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Purpose: Isogeometric analysis (IGA) seamlessly integrates computer-aided design (CAD) with finite element analysis (FEA), streamlining the transition from geometric modeling to structural analysis. Methods: This paper introduces a high-order isogeometric collocation method (IGA-C) specifically designed for analyzing complex multi-patch geometric structures. The method effectively resolves two primary challenges: the optimal selection of collocation points in high-order elements and ensuring stability in computations under complex geometric boundary conditions. Results: Our contributions are threefold: first, we develop high-order basis function elements featuring local adaptive refinement tailored for IGA-C. Second, we investigate the optimal placement of collocation points across elements of varying orders, emphasizing their strategic distribution within non-uniform grids. Third, we present the Gaussian collocation method (IGA-GC) that employs advanced PHT-spline elements to facilitate the analysis of intricate multipatch structures. Conclusions: Additionally, this work expands the algorithm to support elements of arbitrarily high order within the IGA-GC framework. Our numerical experiments validate that the proposed method not only achieves optimal convergence rates but also adeptly navigates the complexities associated with multi-patch geometric structures. [ABSTRACT FROM AUTHOR]

Isogeometric analysis (IGA) is an effective numerical method for connecting computer-aided design and engineering, which has been widely applied in various aspects of computational mechanics. IGA involves Galerkin and collocation formulations. Exploiting the same high-order non-uniform rational B-spline (NURBS) bases that span the physical domain and the solution space leads to increased accuracy and fast computation. Although IGA Galerkin provides optimal convergence, IGA collocation performs better in terms of the ratio of accuracy to computational time. Without numerical integration, by working directly with the strong form of the partial differential equation over the physical domain defined by NURBS geometry, the derivatives of the NURBS-expressed numerical solution at some chosen collocation points can be calculated. In this study, we survey the methodological framework and the research prospects of IGA. The collocation schemes in the IGA collocation method that affect the convergence performance are addressed in this paper. Recent studies and application developments are reviewed as well. [ABSTRACT FROM AUTHOR]

We propose a frictionless contact formulation for isogeometric analysis, which combines a collocated formulation for the contact surfaces with a standard Galerkin treatment of the bulk. We denote it as isogeometric Collocated Contact Surface (CCS) formulation. The approach is based on a simple pointwise enforcement of the contact constraints, performed in this study with the penalty method. Unlike pointwise (node-to-surface or point-to-surface) contact algorithms in the Galerkin framework, the CCS formulation passes the contact patch test to machine precision by naturally exploiting the favorable properties of isogeometric collocation. Compared with approaches where the discretization of both bulk and contact surfaces is based on collocation, the CCS approach does not need enhancements to remove oscillations for highly non-uniform meshes. With respect to integral contact approaches, the CCS algorithm is less computationally expensive, due to the reduced amount of contact evaluation points. In addition, the CCS approach is easy to code and can be added to a pre-existing isogeometric analysis code with minimal effort. Numerical examples in both small and large deformations are investigated to compare the CCS approach with some available contact formulations and to demonstrate its accuracy. [ABSTRACT FROM AUTHOR]

We propose an isogeometric approach to model the deformation of active thin films using layered, nonlinear, Kirchhoff–Love shells. Isogeometric Collocation and Galerkin formulations are employed to discretize the electrophysiological and mechanical sub-problems, respectively, with the possibility to adopt different element and time-step sizes. Numerical tests illustrate the capabilities of the active-stress-based approach to effectively simulate the contraction of thin films in both quasi-static and dynamic conditions. [ABSTRACT FROM AUTHOR]

Although the isogeometric collocation (IGA-C) method has been successfully utilized in practical applications due to its simplicity and efficiency, only a little theoretical results have been established on the numerical analysis of the IGA-C method. In this paper, we deduce the convergence rate of the consistency of the IGA-C method. Moreover, based on the formula of the convergence rate, the necessary and sufficient condition for the consistency of the IGA-C method is developed. These results advance the numerical analysis of the IGA-C method. [ABSTRACT FROM AUTHOR]

We investigate primal and mixed u- p isogeometric collocation methods for application to nearly-incompressible isotropic elasticity. The primal method employs Navier's equations in terms of the displacement unknowns, and the mixed method employs both displacement and pressure unknowns. As benchmarks for what might be considered acceptable accuracy, we employ constant-pressure Abaqus finite elements that are widely used in engineering applications. As a basis of comparisons, we present results for compressible elasticity. All the methods were completely satisfactory for the compressible case. However, results for lowdegree primal methods exhibited displacement locking and in general deteriorated in the nearly-incompressible case. The results for the mixed methods behaved very well for two of the problems we studied, achieving levels of accuracy very similar to those for the compressible case. The third problem, which we consider a "torture test" presented a more complex story for the mixed methods in the nearly-incompressible case. [ABSTRACT FROM AUTHOR]

We present a displacement-based and a mixed isogeometric collocation (IGA-C) formulation for free-form, three-dimensional, shear-deformable beams with high and rapidly-varying curvature and torsion. When such complex shapes are concerned, the approach used to build the IGA geometric model becomes relevant. Although IGA-C has been so far successfully applied to a wide range of problems, the effects that different parameterization and knot placement techniques may have on the accuracy of collocation-based formulations is still an unexplored field. To fill this gap, primal and mixed formulations are used combining two parameterization methods (chord-length and equally spaced) with two knot placement techniques (uniformly spaced and De Boor). With respect to the space-varying Frenet local frame, we derive the strong form of the governing equations in a compact form through the definition of two matrix operators conveniently used to perform first and second order derivatives of the vector fields involved in the formulations. This approach is very efficient and easy to implement within a collocation-based scheme. Several challenging numerical experiments allow to test the different considered parameterizations and knot placement techniques, revealing in particular that with the primal formulation an equally spaced parameterization is definitively the most recommended choice and it should always be used with an approximation degree of, at least, p = 6 , although some caution must be adopted when very high Jacobians and small curvatures occur. The same holds for the mixed formulation, with the difference that p = 4 is enough to yield accurate results. [ABSTRACT FROM AUTHOR]

We propose a novel approach to the implicit dynamics of shear-deformable geometrically exact beams, based on the isogeometric collocation method combined with the Newmark time integration scheme extended to the rotation group SO(3). The proposed formulation is fully consistent with the underlying geometric structure of the configuration manifold. The method is highly efficient, stable, and does not suffer from any singularity problem due to the (material) incremental rotation vector employed to describe the evolution of finite rotations. Consistent linearization of the governing equations, variables initialization and update procedures are the most critical issues which are discussed in detail in the paper. Numerical applications involving very large motions and different boundary conditions demonstrate the capabilities of the method and reveal the critical role that the high-order approximation in space may have in improving the accuracy of the solution. [ABSTRACT FROM AUTHOR]

Isogeometric collocation methods have been introduced as an alternative to isogeometric Galerkin formulations, aiming at improving the computational cost of simulation by reducing the cost of assembly of the corresponding matrices. However, in contrast to their Galerkin counterparts, collocation formulations result in non-symmetric matrices of much higher dimensions, for a specified level of accuracy, which require special attention when addressing large-scale simulations. Furthermore, the presence of mixed boundary conditions may lead to indefinite collocation matrices, which hinder the convergence properties of domain decomposition-based iterative solution methods of the corresponding algebraic equations. To address these shortcomings, two-level hybrid, domain decomposition-based preconditioners are proposed, related to both the basic and the enhanced collocation formulations, in conjunction with the generalized minimal residual method and Schwarz-based preconditioners. The proposed solution schemes improve the computational efficiency of the isogeometric collocation simulations and exhibit numerical scalability for an increased number of subdomains. This is attributed to the fact that the iterations are performed on the Schur complement level of the corresponding matrices, leading to stable, computationally efficient, and robust solutions of the corresponding non-symmetric algebraic equations. [ABSTRACT FROM AUTHOR]

Neuron growth is a complex, multi-stage process that neurons undergo to develop sophisticated morphologies and interwoven neurite networks. Recent experimental research advances have enabled us to examine the effects of various neuron growth factors and seek potential causes for neurodegenerative diseases, such as Alzheimer's disease, Parkinson's disease, and amyotrophic lateral sclerosis. A computational tool that studies the neuron growth process could shed crucial insights on the effects of various factors and potentially help find a cure for neurodegeneration. However, there is a lack of computational tools to accurately and realistically simulate the neuron growth process within reasonable time frames. Bio-phenomenon-based models ignore potential neuron growth factors and cannot generate realistic results, and bio-physics-based models require extensive, high-order governing equations that are computationally expensive. In this paper, we incorporate experimental neurite features into a phase field method-based neuron growth model using an isogeometric analysis collocation (IGA-C) approach. Based on a semi-automated quantitative analysis of neurite morphology, we obtain relative turning angle, average tortuosity, neurite endpoints, average segment length, and the total length of neurites. We use the total neurite length to determine the evolving days in vitro (DIV) and select corresponding neurite features to drive and constrain the neuron growth. This approach archives biomimetic neuron growth patterns with automatic growth stage transitions by incorporating corresponding DIV neurite morphometric data based on the total neurite length of the evolving neurite morphology. Furthermore, we built a convolutional neural network (CNN) to significantly reduce associated computational costs for predicting complex neurite growth patterns. Our CNN model adopts a customized convolutional autoencoder as the backbone that takes neuron growth simulation initializations and target iteration as the input and predicts the corresponding neurite patterns. This approach achieves high prediction accuracy (97.77%) while taking 7 orders of magnitude less computational times when compared with our IGA-C neuron growth solver. [ABSTRACT FROM AUTHOR]

When dealing with timber structures, the characteristic strength and stiffness of the material are made highly variable and uncertain by the unavoidable, yet hardly predictable, presence of knots and other defects. In this work, we apply the sparse grids stochastic collocation method to perform uncertainty quantification for structural engineering in the scenario described above. Sparse grids have been developed by the mathematical community in the last decades, and their theoretical background has been rigorously and extensively studied. The document proposes a brief practice-oriented introduction with minimal theoretical background, provides detailed instructions for the use of the off-the-shelf Sparse Grid Matlab kit (freely available online and straightforward to use) and discusses two preliminary examples inspired from timber engineering problems that highlight how sparse grids exhibit superior performances compared to the plain Monte Carlo method. • An Uncertainty Quantification methodology is applied to timber-like structures. • IGA collocation is used for solving the equilibrium equations. • Smolyak and adaptive sparse grids are used for managing random parameters. • This methodology requires no coding effort and outperforms Monte Carlo. [ABSTRACT FROM AUTHOR]

A novel isogeometric collocation method is proposed for the static limit analysis of axially-symmetric masonry domes subject to their self-weight. A shell-based static formulation is employed, alongside Heyman's assumptions on masonry, to characterize the equilibrated and statically admissible stress states in the dome. As a distinctive feature of the approach, a vector stress function is introduced, generating point-wise self-equilibrated shell stress resultants in the dome. Accordingly, the classical minimum-thrust problem is formulated in terms of the unknown vector stress function, and the static admissibility conditions are enforced as the only optimization constraint. NURBS-based isogeometric analysis is adopted to accomplish the need for an accurate geometric description of the dome and a high-order continuous interpolation of the vector stress function. A discrete minimum-thrust problem is derived as a linear programming problem, with the static admissibility conditions checked at suitable collocation points. Instrumental to its solution is the computation of a particular solution of the equilibrium equations, which is obtained by an isogeometric collocation method. By a mechanical interpretation of the dual optimization problem, the settlement mechanism of the dome corresponding to its minimum-thrust state is also computed. Numerical results, dealing with a thorough convergence analysis, parametric analyses on spherical and ogival domes with parameterized geometry, and the real case of the Taj Mahal central dome are presented to prove the computational merit of the proposed approach. • Axially-symmetric masonry domes subject to their self-weight are considered. • A shell-based static limit analysis approach exploiting stress functions is proposed. • NURBS-based isogeometric collocation is adopted for the problem discretization. • The discrete minimum-thrust problem is solved through linear programming. • Numerical results are presented to prove the computational merit of the method. [ABSTRACT FROM AUTHOR]

Abstract We initiate the study of three-dimensional shear-deformable geometrically exact beam dynamics through explicit isogeometric collocation methods. The formulation we propose is based on a natural combination of the chosen finite rotations representation with an explicit, geometrically consistent Lie group time integrator. We focus on extending the integration scheme, originally proposed for rigid body dynamics, to our nonlinear initial–boundary value problem, where special attention is required by Neumann boundary conditions. The overall formulation is simple and only relies on a geometrically consistent procedure to compute the internal forces once control angular and linear accelerations of the beam cross sections are obtained from the previous time step. The capabilities of the method are shown through numerical applications involving very large displacements and rotations and different boundary conditions. Highlights • We initiate the study of geometrically exact beam dynamics through explicit IGA-C. • Finite rotations are represented through orthogonal matrices. • Configuration updates are performed exploiting spatial incremental rotation vectors. • A geometrically consistent explicit time integration scheme for SO(3) is employed. • Applications reveal the capabilities of the method for very large and fast motions. [ABSTRACT FROM AUTHOR]

The objective of the present work is to develop efficient, higher-order space- and time-accurate, methods for structural dynamics. To this end, we present a family of explicit isogeometric collocation methods for structural dynamics that are obtained from predictor–multicorrector schemes. These methods are very similar in structure to explicit finite-difference time-domain methods, and in particular, they exhibit similar levels of computational cost, ease of implementation, and ease of parallelization. However, unlike finite difference methods, they are easily extended to non-trivial geometries of engineering interest. To examine the spectral properties of the explicit isogeometric collocation methods, we first provide a semi-discrete interpretation of the classical predictor–multicorrector method. This allows us to characterize the spatial and modal accuracy of the isogeometric collocation predictor–multicorrector method, irrespective of the considered time-integration scheme, as well as the critical time step size for a particular explicit time-integration scheme. For pure Dirichlet problems, we demonstrate that it is possible to obtain a second-order-in-space scheme with one corrector pass, a fourth-order-in-space scheme with two corrector passes, and a fifth-order-in-space scheme with three corrector passes. For pure Neumann and mixed Dirichlet–Neumann problems, we demonstrate that it is possible to obtain a second-order-in-space scheme with one corrector pass and a third-order-in-space scheme with two corrector passes, and we observe that fourth-order-in-space accuracy may be obtained pre-asymptotically with three corrector passes. We then present second-order-in-time, fourth-order-in-time, and fifth-order-in-time fully discrete predictor–multicorrector algorithms that result from the application of explicit Runge–Kutta methods to the semi-discrete isogeometric collocation predictor–multicorrector method. We confirm the accuracy of the family of explicit isogeometric collocation methods using a suite of numerical examples. [ABSTRACT FROM AUTHOR]

Isogeometric collocation is for the first time considered as a simulation tool for fluid-saturated porous media. Accordingly, with a focus on one-dimensional problems, a mixed collocation approach is proposed and tested in demanding situations, on both quasi-static and dynamic benchmarks. The developed method is proven to be very effective in terms of both stability and accuracy. In fact, the peculiar properties of the spline shape functions typical of isogeometric methods, along with the ease of implementation and low computational cost guaranteed by the collocation framework, make the proposed approach very attractive as a viable alternative to Galerkin-based approaches classically adopted in computational poromechanics. [ABSTRACT FROM AUTHOR]

With the emergence of isogeometric analysis (IGA), the Galerkin rotation-free discretization of Kirchhoff–Love shells is facilitated, enabling more efficient thin shell structural analysis. High-order shape functions used in IGA also allow the collocation of partial differential equations, avoiding the time-consuming numerical integration of the Galerkin technique. The goal of the present work is to apply this method to NURBS-based isogeometric Kirchhoff–Love plates and shells, under the assumption of small deformations. Since Kirchhoff–Love plate theory yields a fourth-order formulation, two boundary conditions are required at each location on the contour, generating some conflicts at the corners where there are more equations than needed. To remedy this overdetermination, we provide priority and averaging rules that cover all the possible combinations of adjacent edge boundary conditions (i.e. the clamped, simply-supported, symmetric and free supports). Greville and alternative superconvergent points are used for NURBS basis of even and odd degrees, respectively. For square, circular, and annular flat plates, convergence orders are found to be in agreement with a-priori error estimates. The proposed isogeometric collocation method is then validated and benchmarked against a Galerkin implementation by studying a set of problems involving Kirchhoff–Love shells. [ABSTRACT FROM AUTHOR]

We present an extension of isogeometric collocation to coupled cardiac electromechanical problems. We develop a staggered solution scheme that takes advantage of isogeometric collocation to reduce the computational effort in the simulation of the mechanical step, guaranteeing high accuracy for all field variables. We mainly focus on (i) the strategy adopted to couple the electrical and mechanical sub-problems, (ii) the possibility of handling different meshes to better represent the spatial scales, (iii) and the mitigation of volumetric locking. To this end, we propose a suitable mixed formulation for finite elasticity. Several numerical tests demonstrate that the mixed formulation retrieves the expected convergence rates under h -refinement and the effectiveness of the proposed solution scheme for electromechanics. [ABSTRACT FROM AUTHOR]

We extend the isogeometric collocation method to the geometrically nonlinear beams. An exact kinematic formulation, able to represent three-dimensional displacements and rotations without any restriction in magnitude, is presented without the introduction of the moving frame concept. A displacement-based formulation is adopted. Full linearization of the strong form of the governing equations is derived consistently with the underlying geometric structure of the configuration manifold. Incremental rotations are parametrized through Eulerian rotation vectors and configuration updates are performed by means of the exponential map. Numerical tests demonstrate that the proposed combination of isogeometric collocation method with the chosen rotations parametrization results in an efficient computational scheme able to model complex problems with high accuracy. [ABSTRACT FROM AUTHOR]

The present paper presents a robust multi-patch formulation based on the isogeometric collocation (IGA-C) method for the solution of linear, spatial Timoshenko beam structures with complex geometry. The proposed approach is based on the combination of the (local) Bishop frame with the exponential map for SO (3) (Rodrigues formula) to compute the beam curvature and its derivative. This choice permits bypassing known issues related to the Serret–Frenet (SF) frame (in cases where the local beam curvature vanishes) and does not require the Darboux vector and its derivative, which are both affected by limitations of the SF frame. Moreover, in contrast to existing formulations mostly derived in the local SF frame, here the formulation is consistently derived by linearizing the governing equations in the material setting, where the multi-patch coupling can be enforced in a straightforward way. Numerical tests are performed on complex spatial curved beams, including a demanding biomechanical problem of a braided stent, proving the accuracy and the robustness of the formulation. [ABSTRACT FROM AUTHOR]

Isogeometric collocation (IGA-C) method is a computational approach to solve boundary value problems. In this method (IGA-C), the differential equations are solved in strong form instead of the weak form approach adopted by Galerkin based formulations. IGA-C method is computationally efficient in comparison to conventional finite element method and Galerkin-Isogeometric approaches. IGA-C method does not involve the process of assembling global stiffness matrix from element stiffness matrix. Another advantage of IGA-C is that it requires a single integration point per element irrespective of the order of Non-Uniform Rational B-Spline functions (NURBS) adopted. Isogeometric collocation has also been demonstrated as a stable, efficient and accurate higher order computational method for explicit problems. For a wider adoption of isogeometric collocation method, beam/plate/shell finite elements within the framework of IGA-C method need to be formulated. Owing to the extensive adoption of laminated composites in structural components, development of beam finite elements for laminated composite beams based on isogeometric collocation method will prove useful during analysis of composite structures. IGA-C method is proposed in this study for the static bending, free vibration and buckling analysis of laminated composite beams. Classical laminated plate theory (CLPT), first order shear deformation theory (FSDT) and higher order shear deformation theory (HSDT) are considered for all the three analyses. The computational approach proposed for laminated beam based on HSDT contains two Degrees of Freedom (DOF) per node. Computational approach for analysing laminated composite beams based on each of these kinematic theories and using IGA-C method is presented. Accuracy of the proposed computational approaches is checked by solving different numerical examples. Values of normalized transverse displacement, normalized stresses, normalized natural frequencies and normalized critical buckling loads are compared with results from the literature and are found to be accurate. • Analysis of laminated composite beams using isogeometric collocation method. • HSDT based 2 DOF per node approach for symmetric laminated beams. • CLPT based 1 DOF per node approach for symmetric laminated beams. • Assessed for bending, buckling and free vibrational analysis. [ABSTRACT FROM AUTHOR]

Phase-field models based on the variational formulation for brittle fracture have recently been shown capable of accurately and robustly predicting complex crack behavior. Their numerical implementation requires costly operations at the quadrature point level, which may include finding eigenvalues and forming tensor projection operators. We explore the application of isogeometric collocation methods for the discretization of second-order and fourth-order phase-field fracture models. We show that a switch from isogeometric Galerkin to isogeometric collocation methods has the potential to significantly speed up phase-field fracture computations due to a reduction of point evaluations. We advocate a hybrid collocation–Galerkin formulation that provides a consistent way of weakly enforcing Neumann boundary conditions and multi-patch interface constraints, is able to handle the multiple boundary integral terms that arise from the weighted residual formulation, and offers the flexibility to adaptively improve the crack resolution in the fracture zone. We present numerical examples in one and two dimensions that illustrate the advantages of our approach. [ABSTRACT FROM AUTHOR]

A geometrically nonlinear, shear-deformable 3D beam formulation with inelastic material behavior and its numerical discretization by a mixed isogeometric collocation method are presented. In particular, the constitutive model captures elasto-visco-plasticity with damage/softening from Mullin's effect, which applies to the modeling of metallic and polymeric materials, e.g., in additive manufacturing applications and metamaterials. The inelastic material behavior is formulated in terms of thermodynamically consistent internal variables for viscoelastic and plastic strains and isotropic and kinematic hardening variables, as well as accompanying evolution equations. A mixed isogeometric collocation method is applied for the discretization of the strong form of the quasi-static nonlinear differential equations. Thus, the displacements of the centerline curve, the cross-section orientations, and the stress resultants (forces and moments) are discretized as B-spline or NURBS curves. The internal variables are defined only locally at the collocation points, and an implicit return-mapping algorithm is employed for their time discretization. The method is verified in comparison to 1D examples as well as reference results for 3D beams. Furthermore, its applicability to the simulation of beam lattice structures subject to large deformations and instabilities is demonstrated. [Display omitted] • Geometrically exact, shear-deformable 3D beam model with inelastic material behavior. • Thermodynamically consistent inelastic material model using internal variables. • Discretization by a mixed isogeometric collocation method using spline curves. • Mixed collocation cures locking and avoids higher derivatives of internal variables. • Applicability to beams and structures subject to large deformations and instabilities. [ABSTRACT FROM AUTHOR]

A collocation technique based on the use of Bernstein polynomials to approximate the field variable is assessed in Boundary Value Problems (BVPs) of beams with governing non-linear differential equations. The BVPs are transformed into unconstrained optimization problems by means of an extended cost function which leverages the properties of the Bernstein basis to enforce the boundary conditions. The minimization of the squared error cost function is conducted by means of the quasi-Newton Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm. The method is tested in benchmarks of various types of non-linearities, including materials with Ludwick stress–strain curves, follower loads and beams on Winkler foundation. The approach is compared with Isogeometric collocation (IGA-c) and straightforward (pseudospectral) Bernstein collocation in terms of performance and computational effort. Moreover, the accuracy and convergence of the method is discussed to ease its successful application to other non-linear beam problems. • Bernstein polynomials and Broyden–Fletcher–Goldfarb–Shanno optimization are combined to solve non-linear beam boundary value problems. • Number of collocation points and unknown parameters are decoupled. • Boundary conditions are enforced via an extended cost function with penalty parameter. • The numerical performance is compared with Isogeometric collocation and standard Bernstein collocation. [ABSTRACT FROM AUTHOR]

We propose a mixed stress-displacement isogeometric collocation method for nearly incompressible elastic materials and for materials exhibiting von Mises plasticity. The discretization is based on isogeometric analysis (IGA) with non-uniform rational B-Splines (NURBS) as basis functions. As compared to conventional IGA Galerkin formulations, isogeometric collocation methods offer a high potential of computational cost reduction for higher-order discretizations as they eliminate the need for quadrature. In the proposed mixed formulation, both stress and displacement fields are approximated as primary variables with the aim of treating volumetric locking and instability issues, which occur in displacement-based isogeometric collocation for nearly incompressible elasticity and von Mises plasticity. The performance of the proposed approach is demonstrated by several numerical examples. • Mixed stress-displacement isogeometric collocation method. • Application to elastoplastic and nearly incompressible elastic materials. • Successful treatment of volumetric locking and instability issues of displacement-based isogeometric collocation method. [ABSTRACT FROM AUTHOR]