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Isogeometric Analysis generalizes classical finite element analysis and intends to integrate it with the field of Computer-Aided Design. A central problem in achieving this objective is the reconstruction of analysis-suitable models from Computer-Aided Design models, which is in general a non-trivial and time-consuming task. In this article, we present a novel spline construction, that enables model reconstruction as well as simulation of high-order PDEs on the reconstructed models. The proposed almost-C1 splines are biquadratic splines on fully unstructured quadrilateral meshes (without restrictions on placements or number of extraordinary vertices). They are C1 smooth at all regular and extraordinary vertices. Moreover, they are C1 smooth across all edges between regular vertices and C0 smooth across all edges that are adjacent to an extraordinary vertex. The splines thus form H2-nonconforming analysis-suitable discretization spaces. This is the lowest-degree unstructured spline construction that can be used to solve fourth-order problems. The associated spline basis is non-singular and has several B-spline-like properties (e.g., partition of unity, non-negativity, local support), the almost-C1 splines are described in an explicit Bézier-extraction-based framework that can be easily implemented. Numerical tests suggest that the basis is well-conditioned and exhibits optimal approximation behaviour.

Easy to construct and optimally convergent generalisations of B-splines to unstructured meshes are essential for the application of isogeometric analysis to domains with non-trivial topologies. Nonetheless, especially for hexahedral meshes, the construction of smooth and optimally convergent isogeometric analysis basis functions is still an open question. We introduce a simple partition of unity construction that yields smooth blended B-splines, referred to as SB-splines, on semi-structured quadrilateral and hexahedral meshes, namely on mostly structured meshes with a few sufficiently separated unstructured regions. To this end, we first define the mixed smoothness B-splines that are $C^0$ continuous in the unstructured regions of the mesh but have higher smoothness everywhere else. Subsequently, the SB-splines are obtained by smoothly blending in the physical space the mixed smoothness B-splines with Bernstein bases of equal degree. One of the key novelties of our approach is that the required smooth weight functions are assembled from the available smooth B-splines on the unstructured mesh. The SB-splines are globally smooth, non-negative, have no breakpoints within the elements and reduce to conventional B-splines away from the unstructured regions of the mesh. Although we consider only quadratic mixed smoothness B-splines in this paper, the construction generalises to arbitrary degrees. We demonstrate the excellent performance of SB-splines studying Poisson and biharmonic problems on semi-structured quadrilateral and hexahedral meshes, and numerically establishing their optimal convergence in one and two dimensions., Comment: 28 pages, 26 figures, changes to title, abstract and body text after peer-review

Starting from a highly continuous isogeometric analysis discretization, we introduce hyperplanes that partition the domain into subdomains and reduce the continuity of the discretization spaces at these hyperplanes. As the continuity is reduced, the number of degrees of freedom in the system grows. The resulting discretization spaces are finer than standard maximal continuity IGA spaces. Despite the increase in the number of degrees of freedom, these finer spaces deliver simulation results faster with direct solvers than both traditional finite element and isogeometric analysis for meshes with a fixed number of elements. In this work, we analyze the impact of continuity reduction on the number of Floating Point Operations (FLOPs) and computational times required to solve fluid flow and electromagnetic problems with structured meshes and uniform polynomial orders. Theoretical estimates show that for sufficiently large grids, an optimal continuity reduction decreases the computational cost by a factor of O ( p 2 ) . Numerical results confirm these theoretical estimates. In a 2D mesh with one million elements and polynomial order equal to five, the discretization including an optimal continuity pattern allows to solve the vector electric field, the scalar magnetic field, and the fluid flow problems an order of magnitude faster than when using a highly continuous IGA discretization. 3D numerical results exhibit more moderate savings due to the limited mesh sizes considered in this work.

The boundary element method (BEM) is a powerful tool in computational acoustics, because the analysis is conducted only on structural surfaces, compared to the finite element method (FEM) which resorts to special techniques to truncate infinite domains. The isogeometric boundary element method (IGABEM) is a recent progress in the category of boundary element approaches, which is inspired by the concept of isogeometric analysis (IGA) and employs the spline functions of CAD as basis functions to discretize unknown physical fields. As a boundary representation approach, IGABEM is naturally compatible with CAD and thus can directly perform numerical analysis on CAD models, avoiding the cumbersome meshing procedure in conventional FEM/BEM and eliminating the difficulty of volume parameterization in isogeometric finite element methods. The advantage of tight integration of CAD and numerical analysis in IGABEM renders it particularly attractive in the application of structural shape optimization because (1) the geometry and the analysis can be interacted, (2) remeshing with shape morphing can be avoided, and (3) an optimized solution returns a CAD geometry directly without postprocessing steps. In the present paper, we apply the IGABEM to structural shape optimization of three dimensional exterior acoustic problems, fully exploiting the strength of IGABEM in addressing infinite domain problems and integrating CAD and numerical analysis. We employ the Burton–Miller formulation to overcome fictitious frequency problems, in which hyper-singular integrals are evaluated explicitly. The gradient-based optimizer is adopted and shape sensitivity analysis is conducted with implicit differentiation methods. The design variables are set to be the positions of control points which directly determine the shape of structures. Finally, numerical examples are provided to verify the algorithm.

This paper aims at investigating fracture behavior of single and multiple cracks in two-dimensional solids by an adaptive extended isogeometric analysis (XIGA) based on locally refined (LR) B-splines. The adaptive XIGA is capable of modeling cracks without considering the location of crack faces due to the local enrichment technique based on partition-of-unity concept. The XIGA approximation is locally enriched by Heaviside function and crack tip enrichment functions to capture the discontinuity across crack faces and singularity in the vicinity of crack tips. The LR B-splines, which are generalized by B-splines and NURBS, not only inherent desirable properties of the B-splines and NURBS but also can be locally refined, ideally suitable for adaptive analysis. Structured mesh refinement strategy is applied to perform local refinement for LR B-splines based on a posteriori error estimator. According to the recovery technique proposed by Zienkiewicz and Zhu, the smoothed strain field is obtained to construct the posteriori error estimation based local refinement. The stress intensity factors (SIFs) are evaluated using the contour interaction integral technique. Several benchmark numerical examples are illustrated in comparison to analytical or reference solutions to verify the accuracy and efficiency of the developed approach. The proposed adaptive XIGA method is also applied to a curved crack, multiple cracks and complicated structure with a crack, which sufficiently presents the applicability of the proposed method in crack modeling. In addition, the convergence rate of the adaptive local refinement strategy is numerically studied and compared with that of the global refinement approach. The convergence rate of the adaptive local refinement is shown to be faster than that of the global refinement.

Composite shells with complex geometry are widely used in aerospace structures. Due to the complexity of geometry and curvilinear fiber path, the analysis and optimization based on finite element analysis (FEA) for complex variable-stiffness (VS) shells is extremely time-consuming. By comparing with FEA, isogeometric analysis (IGA) exhibits higher prediction efficiency of buckling load. In this work, the formula of geometric stiffness matrix for complex VS shells is derived for the first time based on degenerated shell method using IGA, which is the basis of performing linear buckling analysis. Then, a new variable curvature quasi-linear function (VCQLF) to describe curvilinear fiber path is proposed, which can further expand the design space of VS shells. After that, two frameworks for shape optimization of complex shells are put forward and then compared, and it is found that the one based on LOFT function can provide representative control variables of shape and effectively reduces the number of design variables for complex shells. Finally, a novel collaborative optimization framework of fiber path and shell shape using IGA is established. By comparison of traditional methods, it is demonstrated that the proposed framework can greatly improve the efficiency of optimization and fully explore the buckling load of complex VS shells.

A three-dimensional (3D) isogeometric/discrete-element coupling method is presented for modeling contact/impact between structures and particles. This method takes advantages of the geometry smoothness and exactness of isogeometric analysis (IGA) for continuous solid media and the effectiveness and flexibility of the discrete element method (DEM) for particulate matters. The coupling procedure for handling interactions between IGA elements and discrete elements (DEs) includes global search, local search and interaction calculation. In the global search, the CGRID method is modified to detect potential contact pairs between IGA elements and DEs based on their bounding box representations. The strong convex hull property of a NURBS control mesh plays an important part in the bounding box representation of IGA elements. In the local search, the proposed approach treats each spherical DE centroid as a slave node and the contact surface of each IGA element as the master surface. The projection of a DE centroid onto an IGA element contact surface is solved by modifying the simplex method and Brent iterations. The contact force between an IGA element and a DE is determined from their penetration by using a (nonlinear) penalty function based method. The whole coupled system is solved by the explicit time integration within a updated Lagrangian scheme. Finally, three impact examples, including the impact of two symmetric bars, a tube onto a footing strip, and an assembly of granular particles to a tailor rolled blank, are simulated in elastic regime to assess the accuracy and applicability of the proposed method.

This paper introduces a new non-uniform subdivision surface representation, called hybrid non-uniform subdivision surface (for short, HNUSS). The subdivision scheme is constructed through two steps. The first step inserts a set of edges and converts a valence- n extraordinary point into a valence- n face. The second step combines both primal and dual subdivision schemes to define the subdivision rules. The developed subdivision scheme generalizes bi-cubic NURBS to arbitrary topology and is proved to be G 1 -continuous for any valence extraordinary points and any non-negative knot intervals. The HNUSS limit surface has comparable shape quality as non-uniform subdivision via eigen-polyhedron (Li et al., 2016) and has better shape quality than all the other subdivision schemes. In addition, numerical experiments show that HNUSS based isogeometric analysis yields improved convergence rates compared to any existing non-uniform subdivision schemes.

In this paper, an effective and efficient topology optimization method, termed as Isogeometric Topology Optimization (ITO), is proposed for systematic design of both 2D and 3D auxetic metamaterials based on isogeometric analysis (IGA). Firstly, a density distribution function (DDF) with the desired smoothness and continuity, to represent the topological changes of structures, is constructed using the Shepard function and non-uniform rational B-splines (NURBS) basis functions. Secondly, an energy-based homogenization method (EBHM) to evaluate material effective properties is numerically implemented by IGA, with the imposing of the periodic boundary formulation on material microstructure. Thirdly, a topology optimization formulation for 2D and 3D auxetic metamaterials is developed based on the DDF, where the objective function is defined as a combination of the homogenized elastic tensor and the IGA is applied to solve the structural responses. A relaxed optimality criteria (OC) method is used to update the design variables, due to the non-monotonic property of the problem. Finally, several numerical examples are used to demonstrate the effectiveness and efficiency of the proposed method. A series of auxetic microstructures with different deformation mechanisms (e.g. the re-entrant and chiral) can be obtained. The auxetic behavior of material microstructures are numerically validated using ANSYS, and the optimized designs are prototyped using the Selective Laser Sintering (SLS) technique.

In this paper, we present an isogeometric analysis (IGA) for phase-field models of three different yet closely related classes of partial differential equations (PDEs): geometric PDEs, high-order PDEs on stationary surfaces, and high-order PDEs on evolving surfaces; the latter can be a coupling of the former two classes. In the context of geometric PDEs, we consider mean curvature flow and Willmore flow problems and their corresponding phase-field approximations which yield second-order and fourth-order nonlinear parabolic PDEs. Through some numerical examples, we study the convergence behavior of isogeometric analysis for these equations using the method of manufactured solutions. Moreover, we study numerically the convergence of these phase-field approximations to the sharp interface solutions . As for the high-order PDEs on stationary surfaces, we consider a model problem which is the Cahn–Hilliard equation on a unit sphere, where the surface is modeled using a diffuse-interface approach. Finally, as a model problem for high-order PDEs on evolving surfaces, we consider a phase-field model of a deforming multicomponent vesicle which couples the vesicle shape changes with the phase separation process on the vesicle surface. The model consists of two fourth-order nonlinear PDEs which their direct finite element formulation in a Galerkin framework necessitates smooth basis functions with at least global C 1 continuity; a condition that can be easily satisfied using spline bases in IGA. We solve the coupled equations both in two dimensions, where the vesicle is a curve, and in three dimensions, where the vesicle is a surface. The simulation results agree with the numerical and experimental results from the literature.

In this paper, using an isogeometric approach, a continuum-based adjoint configuration design sensitivity analysis (DSA) method is presented for three-dimensional finite deformation shear-deformable beam structures. A geometrically exact beam model together with a multiplicative update of finite rotation by an exponential map of a skew-symmetric matrix is utilized. The material derivative of the orthogonal transformation matrix can be evaluated at final equilibrium configuration , which enables to compute design sensitivity using the tangent stiffness at the equilibrium without further iterations. We also present a procedure of explicit parameterization of initial orthonormal frame using the smallest rotation (SR) method within the isogeometric analysis framework. Furthermore, it is shown that for curve entities embedded to a smooth surface, the convected basis of the surface can be effectively utilized for reference orthonormal frames in the SR method. Various numerical examples including pressure loads and nonhomogeneous kinematic boundary conditions in built-up structures demonstrate the effectiveness of the developed DSA method.

Volumetric spline parameterization of complex geometry plays a key role in isogeometric analysis (IGA). In this paper, we propose a general framework to construct volumetric parameterization from complex shapes based on directed graph simplification of the l 1 polycube structure. By minimizing the l 1 -norm of the normals on the input triangular meshes, a polycube structure can be generated with robustness, efficiency, and controllability. Then an algorithm is proposed for l 1 polycube structure simplification with a directed graph, which is an abstract structure generated from the polycube. After simplification, the vertex number of the directed graph will decrease, and the polycube structure will become more simple. From the simplified polycube structure, we construct a segmented surface by spline fitting techniques, and finally we fill each block with a trivariate B-spline volume with C 0 -constraints. The proposed method can generate volumetric parameterization without an interior extraordinary vertex, and it has very high potential for constructing volumetric parameterization in IGA simulations. Some volume parameterization examples from complex shapes are presented to show the robustness and efficiency of the proposed method.

Isogeometric analysis (IGA) has been used with great success when combined with incremental methods to simulate sheet metal forming. In this paper, we present the development of one-step inverse IGA based on the total deformation theory of plasticity . For a large number of industrial stamping parts, the membrane effects are dominant. Thus, we adopted an isogeometric membrane element to predict the flattened contour of the initial blank from the energy-based initial solution estimation approach. In addition, we used the Newton–Raphson algorithm for nonlinear plastic iterations to evaluate the thickness and equivalent strain and stress of the final stamping parts. We applied our framework to square box and S-rail surface models for demonstration. The results for these two examples illustrate the performance of one-step inverse IGA and its applicability to the integrated design of sheet metal forming.

International audience; Isogeometric shape optimization uses a unique model for the geometric description and for the analysis. The benefits are multiple: in particular, it avoids tedious procedures related to mesh updates. However, although the analysis of complex multipatch structures now becomes tractable with advanced numerical tools, isogeometric shape optimization has not yet been proven to be applicable for designing such structures. Based on the initial concept of integrating design and analysis, we develop a new approach that deals with the shape optimization of non-conforming multipatch structures. The model is built by employing the Free-Form Deformation principle. Introducing NURBS composition drastically simplifies the imposition of the shape updates in case of a non-conforming multipatch configuration. In the case of stiffened structures, the use of embedded surfaces enables to tackle the geometric constraint of connecting interfaces between the panel and the stiffeners during shape modifications. For the analysis, we introduce the embedded Kirchhoff-Love shell formulation. The NURBS composition defines the geometry of the shell while the displacement field is approximated using the same spline functions as for the embedded surface. We also formulate a new mortar method to couple non-conforming Kirchhoff-Love shells which intersect with any angle. We apply the developed method on different examples to demonstrate its efficiency and its potential to optimize complex industrial structures in a smooth manner.

It is well known that Galerkin finite element methods suffer from pollution error when solving wave problems. To reduce the pollution impact on the solution different approaches were proposed to enrich the finite element method with wave-like functions so that the exact wavenumber is incorporated into the finite element approximation space. Solving wave problems with isogeometric analysis was also investigated in the literature where the superior behaviour of isogeometric analysis due to higher continuity in the underlying basis has been studied. Recently, a plane wave enriched isogeometric analysis was introduced for acoustic problems. However, it remains unquantified the impact of these different approaches on the pollution or how they perform compared to each other. In this work, we show that isogeometric analysis outperforms finite element method in dealing with pollution. We observe similar behaviour when both the methods are enriched with plane waves. Using higher order polynomials with fewer enrichment functions seems to improve the pollution compared to lower order polynomials with more functions. However, the latter still leads to smaller errors using similar number of degrees of freedom. In conclusion, we propose that partition of unity isogeometric analysis can be an efficient tool for wave problems as enrichment eliminates the need for domain re-meshing at higher frequencies and also due to its ability to capture the exact geometry even on coarse meshes as well as its improved pollution behaviour.

This paper introduces S-spline curves and surfaces. Local refinement of S-spline surfaces is much simpler to understand and to implement than T-spline refinement. Furthermore, no unwanted control points arise in S-spline refinement, unlike T-spline refinement. The refinement algorithm assures linear independence of blending functions . Thus, for isogeometric analysis, S-spline surfaces provide optimal degrees of freedom during adaptive local refinement. S-splines are compatible with NURBS and T-splines, and can easily be added to existing T-spline implementations.

We formulate a methodology to enforce interface conditions preserving higher-order continuity across the interface. Isogeometrical methods (IGA) naturally allow us to deal with equations of higher-order omitting the usage of mixed approaches. For multi-patch analysis of Kirchhoff–Love shell elements, G 1 continuity at the interface is required and serve here as a prototypical example for a higher-order coupling conditions. When working with this class of shell elements, two different types of constraints arise: Higher-order Dirichlet conditions and higher-order patch coupling conditions. A basis modification approach is presented here, based on a least-square formulation and the incorporation of the constraints into the IGA approximation space. An alternative formulation using Lagrange multipliers which are statically condensed via a discrete Null-Space method provides additional insight into the proposed formulation. A detailed comparison with a classical mortar approach shows the similarities and differences. Eventually, numerical examples demonstrate the capabilities of the presented formulation.

Soft, active materials have been widely studied due to their ability to undergo large, complex shape changes in response to both mechanical and non-mechanical external stimuli. However, the vast majority of such studies has focused on investigating the forward problem, i.e. determining the shape changes that result from the applied stimuli. In contrast, very little work has been done to solve the inverse problem, i.e. that of identifying the external loads and stimuli that are needed to generate desired shapes and morphological changes. In this work, we present a new inverse methodology to study residual thermal expansion induced morphological changes in geometric composites made of soft, thin shells. In particular, the method presented in this work aims to determine the prescribed external stimuli needed to reconstruct a specific target shape, with a specific focus and interest in morphological changes from two-dimensional (2D) to three-dimensional (3D) shapes by considering the external stimuli within a thermohyperelastic framework. To do so, we utilize a geometrically exact, rotation-free Kirchhoff–Love shell formulation discretized by NURBS-based shape functions. We show that the proposed method is capable of identifying the stimuli, including cases where thermal expansion induced shape changes involving elastic softening occur in morphing from the initially flat 2D to non-planar 3D shapes. Validation indicates that the reconstructed shapes are in good agreement with the target shape.

The use of modified couple stress theory to simulate the size-dependent phenomenon of composite laminate microplate is commonly limited to simple boundary conditions and mechanical bending load. The small-scale effects on bending and buckling on composite laminate microplate under complex boundary conditions in thermal environment have not been understood fully in the literature. Hence, this research develops, for the first time, a model to overcome the above limitation through the combination of a new modified couple stress theory and isogeometric analysis (IGA). By solving the governing equation using IGA, the thermal displacement, stress and thermal buckling load for various material length scale parameters are obtained. To satisfy the continuous shear stress condition at the layer interfaces, the equilibrium equations as integrated in-plane stress derivatives over the thickness are imposed. In addition, the non-uniform rational B-splines (NURBS) satisfy the higher-order derivative of shape function using the equilibrium equation. Furthermore, to show the effectiveness of presented model for capturing the size effect on thermal bending and thermal buckling of multi-ply laminate microplate, the influences of fiber orientation, thickness ratio, boundary condition and the variation in material length scale parameter are investigated.

Structural stochastic analysis is vital to engineering. However, current material related uncertainty methods are mostly limited to low dimension, and they mostly remain unable to account for spatially uncorrelated material uncertainties. They are not representative of realistic and practical engineering situations. In particular, it is more serious for composite structures comprised of dissimilar materials. Therefore, we propose a novel model order reduction via proper orthogonal decomposition accelerated Monte Carlo stochastic isogeometric method (IGA-POD-MCS) for stochastic analysis of exactly represented (composite) structures. This approach particularly enables high-dimensional material uncertainties wherein the characteristics of each element are independent. And the novelties include: (1) the structural geometry is exactly modeled thanks to isogeometric analysis (IGA), as well as providing more accurate deterministic and stochastic solutions, (2) we innovatively consider high-dimensional and independent material uncertainties by separating the stochastic mesh from the IGA mesh, and modeling different stochastic elements to have different (independent) uncertainty behaviors, (3) the classical Monte Carlo simulation (MCS) is employed to universally solve the high-dimensional uncertainty problem. However, to circumvent its computational expense, we employ model order reduction via proper orthogonal decomposition (POD) into the IGA coupled MCS stochastic analysis. In particular, we observe that this work decouples all IGA elements and hence permits independent uncertainty models easily, thereby the engineering problem is modeled to be more realistic and authentic. Several illustrative numerical examples verify the proposed IGA-POD-MCS approach is effective and efficient; and the larger the scale of the problem is, the more advantageous the method will become.

We present a systematic study on higher-order penalty techniques for isogeometric mortar methods. In addition to the weak-continuity enforced by a mortar method, normal derivatives across the interface are penalized. The considered applications are fourth order problems as well as eigenvalue problems for second and fourth order equations. The hybrid coupling, which combines mortar and penalty methods, enables the discretization of fourth order problems in a multi-patch setting as well as a convenient implementation of natural boundary conditions. For second order eigenvalue problems, the pollution of the discrete spectrum – typically referred to as “outliers” – can be avoided. Numerical results illustrate the good behaviour of the proposed method in simple systematic studies as well as more complex multi-patch mapped geometries for linear elasticity and Kirchhoff plates.

Numerical analysis of physical/mathematical problems based on generalized/extended isogeometric analysis suffers from the major drawbacks of sub optimal convergence rates and ill conditioning of system matrices. Blending elements and linear dependency of basis functions are some of the causes attributed to these drawbacks. The presence of blending elements reduces the overall convergence rates and the ill conditioning of system matrices results in either increasing computational time when iterative solvers are used or erroneous results when direct solvers are employed. In order to alleviate these drawbacks, three different Stable Generalized IsoGeometric Analysis (SGIGA) methods are proposed in this paper. In SGIGA, the enrichment functions are modified by shifting the enrichment function using linear or least square interpolant of the enrichment function. Problems with weak and strong discontinuities, singularities and combination of both discontinuities and singularities are considered as case studies to illustrate the performance of the proposed SGIGA methods. From the results, it is observed that SGIGA yields optimal convergence rates as well as better conditioning of system matrices. The results obtained from the proposed SGIGA methods are also compared with the results from the established methods, Stable Generalized Finite Element Method (SGFEM) and eXtended IsoGeometric Analysis (XIGA), to study the relative performances with respect to accuracy and conditioning.

We consider the numerical approximation of lipid biomembranes at equilibrium described by the Canham-Helfrich model, according to which the bending energy is minimized under area and volume constraints. Energy minimization is performed via L-2-gradient flow of the Canham-Helfrich energy using two Lagrange multipliers to weakly enforce the constraints. This yields a highly nonlinear, high order, time dependent geometric Partial Differential Equation (PDE). We represent the biomembranes as single-patch NURBS closed surfaces. We discretize the geometric PDEs in space with NURBS-based Isogeometric Analysis and in time with Backward Differentiation Formulas. We tackle the nonlinearity in our formulation through a semi-implicit approach by extrapolating, at each time level, the geometric quantities of interest from previous time steps. We report the numerical results of the approximation of the Canham-Helfrich problem on ellipsoids of different aspect ratio, which leads to the classical biconcave shape of lipid vesicles at equilibrium. We show that this framework permits an accurate approximation of the Canham-Helfrich problem, while being computationally efficient. (C) 2019 Elsevier B.Y. All rights reserved.

We propose a method to obtain diagonal mass matrices for NURBS-based approximation spaces by a “dual lumping” method. The use of lumped mass matrices is of great importance in elastodynamics problems, as they can be employed in explicit time integration schemes which do not require the solution of a linear system. In finite elements, several well-established methods, such as row-sum, diagonal scaling, or nodal quadrature methods have been used to obtain lumped mass matrices for different applications. However, for higher-order and higher continuity approximation spaces such as those derived from NURBS, these approaches have only limited (second-order) accuracy. In this work, we derive a dual basis which has optimal approximation and dispersion properties, while maintaining local support. The dual space has discontinuities at the element boundaries (knots) and it is used to provide the test functions in the context of a Petrov–Galerkin method. This results in a general framework for the study of lumped mass matrices which can be employed in explicit time integration schemes with high-order accuracy. Numerical experiments are presented to demonstrate the applicability of the method to problems with smooth solutions as well as to wave propagation problems with reduced regularity.

This article presents application of rational triangular Bezier splines (rTBS) for developing Kirchhoff–Love shell elements in the context of isogeometric analysis. Kirchhoff–Love shell formulation requires high continuity between elements because of higher order PDEs in the description of the problem. Non-uniform rational B-spline (NURBS)-based IGA has been extensively used for developing Kirchhoff–Love shell elements, as NURBS-based IGA can provide high continuity between and within elements. However, NURBS-based IGA has some limitations; such as, analysis of a complex geometry might need multiple NURBS patches and imposing higher continuity constraints over interfaces of patches is a challenging issue. Addressing these limitations, isogeometric analysis based on rTBS can provide C1 continuity over the mesh including element interfaces, a necessary condition in finite elements formulation of Kirchhoff–Love shell theory. Based on this technology, we use Cr smooth rational triangular Bezier spline as the basis functions for representing both geometry and solution field. In addition to providing higher continuity for Kirchhoff–Love formulation, using rTBS elements we can achieve three significant challenging goals: optimal convergence rate, efficient local mesh refinement and analysis of geometric models of complex topology. The proposed method is applied on several examples; first, this technique is verified against multiple plate and shell benchmark problems; investigating the convergence rate on the benchmark problems demonstrates that the optimal convergence rate can be obtained by the proposed technique. We also apply our method on geometric models of complex topology or geometric models in which efficient local refinement is required. Moreover, a car hood is modeled with rTBS and structurally analyzed by using the proposed framework.

Isogeometric analysis (IGA) has been a particularly impactful development in the realm of Kirchhoff–Love thin-shell analysis because the high-order basis functions employed naturally satisfy the requirement of C 1 continuity. Still, engineering models of appreciable complexity, such as wind turbine blades, are typically modeled using multiple surface patches and, often, neither rotational continuity nor conforming discretization can be practically obtained at patch interfaces. A penalty approach for coupling adjacent patches is therefore presented. The proposed method imposes both displacement and rotational continuity and is applicable to either smooth or non-smooth interfaces and either matching or non-matching discretization. The penalty formulations require only a single, dimensionless penalty coefficient for both displacement and rotation coupling terms, alleviating the problem-dependent nature of the penalty parameters. Using this coupling methodology, numerous benchmark problems encapsulating a variety of analysis types, geometrical and material properties, and matching and non-matching interfaces are addressed. The coupling methodology produces consistently accurate results throughout all tests. Furthermore, the suggested penalty coefficient of α = 1 0 3 is shown to be effective for the wide range of problem configurations addressed. Finally, a realistic wind turbine blade model, consisting of 27 patches and 51 coupling interfaces and having a chordwise- and spanwise-variant composite material definition, is subjected to buckling, vibration, and nonlinear deformation analyses using the proposed approach.

A superconvergent isogeometric formulation is presented to accurately analyze the natural frequencies for elastic continua. This formulation is realized by a set of superconvergent quadrature rules which are designed for the numerical integration of isogeometric mass and stiffness matrices. In order to obtain these quadrature rules, a natural frequency error measure for elastic continua is systematically deduced using the quadratic basis functions, where the mass and stiffness matrices are formulated by the assumed quadrature rules. In contrast to the quadrature-based superconvergent isogeometric formulation for the scalar-valued wave equations, it is shown that herein different quadrature rules are required for the mass and stiffness matrices to achieve the superconvergence of natural frequency computation for the vector-valued elastic continuum problems . Consequently, the superconvergent quadrature rules are established through optimizing the natural frequency accuracy. It turns out that with these quadrature rules, the accuracy of natural frequencies for elastic continua is improved by two orders compared with the standard isogeometric formulation employing the consistent mass matrix . Meanwhile, it is found that the superconvergent quadrature rules involve the wave propagation angle and consequently simplified quadrature rules without the angle dependence are further proposed for straightforward practical applications. Numerical results reveal the superconvergence of the proposed method regarding the natural frequencies for elastic continua.

Nanowire based semiconductors are promising for nanogenerators. However, there exist limited numerical tools to analyze these type of structures taking into account effects which are of particular importance at nanoscale . Therefore, we present a finite deformation NURBS based formulation to model a multifunctional material that couples strain, strain gradient , polarization and free charge carriers simultaneously. Specifically, the weak form and consistent linearization of the piezoelectric semiconductor including flexoelectricity and non-local elasticity are introduced. The nonlinear equations are then discretized and solved by utilizing isogeometric analysis (IGA) which fulfills the C 1 continuity requirement. Several numerical examples are performed to investigate the influence of flexoelectricity and non-local elasticity in ZnO piezoelectric semiconductor nanowires under large deformation . The formulation developed in this work can contribute to the development of novel nanoelectromechanical coupling devices such as flexoelectric nanogenerators.

The contribution herein proposes an isogeometric generalized n th order perturbation-based stochastic method for exactly modeling/representing composite structures comprising of different materials with particular attention to both static and dynamic analysis of structures with random material characteristics. Herein, we exactly represent the geometric model via isogeometric analysis (IGA), such as exactly modeling the interfaces in composite structures with dissimilar materials and also continuous variable thicknesses that cannot be achieved by existing traditional methods such as FEM. Besides, only a limited or scant work has been conducted thus far with IGA and stochastic methods. We consider the uncertainties of elastic modulus and mass density into account as stochastic inputs in static and dynamic isogeometric stochastic analyses. Moreover, we derive and expand the IGA based random-input parameter and all state functions included in static and dynamic equilibrium equations around their expectations via a generalized n th order Taylor series using a small perturbation parameter e . In addition, we determine the probabilistic moments of the stochastic solution that satisfy the given accuracy requirement by expanding to n th order. The results obtained by the proposed method, the finite element method (wherever feasible), and Monte Carlo simulations for both benchmark and engineering applications verify the following: (a) the proposed methodology can achieve more accurate deterministic solutions with improved efficiency, thereby strengthening the effectiveness and efficiency brought by the stochastic method. This is in contrast to the FEM based method which weakens them, (b) on the other hand, it can more efficiently acquire reliable and more accurate stochastic results, both for expected values and standard deviations in the static and dynamic (free vibration) analyses. Note that the larger the problem size is, the more efficient the proposed method will be.

We extend a recently-developed framework for isogeometric analysis of composite laminates to drive material damage evolution with a smoothed strain field. This builds on ideas from gradient-enhanced continuum damage modeling, and is intended to limit the dependence of damage predictions on the choice of discrete mesh. The resulting enhanced framework models each lamina of a composite shell structure as a Kirchhoff–Love thin shell. To account for the anisotropic damage modes of laminae, we smooth a tensor-valued strain by solving an elliptic partial differential equation (PDE) system on each lamina. This strain-smoothing PDE system is formulated to be independent of the choice of coordinates and is applicable to general manifold shell geometries. Numerical examples illustrate the enhanced damage model’s validity, mesh-independence, and applicability to complex industrial geometries.

The aim of the paper is to show that the isogeometric Boundary Element Method (isoBEM) has advantages over other numerical methods when applied to problems in potential flow. The problems presented here range from the flow past an obstacle to confined and unconfined seepage problems in isotropic and anisotropic media. It is shown how accurate results can be obtained with very few unknowns. The superior capability of NURBS to describe geometry and the variation of the unknowns is exploited.

Reliability-based design optimization (RBDO) is a powerful tool to handle the influence of various uncertainties during optimization. However, unbearable computation cost is one of the largest barriers for its application, especially for the finite element method (FEM)-based RBDO. In this paper, an efficient and accurate RBDO framework is established based on isogeometric analysis (IGA) for complex engineering problems. Furthermore, an enhanced step length adjustment (ESLA) iterative algorithm and a second-order reliability method-based stepped-up sequential optimization and reliability assessment approach (SSORA-SORM) are proposed to boost the efficiency of RBDO. According to the situation of iteration process, the step length of search the most probable target point can be adaptively updated by the proposed criterion to improve the robustness in ESLA. In the proposed framework, the analytical first-order sensitivity is derived based on IGA in the optimization process to substitute the time-consuming finite difference method. The robustness, accuracy and efficiency of proposed methods are verified via several numerical benchmarks. Besides, three complex IGA-based examples demonstrate that the proposed method is able to save much computational cost without losing accuracy, which is inherently suitable for the RBDO of complex engineering problems.

In this paper, we propose an enhanced isogeometric analysis (IGA) collocation method. It is well known that the location of the collocation points plays an important role in the accuracy and stability of IGA collocation methods. This is particularly true for non-uniform meshes and domains generated from multi-patch geometries. We present an enhanced collocation method based on Gauss points , which has improved accuracy as compared to using C 1 splines and a recovery-based error estimator that can be derived by sampling the solution at particular points in the domain. Adaptivity is implemented using a hierarchical spline basis, which satisfies the C 1 continuity requirement. The proposed approach has been tested by several benchmark problems, including multipatch domains and geometries with re-entrant corners.

A strain gradient elasticity model for shells of arbitrary geometry is derived for the first time. The Kirchhoff–Love shell kinematics is employed in the context of a one-parameter modification of Mindlin’s strain gradient elasticity theory . The weak form of the static boundary value problem of the generalized shell model is formulated within an H 3 Sobolev space setting incorporating first-, second- and third-order derivatives of the displacement variables. The strong form governing equations with a complete set of boundary conditions are derived via the principle of virtual work. A detailed description focusing on the non-standard features of the implementation of the corresponding Galerkin discretizations is provided. The numerical computations are accomplished with a conforming isogeometric method by adopting C p − 1 -continuous NURBS basis functions of order p ≥ 3 . Convergence studies and comparisons to the corresponding three-dimensional solid element simulation verify the shell element implementation. Numerical results demonstrate the crucial capabilities of the non-standard shell model: capturing size effects and smoothening stress singularities .

This paper introduces tIGAr , a library for using the open-source finite element (FE) automation software FEniCS to perform isogeometric analysis (IGA). The library uses a global variant of Bezier extraction to avoid modifying the finite element assembly procedures of FEniCS . This makes much of the convenient functionality of FEniCS directly available for IGA. General rational splines are implemented through an abstraction that sees only an extracted representation of the IGA function space. Through this abstraction, an enormous variety of spline spaces can be used for analysis, so long as a corresponding preprocessor is developed for each one, implementing a simple interface. As examples, we discuss preprocessors for B-splines specified analytically, non-uniform rational B-splines (NURBS) designed using the open-source software igakit , and T-splines designed using commercial software. We then demonstrate the implementation of solvers for several non-trivial partial differential equations that benefit from IGA. We also evaluate the parallel performance of tIGAr on a distributed-memory supercomputer . Finally, we outline possibilities for further development of IGA in FEniCS . Source code for tIGAr is continuously updated online at https://github.com/david-kamensky/tIGAr .

In this paper, 2-D hp local adaptive procedure is developed based on Control Volume Isogeometric Analysis (CV-IGA) and Hierarchical Fup (HF) basis functions. Contrary to the most common truncated hierarchical splines, HF enables local hp adaptation because higher resolution levels do not include only basis with smaller compact support or higher frequencies, but also with higher order. Consequence of this property is spectral convergence of the proposed adaptive algorithm which is presented on classical benchmarks such as L- shape benchmark and advection dominated problems. Even in non-smooth problems, spectral convergence is achieved contrary to the application of uniform grid. CV- IGA ensures local and global mass conservation which is potentially very important for fluid mechanics problems. 2-D proposed algorithm chooses regular control volumes in parametric space at all resolution levels closely related to the Greville points (vertices) of basis functions. Therefore, methodology is very simple requiring only overlapping of control volumes in the areas where different levels are connected, while its computational cost lies between Galerkin and collocation formulations.

Approximately 1.6 million patients in the United States are affected by tricuspid valve regurgitation , which occurs when the tricuspid valve does not close properly to prevent backward blood flow into the right atrium. Despite its critical role in proper cardiac function, the tricuspid valve has received limited research attention compared to the mitral and aortic valves on the left side of the heart. As a result, proper valvular function and the pathologies that may cause dysfunction remain poorly understood. To promote further investigations of the biomechanical behavior and response of the tricuspid valve, this work establishes a parameter-based approach that provides a template for tricuspid valve modeling and simulation . The proposed tricuspid valve parameterization presents a comprehensive description of the leaflets and the complex chordae tendineae for capturing the typical three-leaflet structural deformation observed from medical data. This simulation framework develops a practical procedure for modeling tricuspid valves and offers a robust, flexible approach to analyze the performance and effectiveness of various valve configurations using isogeometric analysis. The proposed methods also establish a baseline to examine the tricuspid valve’s structural deformation, perform future investigations of native valve configurations under healthy and disease conditions, and optimize prosthetic valve designs.

We present a novel method to perform numerical integration over curved polyhedra enclosed by high-order parametric surfaces. Such a polyhedron is first decomposed into a set of triangular and/or rectangular pyramids, whose certain faces correspond to the given parametric surfaces. Each pyramid serves as an integration cell with a geometric mapping from a standard parent domain (e.g., a unit cube), where the tensor-product Gauss quadrature is adopted. As no constraint is imposed on the decomposition, certain resulting pyramids may intersect with themselves, and thus their geometric mappings may present negative Jacobian values. We call such cells the folded cells and refer to the corresponding decomposition as a folded decomposition. We show that folded cells do not cause any issues in practice as they are only used to numerically compute certain integrals of interest. The same idea can be applied to planar curved polygons as well. We demonstrate both theoretically and numerically that folded cells can retain the same accuracy as the cells with strictly positive Jacobians. On the other hand, folded cells allow for a much easier and much more flexible decomposition for general curved polyhedra, on which one can robustly compute integrals. In the end, we show that folded cells can flexibly and robustly accommodate real-world complex geometries by presenting several examples in the context of immersed isogeometric analysis, where involved sharp features can be well respected in generating integration cells., 23 pages, 18 figures

We present the construction of additive multilevel preconditioners, also known as BPX preconditioners, for the solution of the linear system arising in isogeometric adaptive schemes with (truncated) hierarchical B-splines. We show that the locality of hierarchical spline functions, naturally defined on a multilevel structure, can be suitably exploited to design and analyze efficient multilevel decompositions. By obtaining smaller subspaces with respect to standard tensor-product B-splines, the computational effort on each level is reduced. We prove that, for suitably graded hierarchical meshes, the condition number of the preconditioned system is bounded independently of the number of levels. A selection of numerical examples validates the theoretical results and the performance of the preconditioner. (C) 2021 Elsevier B.V. All rights reserved.

We introduce a boundary penalization technique to improve the spectral approximation of isogeometric analysis (IGA). The technique removes the outliers appearing in the high-frequency region of the approximate spectrum when using the $C^{p-1}, p$-th ($p\ge3$) order isogeometric elements. We focus on the classical Laplacian (Dirichlet) eigenvalue problem in 1D to illustrate the idea and then use the tensor-product structure to generate the stiffness and mass matrices for multiple dimensional problems. To remove the outliers, we penalize the product of the higher-order derivatives from both the solution and test spaces at the domain boundary. Intuitively, we construct a better approximation by weakly imposing features of the exact solution. Effectively, we add terms to the variational formulation at the boundaries with minimal extra computational cost. We then generalize the idea to remove the outliers for the isogeometric analysis to the Neumann eigenvalue problem (for $p\ge2$). The boundary penalization does not change the test and solution spaces. In the limiting case when the penalty goes to infinity, we perform the dispersion analysis of $C^2$ cubic elements for Dirichlet eigenvalue problem and $C^1$ quadratic elements for Neumann eigenvalue problem. We obtain the analytical eigenpairs for the resulting matrix eigenvalue problems. Numerical experiments show optimal convergence rates for the eigenvalues and eigenfunctions of the discrete operator., 30 pages

We propose an isogeometric approximation of the equations describing the propagation of an electrophysiologic stimulus over a thin cardiac tissue with the subsequent muscle contraction. The underlying method relies on the monodomain model for the electrophysiological sub-problem. This requires the solution of a reaction–diffusion equation over a surface in the three-dimensional space. Exploiting the benefits of the high-order NURBS basis functions within a curvilinear framework, the method is found to reproduce complex excitation patterns with a limited number of degrees of freedom. Furthermore, the curvilinear description of the diffusion term provides a flexible and easy-to-implement approach for general surfaces. At the discrete level, two different approaches for integrating the ionic current are investigated in the isogeometric analysis framework. The electrophysiological stimulus is converted into a mechanical load employing the well-established active strain approach. The multiplicative decomposition of the deformation gradient tensor is grafted into a classical finite elasticity weak formulation, providing the necessary tensor expressions in curvilinear coordinates. The derived expressions provide what is needed to implement the active strain approach in standard finite-element solvers without resorting to dedicated formulations. Such a formulation is valid for general three-dimensional geometries and isotropic hyperelastic materials. The formulation is then restricted to Kirchhoff–Love shells by means of the static condensation of the material tensor. The purely elastic response of the structure is investigated with simple static test-cases of thin shells undergoing different active strain patterns. Eventually, various numerical tests performed with a staggered scheme illustrate that the coupled electromechanical model can capture the excitation–contraction mechanism over thin tissues and reproduce complex curvature variations.

This article firstly presents a novel numerical methodology to concurrently optimize material distribution (size) and thickness variation (shape) of multidirectional functionally graded (MFG) plates under free vibration within the isogeometric analysis (IGA) framework. An isogeometric multimesh design (IMD) approach is proposed to generate two distinct non-uniform rational B-spline (NURBS) surfaces via the k -refinement strategy. A finer analysis one relied upon a combination of the IGA and a generalized shear deformation theory (GSDT) is utilized for the unknown solution approximation in finite element analyses (FEAs). Whilst the other coarser design one is employed for the exact geometry representation as well as the optimal material and thickness depiction. Size and shape design variables are in turn the ceramic volume fraction and z -axis coordinate of the top side of the MFG plate coincidentally assigned to each of control points on this surface. Flexibly utilizing such two surfaces helps diminish a large number of design variables and considerably save the computational cost in optimization problems, yet still appropriately manifesting optimal material and thickness profiles. Additionally, this definition accurately simulates mechanical behavior of MFG plates in analysis ones as well. A recently developed derivative-free adaptive hybrid evolutionary firefly algorithm (AHEFA) is used to solve constrained frequency maximization problems. Several numerical examples are executed to verify the effectiveness and robustness of the present paradigm.

The dynamic analysis of planar mechanisms with multiple revolute clearance joints is performed using the isogeometric analysis for the first time. A general multiple patches based revolute clearance joint model, which can be integrated into other parts or structures without changing the procedure of contact treatment, is proposed. A stable and optimized contact detection algorithm that accompanies the joint model is also introduced. Benefit from the high order NURBS basis functions, the geometrically exact covariant formulations of contact geometry , contact force and its Jacobian matrix are implemented. The implicit generalized- α method is employed to solve the time dependent nonlinear system equations. The effect of joint clearance on rigid and flexible mechanisms is studied through numerical examples. The accuracy and efficiency of the proposed approach are verified through a comparison with other conventional methods. The proposed approach exhibits great stability in long-term simulation regardless of the number of clearance joints, the size of clearance, the flexibility of the link and bearing. Results show that isogeometric analysis can provide a unified framework for joint clearance analysis of rigid and flexible systems. In the field of analyzing the combined effect of joint clearance and bodies’ large flexibility, the proposed method has significant advantages.