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We are witnessing a rapid transition from Software 1.0 to 2.0. Software 1.0 focuses on manually designed algorithms, while Software 2.0 leverages data and machine learning algorithms (or artificial intelligence) for optimized, fast, and accurate solutions. For the past few years, we have been developing Convolution Hierarchical Deep-learning Neural Network Artificial Intelligence (C-HiDeNN-AI), which enables the realization of Engineering Software 2.0 by opening the next-generation neural network-based computational tools that can simultaneously train data and solve mechanistic equations. This paper focuses on solving partial differential equations with C-HiDeNN. Still, the same neural network can be used for training and calibration with experimental data, which will be discussed in a separate paper. This paper presents a computational framework combining the C-HiDeNN theory with isogeometric analysis (IGA), called Convolution IGA (C-IGA). C-IGA has five key features that advance IGA: (1) arbitrarily high-order smoothness and convergence rates without increasing degrees of freedom; (2) a Kronecker delta property that enables direct imposition of Dirichlet boundary conditions; (3) automatic and flexible global/local mesh-adaptivity with built-in length scale control and adjustable radial basis functions; (4) ability to handle irregular meshes and triangular/tetrahedral elements; and (5) GPU implementation that speeds up the program as fast as finite element method (FEM). Mathematically, we prove that both IGA and C-IGA mappings are equivalent, and by taking a special design and modified anchors as nodes, C-IGA degenerates to IGA. We demonstrate the accuracy, convergence rates, mesh-adaptivity, and performance of C-IGA with several 1D, 2D, and 3D numerical examples. The future applications of C-IGA from topology optimization to product manufacturing with multi-GPU programming are discussed. [ABSTRACT FROM AUTHOR]

Full waveform inversion (FWI) is an iterative identification process that serves to minimize the misfit of model-based simulated and experimentally measured wave field data. Its goal is to identify a field of parameters for a given physical object. For many years, FWI is very successful in seismic imaging to deduce velocity models of the earth or of local geophysical exploration areas. FWI has also been successfully applied in various other fields, including non-destructive testing (NDT) and biomedical imaging. The inverse optimization process of FWI relies on forward and backward solutions of the (elastic or acoustic) wave equation, as well as on efficient computations of adequate optimization directions. Many approaches employ (low order) finite element or finite difference methods, using parameterized material fields whose resolution is chosen in relation to the elements or nodes of the discretized wave field. In our previous paper (Bürchner et al., 2023), we investigated the potential of using the finite cell method (FCM) as the wave field solver. The FCM offers the advantage that highly complex geometric models can be incorporated easily. Furthermore, we demonstrated that the identification of the model's density outperforms that of the velocity — particularly in cases where unknown voids characterized by homogeneous Neumann boundary conditions need to be detected. The paper at hand extends this previous study in the following aspects: The isogeometric finite cell analysis (IGA-FCM) – a combination of isogeometric analysis (IGA) and FCM – is applied as the wave field solver, with the advantage that the polynomial degree and subsequently also the sampling frequency of the wave field can be increased quite easily. Since the inversion efficiency strongly depends on the accuracy of the forward and backward wave field solutions and of the gradients of the functional, consistent, and lumped mass matrix discretization are compared. The resolution of the grid describing the unknown material density – thus allowing to identify voids in a physical object – is then decoupled from the knot span grid. Finally, we propose an adaptive multi-resolution algorithm that locally refines the material grid using an image processing-based refinement indicator. The developed inversion framework allows fast and memory-efficient wave simulations and object identification. While we study the general behavior of the proposed approach using 2D benchmark problems, a final 3D problem shows that it can also be used to identify void regions in geometrically complex spatial structures. [ABSTRACT FROM AUTHOR]

This paper presents the construction of a correct-energy stabilized finite element method for the incompressible Navier–Stokes equations. The framework of the methodology and the correct-energy concept have been developed in the convective–diffusive context in the preceding paper [M.F.P. ten Eikelder, I. Akkerman, Correct energy evolution of stabilized formulations: The relation between VMS, SUPG and GLS via dynamic orthogonal small-scales and isogeometric analysis. I: The convective–diffusive context, Comput. Methods Appl. Mech. Engrg. 331 (2018) 259–280]. The current work extends ideas of the preceding paper to build a stabilized method within the variational multiscale (VMS) setting which displays correct-energy behavior. Similar to the convection–diffusion case, a key ingredient is the proper dynamic and orthogonal behavior of the small-scales. This is demanded for correct energy behavior and links the VMS framework to the streamline-upwind Petrov–Galerkin (SUPG) and the Galerkin/least-squares method (GLS). The presented method is a Galerkin/least-squares formulation with dynamic divergence-free small-scales (GLSDD). It is locally mass-conservative for both the large- and small-scales separately. In addition, it locally conserves linear and angular momentum. The computations require and employ NURBS-based isogeometric analysis for the spatial discretization. The resulting formulation numerically shows improved energy behavior for turbulent flows comparing with the original VMS method. [ABSTRACT FROM AUTHOR]

This paper investigates the numerical approximation of a two-phase Hele-Shaw fluid flow system, which consists of a phase-field Cahn–Hilliard model coupled with the Darcy flow, confined on a complex curved surface. To avoid geometric errors caused by the surface approximation and additional approximation errors introduced by numerical methods, for spatial discretization, we consider the recently developed strategy of subdivision-based isogeometric analysis (IGA), where we can accurately represent complex surfaces of arbitrary topology due to its high smoothness and hierarchical refinement properties. By incorporating multiple methods of temporal discretizations, including the Explicit-Invariant-Energy-Quadratization (EIEQ) approach for linearizing nonlinear potentials, the Zero-Energy-Contribution (ZEC) method for decoupling of nonlinear couplings, and the projection method for decoupling of linear couplings between the fluid velocity and pressure, and the finite-element method based on IGA for spatial discretizations, we arrive at a fully discrete scheme that possesses desirable properties such as decoupling, linearity, second-order accuracy in time, and unconditional energy stability. At each time step, only several elliptic equations with constant coefficients are needed to be solved. We also present the rigorous proof of unconditional energy stability, accompanied by numerous numerical examples to verify the stability and accuracy of the numerical scheme, such as the Saffman–Taylor fingering instability occurring on various complex curved surfaces induced by rotational forces. [ABSTRACT FROM AUTHOR]

A well-known problem in the field of isogeometric analysis is that of surface-to-volume parameterization. In CAD packages, solid objects are represented by a collection of NURBS or T-spline surfaces, but to perform engineering analysis for many real world problems, we must find a way to parameterize the volumes of these objects as well. This has proven to be difficult using traditional isogeometric methods, as the tensor-product nature of trivariate NURBS and T-splines limit their ability to provide analysis suitable parameterizations of arbitrarily complex volumes. To overcome the limitations of trivariate NURBS and T-splines, we propose the use of unstructured Bernstein–Bézier discretizations. In earlier work, we demonstrated the feasibility of this approach in two dimensions through the construction of an automatic mesh generation environment capable of generating geometrically exact unstructured meshes comprised of rational Bézier triangles. This paper extends the concepts presented in our previous work to unstructured tetrahedral and mixed-element meshes in three dimensions. The main contributions of this paper are three-fold. First, we present a framework for creating geometrically exact meshes comprised of rational Bézier hexahedra, tetrahedra, wedges and pyramids through degree elevation of suitable linear meshes. We additionally discuss how our approach may be applied to higher-order mesh generation for more traditional finite element approaches. Next, we propose two quality metrics for three-dimensional curvilinear meshes, and discuss some important considerations for mesh quality of higher-order meshes. Finally, we demonstrate the analysis suitability of the meshes constructed using our framework through numerical examples. [ABSTRACT FROM AUTHOR]

The efficient representation of random fields on geometrically complex domains is crucial for Bayesian modelling in engineering and machine learning, including Gaussian process regression and statistical finite element analysis. Today's prevalent random field representations are either intended for unbounded domains or are too restrictive in terms of possible field properties. Because of these limitations, new techniques leveraging the historically established link between stochastic PDEs (SPDEs) and random fields have been gaining interest in the statistics and engineering literature. The SPDE representation is especially appealing for engineering applications with complex geometries which already have a finite element discretisation for solving the physical conservation equations. In contrast to the dense covariance matrix of a random field, its inverse, the precision matrix, is usually sparse and equal to the stiffness matrix of an elliptic SPDE. In this paper, we use the SPDE representation to develop a scalable framework for large-scale statistical finite element analysis and Gaussian process (GP) regression on geometrically complex domains. The statistical finite element method (statFEM) introduced by Girolami et al. (2022) is a novel approach for synthesising measurement data and finite element models. In both statFEM and GP regression, we use the SPDE formulation to obtain the relevant prior probability densities with a sparse precision matrix. The properties of the priors are governed by the parameters and possibly fractional order of the SPDE so that we can model on bounded domains and manifolds anisotropic, non-stationary random fields with arbitrary smoothness. We use for assembling the sparse precision matrix the same finite element mesh used for solving the physical conservation equations. The observation models for statFEM and GP regression are such that the posterior probability densities are Gaussians with a closed-form mean and precision. The expressions for the mean vector and the precision matrix do not contain dense matrices and can be evaluated using only sparse matrix operations. We demonstrate the versatility of the proposed framework and its convergence properties with one and two-dimensional Poisson and thin-shell examples. • A statistical approach for synthesising large-scale FE models and data is proposed. • Gaussian random fields are expressed as the solution of fractional stochastic PDEs. • As observation models Gaussian process regression and statFEM are considered. • Posterior mean and covariance are determined using only sparse matrix operations. • Applications to large-scale Poisson and thin-shell problems are demonstrated. [ABSTRACT FROM AUTHOR]

This paper explores the application of maximum-entropy methods (max-ent) to time harmonic acoustic problems. Max-ent basis functions are mesh-free approximants that are constructed observing an equivalence between basis functions and discrete probability distributions and applying Jaynes’s maximum entropy principle. They are C ∞ -continuous and therefore they are particularly suited for the resolution of Helmholtz problems, where classical finite element methods show a poor accuracy in the high frequency region. In addition, it was recently shown that max-ent approximants can be blended with isogeometric basis functions on the boundary of the domain. This preserves the correct representation of the boundary like in Isogeometric Analysis, with the advantage that the discretization of the interior of the domain is straightforward. In this paper the max-ent mathematical formulation is reviewed and then some numerical applications are studied, including a 2D car cavity geometry defined by B-spline curves. In all cases, if the same nodal discretization is used, finite elements results are significantly improved. [ABSTRACT FROM AUTHOR]

This paper presents a hybrid finite-element/boundary-element method for fluid–structure-interaction simulations of inflatable structures. The flow model consists of the steady Stokes equation, which admits a boundary-integral formulation. The structure is represented by a Kirchhoff–Love shell. The boundary-element approximation of the Stokes equation reduces the flow problem to an integral equation on the actual structure configuration, thus obviating the need for volumetric meshing of the strongly deforming fluid domain. The Stokes model moreover exhibits a lubrication effect that acts as an intrinsic mechanism to treat the ubiquitous self-contact that occurs in inflation problems. The aggregated fluid–structure-interaction problem, composed of the boundary-integral equation and the Kirchhoff–Love shell connected by dynamic and kinematic interface conditions, is approximated by means of isogeometric discretizations to accommodate the smoothness requirements on the approximation spaces imposed by the flexural rigidity in the Kirchhoff–Love shell and to provide an accurate and smooth representation of the boundary for the boundary-element method. Auxiliary results presented in this paper are: (1) a parametrization-free Kirchhoff–Love formulation; (2) establishment of a cubic relationship between distance and tractions due to the lubrication effect; and (3) the interpretation of the Lagrange multiplier pertaining to fluid incompressibility as the total excess pressure. [ABSTRACT FROM AUTHOR]

The paper deals with the Isogeometric Analysis (IGA) technology, which has received much attention over the last decade due to its increased flexibility, accuracy, and robustness in many engineering simulations compared to classical Finite Element Analysis (FEA). In this context, we present a verification method, based on duality and the concept of Constitutive Relation Error (CRE), that enables to derive fully computable a posteriori error estimates on the numerical solution provided by IGA. Such estimates, which can be used for a wide class of structural mechanics models, thus constitute effective and practical tools to quantitatively control the numerical accuracy and drive adaptive algorithms. The focus is here on the construction of so-called admissible flux fields which is a key ingredient of the CRE concept, and which was until now almost exclusively addressed in the FEA framework alone. We show that this construction can be performed in a similar way for FEA and IGA, provided some technical issues (due to the use of B-Spline/NURBS basis functions instead of Lagrange polynomials) are carefully addressed. We also use the CRE concept along with adjoint techniques and local enrichments in order to derive accurate goal-oriented error estimates. Two- and three-dimensional numerical experiments are presented, for thermal, linear elasticity and nonlinear damage problems, to illustrate the capabilities and versatility of the proposed approach. • A fully computable a posteriori error estimate is defined to certify IGA solutions. • The approach applies for a large range of computational mechanics models. • The estimate permits the control of global error and error on quantities of interest. • The recovery of equilibrated fields, which is the key point of the approach, is detailed. • The methodology is applied and evaluated on several 2D/3D numerical experiments. [ABSTRACT FROM AUTHOR]

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. • Only the structural surface needs to be discretized for exterior acoustics problems. • Through the whole optimization procedure, meshing is avoided due to seamless integration of Computer-Aided Design (CAD) and numerical analysis in Isogeometric Boundary Element Methods. • The influence of the geometry variation on the physical field is investigated via shape sensitivity analysis. • Control points are selected as design variables directly. [ABSTRACT FROM AUTHOR]

This paper proposes a computational methodology for the integration of Computer Aided Design (CAD) and the Finite Cell Method (FCM) for models with "dirty geometries". FCM, being a fictitious domain approach based on higher order finite elements, embeds the physical model into a fictitious domain, which can be discretized without having to take into account the boundary of the physical domain. The true geometry is captured by a precise numerical integration of elements cut by the boundary. Thus, an effective Point Membership Classification algorithm that determines the inside–outside state of an integration point with respect to the physical domain is a core operation in FCM. To treat also "dirty geometries", i.e. imprecise or flawed geometric models, a combination of a segment-triangle intersection algorithm and a flood fill algorithm being insensitive to most CAD model flaws is proposed to identify the affiliation of the integration points. The present method thus allows direct computations on geometrically and topologically flawed models. The potential and merit for practical applications of the proposed method is demonstrated by several numerical examples. [ABSTRACT FROM AUTHOR]

Abstract This paper deals with a special class of parametrizations for Isogeometric Analysis (IGA). The so-called scaled boundary parametrizations are easy to construct and particularly attractive if only a boundary description of the computational domain is available. The idea goes back to the Scaled Boundary Finite Element Method (SB-FEM), which has recently been extended to IGA. We take here a different viewpoint and study these parametrizations as bivariate or trivariate B-spline functions that are directly suitable for standard Galerkin-based IGA. Our main results are first a general framework for this class of parametrizations, including aspects such as smoothness and regularity as well as a generalization to domains that are not star-shaped. Second, using the Poisson equation as an example, we explain the relation between standard Galerkin-based IGA and the Scaled Boundary IGA by means of the Laplace–Beltrami operator and derive an equivalence theorem. Further results concern the separation of integrals in both approaches and an analysis of the singularity in the scaling center. Among the computational examples we present a rotor geometry that stems from a screw compressor machine and compare different parametrization strategies. [ABSTRACT FROM AUTHOR]

Abstract 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. Highlights • A novel method, Stable Generalized IsoGeometric Analysis, is proposed. • Three different types of SGIGA are employed to overcome the issues arising in XIGA. • The ill conditioning is studied using the angle measure between the spaces. • The proposed SGIGA-LS-C0 showed better results compared to SGFEM and XIGA. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a structured mesh based B-Spline finite elements (BSFE) method combining the isogeometry analysis technique is developed for analyzing crack problems. The BSFE is used for the global approximation and the isogeometry analysis technique is performed on the local crack tip region which is covered by several elements around the crack tip. The geometry of the selected crack tip region is described by the B-Spline basis functions in parameter space. Meanwhile, a specific function with relevant parameters is introduced into geometry description for catching the characters of the solution around the crack tip. In addition, the B-Spline basis functions are also used as basis functions to approximate the displacement field in the local region under the parameter space. The relevant B spline basis functions used for approximation in local region are replaced by the corresponding BSFE basis functions used in the interface elements adjacent to the local region and the seamless connection between the global approximation and the local isogeometry analysis can be easily achieved. For crack problems, no enrichment functions and remeshing are needed in this new method. This newly developed method is applied to the stress analysis of 2D linear elasticity crack problems in order to investigate its performance and study parameters. Numerical results show that the present method is highly accurate and stable. The new method has a promising potential for practical applications. Highlights • A new mapping technique is used to describe the geometry of the selected crack tip region in parameter space. • The seamless connection between the global approximation and the local isogeometry analysis can be easily achieved. • No enrichment functions and remeshing are needed for crack problems in this new method. • The present method is highly accurate, efficient and robust for crack problems. [ABSTRACT FROM AUTHOR]

Abstract 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. [ABSTRACT FROM AUTHOR]

Abstract 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 Bézier 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. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present some novel results and ideas for robust and accurate implicit representation of geometric surfaces in finite element analysis. The novel contributions of this paper are threefold: (1) describe and validate a method to represent arbitrary parametric surfaces implicitly; (2) represent arbitrary solids implicitly, including sharp features using level sets and boolean operations; (3) impose arbitrary Dirichlet and Neumann boundary conditions on the resulting implicitly defined boundaries. The methods proposed do not require local refinement of the finite element mesh in regions of high curvature, ensure the independence of the domain’s volume on the mesh, do not rely on boundary regularization, and are well suited to methods based on fixed grids such as the extended finite element method (XFEM). Numerical examples are presented to demonstrate the robustness and effectiveness of the proposed approach and show that it is possible to achieve optimal convergence rates using a fully implicit representation of object boundaries. This approach is one step in the desired direction of tying numerical simulations to computer aided design (CAD), similarly to the isogeometric analysis paradigm. [ABSTRACT FROM AUTHOR]

This paper presents an approach for automatically generating high-quality high-order parameterizations for isogeometric analysis on triangulations. A B-spline represented boundary geometry is parameterized into a collection of high-order Bézier triangles or tetrahedra in 2D and 3D spaces, respectively. Triangular Bézier splines are used to represent both the geometry and physical fields over the triangulation. By imposing continuity constraints on the Bézier ordinates of the elements, a set of global C r smooth basis is constructed and used as the basis for analysis. To ensure high quality of the parameterization, both the parametric and physical meshes are optimized to reduce the shape distortion of the high-order elements relative to well-defined reference elements. The shape distortion is defined based on the Jacobian of the triangular Bézier splines, and its sensitivity with respect to the location of control points is derived analytically and evaluated efficiently. Moreover, a sufficient condition is derived to guarantee the generated mesh is free of local self-intersection, thanks to the convex hull property of triangular Bézier splines. By using a Heaviside projection function, the non-negative Jacobian determinant constraints are formulated efficiently as a single optimization constraint. Several 2D and 3D numerical examples are presented to demonstrate that high-quality high-order elements are generated using our approach. [ABSTRACT FROM AUTHOR]

Abstract: This paper describes a novel simplification and fairing method for NURBS (Non-Uniform Rational B-Spline) curves and surfaces using orthogonal non-uniform spline wavelets. The compression nature of wavelets is utilized for NURBS simplification. Sufficient conditions for - and -continuity at the shared boundary of multiple NURBS patches are proved and utilized to preserve the boundary continuity. Our fairing method filters out unwanted imperfections and noise on the NURBS curves and surfaces while preserving geometric details via wavelet decomposition and selective reconstruction. The continuity between neighboring patches is preserved automatically during fairing. Our algorithm is accurate, automatic, robust, efficient, and can be extensively applied to any NURBS models in various research fields such as Computer Aided Design (CAD), Reverse Engineering (RE), finite element analysis and isogeometric analysis. Several examples are presented in this paper, including a propeller CAD model, a Navy boat model and a -continuous NURBS model of the thoracic aorta constructed from medical CT (Computed Tomography) data. [Copyright &y& Elsevier]

Abstract: We begin the mathematical study of the k-method utilizing the theory of Kolmogorov n-widths. The k-method is a finite element technique where spline basis functions of higher-order continuity are employed. It is a fundamental feature of the new field of isogeometric analysis. In previous works, it has been shown that using the k-method has many advantages over the classical finite element method in application areas such as structural dynamics, wave propagation, and turbulence. The Kolmogorov n-width and sup–inf were introduced as tools to assess the effectiveness of approximating functions. In this paper, we investigate the approximation properties of the k-method with these tools. Following a review of theoretical results, we conduct a numerical study in which we compute the n-width and sup–inf for a number of one-dimensional cases. This study sheds further light on the approximation properties of the k-method. We finish this paper with a comparison study of the k-method and the classical finite element method and an analysis of the robustness of polynomial approximation. [Copyright &y& Elsevier]

Explicit time integration schemes coupled with Galerkin discretizations of time-dependent partial differential equations require solving a linear system with the mass matrix at each time step. For applications in structural dynamics, the solution of the linear system is frequently approximated through so-called mass lumping, which consists in replacing the mass matrix by some diagonal approximation. Mass lumping has been widely used in engineering practice for decades already and has a sound mathematical theory supporting it for finite element methods using the classical Lagrange basis. However, the theory for more general basis functions is still missing. Our paper partly addresses this shortcoming. Some special and practically relevant properties of lumped mass matrices are proved and we discuss how these properties naturally extend to banded and Kronecker product matrices whose structure allows to solve linear systems very efficiently. Our theoretical results are applied to isogeometric discretizations but are not restricted to them. • General and useful properties of mass lumping are proved. • These properties are extended to banded matrices and Kronecker products. • The nearest Kronecker product approximation is investigated. [ABSTRACT FROM AUTHOR]

A stable basis for isogeometric analysis of trimmed models is obtained by combining extended B-splines with truncated hierarchical B-splines. While extended B-splines guarantee that the condition number of system matrices is independent of the location of a trimming curve, local refinement is used to improve the robustness of the procedure and the accuracy of the numerical results. The present extended B-spline construction works in the context of Galerkin and collocation methods. The paper focuses on the latter and introduces a new collocation scheme for truncated hierarchical B-splines. A proper transition between refinement levels is assured by a novel balancing algorithm that employs a simple criterion. The enhanced performance of the locally refined stabilization is verified by scalar Laplace and linear elasticity problems analyzed by a collocation based isogeometric boundary element method. The proposed approach yields excellent results and requires few refinement levels to improve the stabilization procedure and accuracy along trimming curves. [ABSTRACT FROM AUTHOR]

Engineering design via CAD software relies on Non-Uniform Rational B-Splines (NURBS) as a means for representing and communicating geometry. Therefore, in general, a NURBS description of a given design can be considered the exact description. The development of isogeometric methods has made the geometry available to analysis methods Hughes et al. (2005). Isogeometric analysis has been particularly successful in structural analysis; one reason being the wide-spread use of two-dimensional finite elements in this field. For fluid dynamics, where three-dimensional analysis is usually indispensable, isogeometric methods are more complicated, yet of course not impossible, to apply in a general fashion. This paper describes a method that enables the solution of fluid–structure-interaction with a matching spline description of the interface. On the structural side, the spline is used in an isogeometric setting. On the fluid side, the same spline is used in the framework of a NURBS-enhanced finite element method (extension of Sevilla et al. (2011)). The coupling of the structural and the fluid solution is greatly facilitated by the common spline interface. The use of the identical spline representation for both sides permits a direct transfer of the necessary quantities, all the while still allowing an adjusted, individual refinement level for both sides. [ABSTRACT FROM AUTHOR]

This paper presents the construction of novel stabilized finite element methods in the convective–diffusive context that exhibit correct-energy behavior. Classical stabilized formulations can create unwanted artificial energy. Our contribution corrects this undesired property by employing the concepts of dynamic as well as orthogonal small-scales within the variational multiscale framework (VMS). The desire for correct energy indicates that the large- and small-scales should be H 0 1 -orthogonal. Using this orthogonality the VMS method can be converted into the streamline-upwind Petrov–Galerkin (SUPG) or the Galerkin/least-squares (GLS) method. Incorporating both large- and small-scales in the energy definition asks for dynamic behavior of the small-scales. Therefore, the large- and small-scales are treated as separate equations. Two consistent variational formulations which depict correct-energy behavior are proposed: (i) the Galerkin/least-squares method with dynamic small-scales (GLSD) and (ii) the dynamic orthogonal formulation (DO). The methods are presented in combination with an energy-decaying generalized- α time-integrator. Numerical verification shows that dissipation due to the small-scales in classical stabilized methods can become negative, on both a local and a global scale. The results show that without loss of accuracy the correct-energy behavior can be recovered by the proposed methods. The computations employ NURBS-based isogeometric analysis for the spatial discretization. [ABSTRACT FROM AUTHOR]

Recently, Isogeometric Analysis (IGA) based on incremental methods for simulating the stamping process has been researched. To the best of our knowledge, however, few studies have combined IGA and One-step inverse approach which is based on total deformation theory of plasticity. A key step for One-step inverse IGA is to estimate a good initial solution. Traditional mesh-based initial solution algorithms for One-step inverse approach are not suitable for One-step inverse NURBS-based IGA. In this paper, we presented a method which can rapidly unfold the undevelopable NURBS surface onto a planar domain and obtain a good initial solution estimation for One-step inverse IGA. The key idea of the presented method is unfolding the control net of a NURBS surface for isogeometric analysis by energy-based initial solution estimation algorithm. In addition, we developed a “cutting-stitching” algorithm which can separate a complex control net into several parts with simple shapes. Numerical examples illustrate the initial solutions using the presented method are approaching the final results by One-step inverse finite element method. This implies that the iterative steps and computational time of One-step inverse IGA will be reduced significantly compared with that of One-step inverse finite element method. [ABSTRACT FROM AUTHOR]

In this paper, we propose a general framework for constructing IGA-suitable planar B-spline parameterizations from given complex CAD boundaries. Instead of the computational domain bounded by four B-spline curves, planar domains with high genus and more complex boundary curves are considered. Firstly, some pre-processing operations including Bézier extraction and subdivision are performed on each boundary curve in order to generate a high-quality planar parameterization; then a robust planar domain partition framework is proposed to construct high-quality patch-meshing results with few singularities from the discrete boundary formed by connecting the end points of the resulting boundary segments. After the topology information generation of quadrilateral decomposition, the optimal placement of interior Bézier curves corresponding to the interior edges of the quadrangulation is constructed by a global optimization method to achieve a patch-partition with high quality. Finally, after the imposition of C 1 ∕ G 1 -continuity constraints on the interface of neighboring Bézier patches with respect to each quad in the quadrangulation, the high-quality Bézier patch parameterization is obtained by a local optimization method to achieve uniform and orthogonal iso-parametric structures while keeping the continuity conditions between patches. The efficiency and robustness of the proposed method are demonstrated by several examples which are compared to results obtained by the skeleton-based parameterization approach. [ABSTRACT FROM AUTHOR]

This paper presents a flexible method for coupling NURBS patches in isogeometric Kirchhoff–Love shell analysis. The required C 1 -continuity in such a shell formulation significantly complicates the patch coupling (as compared to typical C 0 -cases). In the present work, the C 0 -part of the coupling is a global coupling in a weak sense, whereas the C 1 -continuity is enforced by a strong point-wise coupling in well-chosen collocation points along the interface. The coupling conditions can be derived using only mesh information, without the need for suitable penalty or stabilisation parameters. They are expressed using a master–slave formulation between the interface variables. A static condensation approach to enforce these continuity constraints results in a reduced system matrix. The proposed method can be employed for both conforming and non-conforming patch configurations, and for G 1 -continuous structures as well as for patches meeting at a kink. This is demonstrated for a set of problems of (dynamic) shell analysis, including both eigenvalue and boundary-value problems. [ABSTRACT FROM AUTHOR]

High-order finite element methods harbor the potential to deliver improved accuracy per degree of freedom versus low-order methods. Their success, however, hinges upon the use of a curvilinear mesh of not only sufficiently high accuracy but also sufficiently high quality. In this paper, theoretical results are presented quantifying the impact of mesh parameterization on the accuracy of a high-order finite element approximation, and a formal definition of shape regularity is introduced for curvilinear meshes based on these results. This formal definition of shape regularity in turn inspires a new set of quality metrics for curvilinear finite elements. Computable bounds are established for these quality metrics using the Bernstein–Bézier form, and a new curvilinear mesh optimization procedure is proposed based on these bounds. Numerical results confirming the importance of shape regularity in the context of high-order finite element methods are presented, and numerical results demonstrating the promise of the proposed curvilinear mesh optimization procedure are also provided. The theoretical results in this paper apply to any piecewise-polynomial or piecewise-rational finite element method posed on a mesh of polynomial or rational mapped simplices and hypercubes. As such, they apply not only to classical continuous Galerkin finite element methods but also to discontinuous Galerkin finite element methods and even isogeometric methods based on NURBS, T-splines, or hierarchical B-splines. • A formal definition of shape regularity is introduced for curvilinear meshes • A set of quality metrics is proposed for curvilinear meshes based on shape regularity • Computable bounds are established for the presented quality metrics • A mesh optimization procedure is proposed based on the quality metric bounds • Numerical results confirm the importance of shape regularity for high-order methods [ABSTRACT FROM AUTHOR]

The paper is to develop stochastic isogeometric analysis for the free vibration of functionally graded plates with spatially varying random material properties. The isogeometric analysis method is employed in the new stochastic analysis scheme which is called Stochastic IsoGeometric Analysis (SIGA). The elastic modulus and mass density are modeled as homogeneous Gaussian random fields along the plane of structure. The governing equation of stochastic isogeometric analysis for free vibration of functionally graded plates is derived in conjunction with perturbation expansions to predict the first and second moments of eigenvalues. In order to verify proposed method, the brute force Monte Carlo simulation is employed. The mean, variance, and COV (Coefficient of Variation) predicted by SIGA and those predicted by Monte Carlo simulation show good agreement. The numerical examples demonstrate that the randomness of material properties affects significantly the structural responses of the functionally graded plates. The correlation between elastic modulus and mass density is also observed to have significant effect on the response COV of eigenvalue. [ABSTRACT FROM AUTHOR]

In this paper a discontinuous, hanging-node, isogeometric analysis (IGA) method is developed and applied to the first-order form of the neutron transport equation with a discrete ordinate (S N ) angular discretisation in two-dimensional space. The complexities involved in upwinding across curved element boundaries that contain hanging-nodes have been addressed to ensure that the scheme remains conservative. A robust algorithm for cycle-breaking has also been introduced in order to develop a unique sweep ordering of the elements for each discrete ordinates direction. The convergence rate of the scheme has been verified using the method of manufactured solutions (MMS) with a smooth solution. Heuristic error indicators have been used to drive an adaptive mesh refinement (AMR) algorithm to take advantage of the hanging-node discretisation. The effectiveness of this method is demonstrated for three test cases. The first is a homogeneous square in a vacuum with varying mean free path and a prescribed extraneous unit source. The second test case is a radiation shielding problem and the third is a 3×3 “supercell” featuring a burnable absorber. In the final test case, comparisons are made to the discontinuous Galerkin finite element method (DGFEM) using both straight-sided and curved quadratic finite elements. [ABSTRACT FROM AUTHOR]

Isogeometric analysis is a method for solving geometric partial differential equations (PDEs). Generating parametrizations of a PDE’s physical domain is a basic and important issue within isogeometric analysis framework. In this paper, we present a global H 1 -parametrization method for a complex planar physical domain. [ABSTRACT FROM AUTHOR]

Numerical integration is a core subroutine in many engineering applications, including the finite element method (FEM). Isogeometric analysis is a FEM technology that uses smooth B-spline and NURBS basis functions. Traditionally, Gaussian quadrature was used for numerical integration, but this yields suboptimal performance. This is attributed to the fact that Gaussian quadrature rules do not take inter-element smoothness of the spline basis into account, resulting in over-integration. The equations for exact quadrature are well known, but prove notoriously hard to solve due to their nonlinear nature. These generalized Gauss rules were first introduced in Hughes et al. (2010) in the context of isogeometric analysis, where newton iteration was utilized to study and tabulate several cases. Later, homotopy continuation (Bartoň and Calo 2015) was used as an alternative strategy for finding these rules. In this paper we describe an algorithm to generalize on both of these techniques. It is optimal in the sense that it will integrate a space of dimension n , using no more than n + 1 2 quadrature points. The algorithm works on arbitrary nonuniform knot vectors of any polynomial degree and continuity, and is demonstrated on polynomial orders up to 15. It extends to 2D and 3D integrals by tensor product. [ABSTRACT FROM AUTHOR]

Thin-walled shells with cutouts are widely used as primary structures in the aerospace field. The mechanical analysis of shell structures using the Finite Element Analysis (FEA) method often needs very fine meshes to achieve a high-fidelity simulation. The advent of isogeometric analysis (IGA) method provides an alternative, yet efficient and accurate means to model complex shell structures with cutouts. In this paper, the trimmed surface analysis (TSA) method is applied in IGA to perform the trimmed elements integration of linear buckling analysis of shell structures. In the shape optimization process, the analytical formulae for the design sensitivities of IGA based shell buckling model are derived, and the sensitivity propagation from the design model to the analysis model is computed using a h -refinement method. Moreover, a novel smart design domain (SDD) method is proposed and implemented to reduce the number of design variables and further enhance the shape optimization efficiency, substantially. SDD can provide useful reference information to guide the selection of control points as design variables. Later, the comparison of optimization examples approve that the SDD method can provide a very efficient means to perform the local shape optimization for the complex shells. Finally, a multi-level progressive optimization framework based on the SDD method is developed, in which the design model and the analysis model are separated and implemented independently. In doing so, the design model is able to provide enough parameterization for optimization, and, in the meantime, the analysis model ensures efficiency and accuracy for simulation. Finally, the numerical examples on the buckling analysis and optimization of complex shells with cutouts demonstrate the greatly improved capacity and efficiency of the design framework proposed in this paper. [ABSTRACT FROM AUTHOR]

Nonlinear finite element methods such as isogeometric analysis and absolute nodal coordinate formulation can be applied to modeling the flexible multibody systems (FMBS) subject to both large rotation and deformation. However, the computational expense is prohibitive for complex systems, especially for a parameterized FMBS. Proper orthogonal decomposition (POD), nonmodal model order reduction (MOR) technique, has been widely applied to nonlinear problems in order to reduce the computational expense of simulating the dynamic responses of FMBSs. This paper presents a component-level POD method, which applies the singular value decomposition process to snapshots of each component of an FMBS and utilizes constraint equations to satisfy the connecting conditions among the components. Furthermore, the paper provides a systematic study of the parameterized MOR (PMOR) for an FMBS, including the reduction of the constraint equations, energy conserving mesh sampling and weighting (ECSW), and the greedy-POD method. The ECSW method aims to further reducing the computational cost of matrix multiplications for evaluating the reduced stiffness matrix. Greedy-POD is applied to get a robust set of reduced order bases with parametric changes in simulation. Finally, three numerical examples validate the feasibility of the component-level POD as well as the whole PMOR process for an FMBS, thereby demonstrating that the component-level POD performs better numerical simulations in accuracy computational costs than the classical POD. • A component-level proper orthogonal decomposition is proposed for multibody systems. • Redundancy of the constraint equations for a reduced order model is investigated. • Efficiency and accuracy can be improved for implementation of reduced order models. • The proposed method is embedded into the greedy algorithm for parametric systems. [ABSTRACT FROM AUTHOR]

This paper presents triangulation-based Isogeometric Analysis of the Cahn–Hilliard phase-field model. The Cahn–Hilliard phase-field model is governed by a time-dependent fourth-order partial differential equation. The corresponding primal variational form involves second-order operators, making it difficult to be directly analyzed with traditional C 0 finite element analysis. In this paper, we construct C 1 Bernstein–Bézier simplicial elements through macro-element techniques, including various triangle-split based macro-elements in both 2D and 3D space. We extend triangulation-based isogeometric analysis to solving the primal variational form of the Cahn–Hilliard equation. We validate our method by convergence analysis, showing the nodal and degree-of-freedom advantages over C 0 Finite Element Analysis. We then demonstrate detailed system evolution from randomly perturbed initial conditions in periodic two-dimensional squares and three-dimensional cubes. We incorporate an adaptive time-stepping scheme in these numerical experiments. Our numerical study demonstrates that triangulation-based isogeometric analysis offers optimal convergence and time step stability, is applicable to complex geometry and allows local refinement. • We use triangular macro-elements for C1 smoothness, rather than Lagrange multipliers. • Our construction of C1 elements is automatic. • Our approach is applicable to problems domains of complex topology. • Our approach allows convenient local refinement. • We have achieved optimal convergence rates for a model problem. [ABSTRACT FROM AUTHOR]

The aim of this paper is to analyze the integration for 3D isogeometric finite element method solvers and its effective scheduling on hierarchical computer architecture. Data necessary for concurrency over elements is independent, so computation on this level is trivially concurrent. However, constructing several layers of concurrency for the integration algorithm is challenging. In this work, we propose a multilevel concurrent integration algorithm associated with scheduling that brings one extra degree of possible speedup. Because of one extra degree of possible speedup, we analyze the concurrent integration inside elements. The scheduling algorithm is intended for strongly related hierarchical architectures of a GPU. Using trace theory and Foata Normal Form, we verify integrity of the proposed solution. Summing up, we propose a general method for analyzing concurrency of the integration algorithm. We instantiate this method on a classical element-based integration algorithm, however, this methodology is possible to apply for other integration algorithms, including sum factorization, fast numerical quadrature, or row-wise integration methods. • Multilayer parallel integrating coefficients of IsoGeometric Analysis IGA equations. • Trace theory model used for the formal verification. • Dikert graph based sub-optimal scheduling on cluster with GPU units. • Efficiency and scalability tests of proposed parallel IGA for L2 projection problem. • Application to sum factorization, fast numerical quadrature, row-wise integration. [ABSTRACT FROM AUTHOR]

Isogeometric analysis (IGA) was introduced as a way to bypass the design-to-analysis bottleneck inherent in the traditional Computer Aided Design (CAD) through Finite Element Analysis (FEA) paradigm. However, an outstanding problem in the field of IGA is that of surface-to-volume parameterization. In CAD packages, solid objects are represented by a collection of NURBS or T-spline bounding surfaces, but to perform engineering analysis on real world problems, we must find a way to parameterize the volumes of these objects as well. This has proven to be difficult using traditional IGA, as the tensor-product nature of trivariate NURBS and T-splines limit their ability to create analysis suitable parameterizations of arbitrarily complex volumes. To overcome the limitations of trivariate NURBS and T-splines, we propose the use of rational Bernstein–Bézier tetrahedra to create analysis suitable volumetric parameterizations for isogeometric analysis. In this paper, which is part one of a two part series, we present the methodology for discretizing two dimensional geometries using rational Bernstein–Bézier triangles. In addition to presenting finite element analysis methodologies based on rational Bernstein–Bézier triangles, we also introduce two new mesh generation strategies for automatically creating high quality, geometrically exact curvilinear meshes. We assess the quality of our mesh generation schemes using a suite of challenging two-dimensional geometries, and we verify the accuracy of our proposed numerical discretization approach using the method of manufactured solutions. [ABSTRACT FROM AUTHOR]

In the paper we present a multi patch isogeometric mixed element with implicit geometric continuity at the ends for the analysis of non polar Kirchhoff space rods. Rotations are introduced at the ends of the element analogously to the Hermitian interpolation, performing a change of basis for the configuration space. A mixed method in which the internal forces are interpolated by means of an L 2 projection leading to a B ̄ formulation is used to cope with the membrane–flexural locking that affects the solution. The efficiency of the method in avoiding locking and instabilities is investigated. The formulation adopted has the advantage with respect to other formulations that a non full global stiffness matrix is obtained. This approach appears very appealing for efficient analyses of general assemblies of 3D non polar elements. [ABSTRACT FROM AUTHOR]

The work presented in this paper concerns the efforts of conducting a computational fluid dynamics (CFD) simulation of air flow past a fixed 2D NACA0015 airfoil at high Reynolds number ( R e = 2.5 × 10 6 ) using an isogeometric finite element methodology with linear, quadratic and cubic spline elements. Flow simulations at such high Reynolds numbers require turbulence models or very high resolution. The present work employs the Spalart–Allmaras turbulence model combined with a Navier–Stokes solver based on a Chorin projection method, the first development of its kind in an isogeometric finite element framework. The obtained results from the simulations are compared with two sets of experimental results available in the literature. [ABSTRACT FROM AUTHOR]

Isogeometric collocation methods have been proposed recently and their accuracy and efficiency demonstrated for elastostatics and explicit dynamics. This paper addresses two important aspects in the development of the isogeometric collocation technology, namely, the imposition of Neumann boundary conditions and the enforcement of contact constraints, which are both treated within the same framework. It is shown that the strong imposition of Neumann boundary conditions may lead to a significant loss of accuracy in some situations, in particular when non-uniform meshes are used. Two possible remedies are proposed to restore the desired level of accuracy while keeping the computational cost virtually unchanged, i.e. a hybrid collocation–Galerkin approach and an enhanced collocation (EC) approach. A frictionless contact formulation suitable for the collocation framework is further proposed and shown to pass the contact patch test to machine precision. When combined with the EC approach, the formulation is shown to deliver accurate results and to perform robustly also for highly non-uniform meshes. For all the collocation formulations, contact pressures are greater than or equal to zero pointwise , in contrast with standard Lagrange finite elements. [ABSTRACT FROM AUTHOR]

In this paper hierarchical analysis-suitable T-splines (HASTS) are developed. The resulting spaces are a superset of both analysis-suitable T-splines and hierarchical B-splines. The additional flexibility provided by the hierarchy of T-spline spaces results in simple, highly localized refinement algorithms which can be utilized in a design or analysis context. A detailed theoretical formulation is presented. Bézier extraction is extended to HASTS simplifying the implementation of HASTS in existing finite element codes. The behavior of a simple HASTS refinement algorithm is compared to the local refinement algorithm for analysis-suitable T-splines demonstrating the superior efficiency and locality of the HASTS algorithm. Finally, HASTS are utilized as a basis for adaptive isogeometric analysis. [ABSTRACT FROM AUTHOR]

This paper proposes a method for the construction of quadratic serendipity element (QSE) shape functions on planar convex and concave polygons. Existing approaches for constructing QSE shape functions are linear combinations of the pair-wise products of generalized barycentric coordinates with linear precision, restricted to the convex polygonal domain or resort to numerical optimization. We extend the construction to general polygons with no more than three collinear consecutive vertices. This is done by defining coefficients of the linear combination as the oriented area of triangles with vertices from the polygonal domain, which can be either convex or concave. The proposed shape functions possess linear to quadratic precision. We prove the interpolation error estimates for mean value coordinate-based QSE shape functions on convex and concave polygonal domains satisfying a set of geometric constraints for standard finite element analysis. We also tailor a polygonal mesh generation scheme that improves the uniformity and avoids short edges of Voronoi diagrams for their use in the QSE-based polygonal finite element computation. Numerical tests for the 2D Poisson equations on various domains are presented, demonstrating the optimal convergence rates in both the L 2 -norm and the H 1 -seminorm. • Construction of quadratic serendipity shape functions on convex/concave polygons. • Interpolation error estimates for MVC-based quadratic serendipity shape functions. • Mesh generation for quadratic serendipity element based finite element computation. [ABSTRACT FROM AUTHOR]

Abstract: Isogeometric analysis (IGA) is emerging as a technology bridging computer aided geometric design (CAGD), most commonly based on Non-Uniform Rational B-Splines (NURBS) surfaces, and engineering analysis. In finite element and boundary element isogeometric methods (FE-IGA and IGA-BEM), the NURBS basis functions that describe the geometry define also the approximation spaces. In the FE-IGA approach, the surfaces generated by the CAGD tools need to be extended to volumetric descriptions, a major open problem in 3D. This additional passage can be avoided in principle when the partial differential equations to be solved admit a formulation in terms of boundary integral equations, leading to boundary element isogeometric analysis (IGA-BEM). The main advantages of such an approach are given by the dimensionality reduction of the problem (from volumetric-based to surface-based), by the fact that the interface with CAGD tools is direct, and by the possibility to treat exterior problems, where the computational domain is infinite. By contrast, these methods produce system matrices which are full, and require the integration of singular kernels. In this paper we address the second point and propose a nonsingular formulation of IGA-BEM for 3D Stokes flows, whose convergence is carefully tested numerically. Standard Gaussian quadrature rules suffice to integrate the boundary integral equations, and carefully chosen known exact solutions of the interior Stokes problem are used to correct the resulting matrices, extending the work by Klaseboer et al. (2012) [27] to IGA-BEM. [Copyright &y& Elsevier]

In this paper, we present a new posteriori method to generate high order curved meshes directly from linear meshes through a recently developed poly-spline isogeometric (IGA) method. Given an input structured or unstructured linear quadrilateral or hexahedral mesh, our method constructs quadratic poly-spline IGA bases for each element and ensures continuity and smoothness across neighboring elements. With these smooth IGA bases, each element can be sampled into a high order element of arbitrary order while maintaining consistently continuous and smooth interfaces between elements. Several fitting algorithms are developed to adjust the IGA control points to ensure the final mesh interpolates or approximates the initial boundary closely. Our method requires no CAD geometry of the initial mesh and guarantee C 1 smoothness in regular regions and C 0 continuity across interfaces of spline-incompatible elements. The method is compared with an elastic analogy based approach and is shown to be superior in terms of computational performance and overall mesh quality and smoothness. The current method can also be extended for higher order IGA basis construction and smooth refinement of meshes that are important for high order physics simulations. • A new isogeometric (IGA) method for posteriori generation of high order curved meshes directly from linear meshes. • The method requires no CAD geometry of the initial mesh and guarantee validity and smoothness of the generated mesh. • The method is much faster than the elastic analogy based approach and produces smoother high order meshes. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we develop a framework for fluid–structure interaction (FSI) modeling and simulation with emphasis on isogeometric analysis (IGA) and non-matching fluid–structure interface discretizations. We take the augmented Lagrangian approach to FSI as a point of departure. Here the Lagrange multiplier field is defined on the fluid–structure interface and is responsible for coupling of the two subsystems. Thus the FSI formulation does not rely on the continuity of the underlying function spaces across the fluid–structure interface in order to produce the correct coupling conditions between the fluid and structural subdomains. However, in deriving the final FSI formulation the interface Lagrange multiplier is formally eliminated and the formulation is written purely in terms of primal variables. Avoiding the use of Lagrange multipliers adds efficiency to the proposed formulation. As an added benefit, the ability to employ non-matching grids for multi-physics simulations leads to significantly relaxed requirements that are placed on the geometry modeling and meshing tools for IGA. We show an application of the proposed FSI formulation to the simulation of the NREL 5MW offshore wind turbine rotor, where the aerodynamics domain is modeled using volumetric quadratic NURBS, while the rotor structure is modeled using a cubic T-spline-based discretization of a rotation-free Kirchhoff–Love shell. We conclude the article by showing FSI coupling of a T-spline shell with a low-order finite element method (FEM) discretization of the aerodynamics equations. This combined use of IGA and FEM is felt to be a good balance between speed, robustness, and accuracy of FSI simulations for this class of problems. [Copyright &y& Elsevier]

Abstract: Finite Element Tearing and Interconnecting (FETI) methods are a powerful approach to designing solvers for large-scale problems in computational mechanics. The numerical simulation problem is subdivided into a number of independent sub-problems, which are then coupled in appropriate ways. NURBS- (Non-Uniform Rational B-spline) based isogeometric analysis (IGA) applied to complex geometries requires to represent the computational domain as a collection of several NURBS geometries. Since there is a natural decomposition of the computational domain into several subdomains, NURBS-based IGA is particularly well suited for using FETI methods. This paper proposes the new IsogEometric Tearing and Interconnecting (IETI) method, which combines the advanced solver design of FETI with the exact geometry representation of IGA. We describe the IETI framework for two classes of simple model problems (Poisson and linearized elasticity) and discuss the coupling of the subdomains along interfaces (both for matching interfaces and for interfaces with T-joints, i.e. hanging nodes). Special attention is paid to the construction of a suitable preconditioner for the iterative linear solver used for the interface problem. We report several computational experiments to demonstrate the performance of the proposed IETI method. [Copyright &y& Elsevier]

Abstract: A methodology is presented which enables isogeometric analysis on trimmed NURBS surfaces. The general limitations of NURBS-based isogeometric analysis are roughly delineated to resolve the associated issues. A local reconstruction technique using a geometric basis is developed and applied to evaluate the finite element constituents of the trimmed knot spans in terms of the underlying control variables. This paper covers the algorithmic treatment of single patches as well as of multi-patches. Moreover, the proposed methodology gives rise to a new feature in the case of load application, what we refer to as Isogeometric Load Design. It provides the possibility to apply an arbitrarily shaped loading area onto a patch. Consequently, the restriction of the alignment of the loading with the underlying parameterization is canceled. In several significant examples the capabilities and effectiveness of this new method are demonstrated and confirm that the proposed method produces compelling results. [Copyright &y& Elsevier]

Abstract: This paper presents the use of Powell–Sabin splines in the context of isogeometric analysis for the numerical solution of advection–diffusion–reaction equations. Powell–Sabin splines are piecewise quadratic functions defined on a given triangulation with a particular macro-structure. We discuss the Galerkin discretization based on a normalized Powell–Sabin B-spline basis. We focus on the accurate detection of internal and boundary layers, and on local refinements. We apply the approach to several test problems, and we illustrate its effectiveness by a comparison with classical finite element and recent isogeometric analysis procedures. [Copyright &y& Elsevier]

Abstract: We study the performance of direct solvers on linear systems of equations resulting from isogeometric analysis. The problem of choice is the canonical Laplace equation in three dimensions. From this study we conclude that for a fixed number of unknowns and polynomial degree of approximation, a higher degree of continuity k drastically increases the CPU time and RAM needed to solve the problem when using a direct solver. This paper presents numerical results detailing the phenomenon as well as a theoretical analysis that explains the underlying cause. [Copyright &y& Elsevier]

Abstract: The use of T-splines in Isogeometric Analysis has been proposed in as a tool to enhance the flexibility of isogeometric methods. If T-splines are a very general concept, their success in isogeometric analysis relies upon some basic properties that needs to be true as e.g. (i) linear independence of blending functions and (ii) polynomial reproducibility at element level. In this paper we study these properties for T-splines of a reduced regularity order, namely, for T-splines of degree p and regularity α = p −1−⌊p/2⌋. Our results are both for odd and even degree. Under mild assumptions on the underlying T-mesh, T-splines are shown to be linearly independent and the space they span is characterized in terms of piecewise polynomials on a topological extension of the T-mesh. Also, as p is odd, we construct a new topological local refinement algorithm and demonstrate its locality properties through numerical examples. [Copyright &y& Elsevier]