weighting functions are modified to be products of B-spline and Bernstein basis functions, multiplied by rational terms. A new local parameterization method is introduced using harmonic functions, that handles periodic hole loops, as well. Interior shape control is adapted to the B-spline representation of the ribbons. Several examples illustrate the capabilities of the proposed scheme. Our implementation is based on a computationally efficient discretization.

A discontinuous isogeometric reanalysis method and its application in closed-loop optimization problems

In this study, a DisContinuous IsoGeometric Reanalysis (DCIGR) method is proposed to analyze and optimize the model of Non-Uniform Rational B-Splines (NURBS) patches with non-matching interfaces. In the DCIGR, the Nitsche method is adapted to couple NURBS patches with gaps and the smooth algorithm is developed to correct singular stresses resulting from gaps. However, the additional stiffness matrices (interfacial stiffness matrices) must be calculated when using the Nitsche method, which will inevitably increase the computational cost. Therefore, Combined Approximation (CA) is introduced to reduce the computational cost of the optimization. Theoretically, the larger and more complex a problem is, the more efficient the DCIGR will be. Moreover, since the modifications of the model can be accurately represented by the IGA, the proposed method can be easily integrated with the closed-loop optimization. Compared with existing methods for non-matching interfaces, the DCIGR fully takes the advantage of the IGA and achieves the fast optimization based on the CA. Numerical examples are given to verify the accuracy and efficiency of the proposed method.

An efficient method to integrate polynomials over polytopes and curved solids

In this paper, we present an efficient approach to compute the integral of monomials and polynomials over polyhedra and regions defined by parametric curved boundary surfaces. We use Euler's theorem for homogeneous functions in combination with Stokes's theorem to reduce the integration of a monomial over a three-dimensional solid to its boundary. If the solid is a polytope, through a recursive application of these theorems, the integral is further reduced to just the evaluation of the monomial and its derivatives at the vertices of the polytope. The present approach is simpler than existing techniques that rely on repeated use of the divergence theorem, which require the antiderivative of the monomials and the projection of these functions onto hyperplanes. For convex and nonconvex polytopes, our approach does not introduce any approximation for the integration of monomials. For curved solid regions bounded by surfaces that admit a parameterization, the same approach yields simplified formulas to compute the integral of any homogeneous function, including monomials. For surfaces parameterized by polynomial surfaces (such as Bézier surface triangles and B-spline patches), the method yields machine-precision accuracy for the volumetric integration of monomials with an appropriate quadrature rule. Numerical examples over regions bounded by polynomial surfaces and rational surfaces are presented to establish the accuracy and efficiency of the method.

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.

Efficient extraction of hierarchical B-Splines for local refinement and coarsening of Isogeometric Analysis

The main motivation for hierarchical B-Splines in Isogeometric Analysis is to perform a local refinement that is globally not a tensor product. Compared to standard knot insertion, this allows to increase refinement locality. Moreover, the implementation of hierarchical refinement on existing codes can be facilitated by Bézier extraction. This approach was initially developed for unrefined patches and then extended to local refinement. In case of B-Spline patches that are not locally refined, extraction and projection algorithms can be optimized by leveraging the tensor structure. However, these optimizations are not straightforwardly applicable to hierarchical B-Splines.In this contribution, we show an approach where all extraction operations are formulated to work directly on the univariate extraction operators, without explicitly computing and storing the full tensor product. In particular, we consider the extraction of functions, and projection of degrees of freedom for refinement and coarsening. A slightly modified Bézier projection is proposed to fully exploit the developed algorithms. We finally compare our approach to an explicit computation of the extraction operator in both a linear transient and a nonlinear example.

A multisided [formula omitted] B-spline patch over extraordinary vertices in quadrilateral meshes

We propose a generalised B-spline construction that extends uniform bicubic B-splines to multisided regions spanned over extraordinary vertices in quadrilateral meshes. We show how the structure of the generalised Bézier patch introduced by Várady et al. can be adjusted to work with B-spline basis functions. We create ribbon surfaces based on B-splines using special basis functions. The resulting multisided surfaces are C2 continuous internally and connect with G2 continuity to adjacent regular and other multisided B-splines patches. We visually assess the quality of these surfaces and compare them to Catmull–Clark limit surfaces on several challenging geometrical configurations.

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.