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Since its inception, isogeometric analysis (IGA) has shown significant advantages over Lagrange polynomials-based finite element analysis (FEA), especially for contact problems. IGA often uses C1-continuous non-uniform rational B-splines (NURBS) as basis functions, providing a smooth description of kinematic variables across the contact interface. This leads to increased accuracy and stability in the numerical solutions. However, from the existing literature on isogeometric contact analysis, it is not yet clear what interpolation order and continuity of NURBS one should employ to accurately capture the distribution of contact forces across the contact interface. The present work aims to fill this gap and provides a comparative assessment of different NURBS-based standard (conventional) refinement strategies for contact problems within the IGA framework. A recently proposed refinement strategy, known as the varying-order (VO) based NURBS discretization, has demonstrated its capability to refine geometry through the implementation of order elevation in a controlled manner. However, a detailed investigation that directly compares the VO based NURBS discretization with the standard NURBS discretization has not yet been carried out. Therefore, a thorough study of the VO based discretization strategy is also conducted, evaluating its effectiveness in comparison with the standard discretization strategy for contact problems. For this, a few examples on contact problems are solved using an in-house MATLAB® code. The solution to these examples shows that quadratic order standard NURBS discretization is sufficient to achieve the desired level of solution accuracy just by increasing the mesh size. It is further demonstrated that VO based discretization can achieve much higher accuracy than standard discretization, even with a coarse mesh, by generating additional degrees of freedom in the contact boundary layer. In addition, VO based discretization makes considerable savings i, QC 20240318

In this contribution, we present a robust and efficient computational framework capable of accurately capturing the dynamic motion and large deformation/deflection responses of highly-flexible rods interacting with an incompressible viscous flow. Within the partitioned approach, we adopt separate field solvers to compute the dynamics of the immersed structures and the evolution of the flow field over time, considering finite Reynolds numbers. We employ a geometrically exact, nonlinear Cosserat rod formulation in the context of the isogeometric analysis (IGA) technique to model the elastic responses of each rod in three dimensions (3D). The Navier–Stokes equations are resolved using a pressure projection method on a standard staggered Cartesian grid. The direct-forcing immersed boundary method is utilized for coupling the IGA-based structural solver with the finite-difference fluid solver. In order to fully exploit the accuracy of the IGA technique for FSI simulations, the proposed framework introduces a new procedure that decouples the resolution of the structural domain from the fluid grid. Uniformly distributed Lagrangian markers with density relative to the Eulerian grid are generated to communicate between Lagrangian and Eulerian grids consistently with IGA. We successfully validate the proposed computational framework against two- and three-dimensional FSI benchmarks involving flexible filaments undergoing large deflections/motions in an incompressible flow. We show that six times coarser structural mesh than the flow Eulerian grid delivers accurate results for classic benchmarks, leading to a major gain in computational efficiency. The simultaneous spatial and temporal convergence studies demonstrate the consistent performance of the proposed framework, showing that it conserves the order of the convergence, which is the same as that of the fluid solver., QC 20231031

Modeling the full vector hysteresis relation provides critical insight in the magnetic behavior of the core of an electric machine. Yet, including a vector hysteresis model comes at the cost of a significant extra computational load, that grows with the size of the electromagnetic problem. Therefore, high-order methods, which achieve similar accuracy as the well-known finite element method for a smaller problem size, are potentially very interesting when modeling vector hysteresis.

Isogeometric analysis (IGA) is a numerical method to solve partial differential equations, which takes the computer-aided design (CAD) model for the analysis of various field variables. Gaussian quadrature rule is generally employed as a numerical integration scheme to compute the quantities, such as stiffness matrix, force vector, for an elemental patch in IGA. With the increase in Gauss quadrature points, the accuracy of the numerical integration increases. However, it increases the computational cost as well. In this work, a machine learning (ML) technique, artificial neural network (ANN), is employed to get the required Gauss quadrature points for a non-uniform rational B-spline (NURBS) geometry. The developed classification model successfully estimates the required Gauss quadrature points for a given patch of geometry., QC 20240924 ; Part of ISBN [9789819959181]

The geometries constructed using the standard CAD software are always boundary-represented (B-Rep). Due to this, analysis-suitable volumetric parameterization of the geometries constructed using the standard CAD software is necessary for three-dimensional isogeometric analysis (IGA). This paper presents a user-friendly, robust volumetric parametrization tool capable of processing the B-Rep of the geometries constructed in one of the most popular CAD software, i.e. Rhinoceros®. The developed tool traverses multiple stages, starting with reading the standard 3DM CAD files of an arbitrarily shaped geometry. It then extracts surface control points, organizes them to establish an outer structural framework, and employs Coons volume parameterization to generate internal control points based on user-defined specifications. We use the NURBS-Python library for reading 3DM files and the NURBS toolbox to visualize and evaluate NURBS functions in the constructed geometries. We construct several baseline and intricate shape geometries to showcase the robustness and generality of the developed tool. Additionally, to demonstrate the effectiveness and applicability of our tool for IGA of non-linear three-dimensional problems, we simulate frictionless contact between two deformable geometries constructed using the developed tool., QC 20240926

Parallel computing is omnipresent in today's scientific computer landscape, starting at multicore processors in desktop computers up to massively parallel clusters. While domain decomposition methods have a long tradition in computational mechanics to decompose spatial problems into multiple subproblems that can be solved in parallel, advancing solution schemes for dynamics or quasi-statics are inherently serial processes. For quasi-static simulations, however, there is no accumulating ‘time’ discretization error, hence an alternative approach is required. In this paper, we present an Adaptive Parallel Arc-Length Method (APALM). By using a domain parametrization of the arc-length instead of time, the multi-level error for the arc-length parametrization is formed by the load parameter and the solution norm. Given coarse approximations of arc-length intervals, finer corrections enable the parallelization of the presented method. This results in an arc-length method that is parallel within a branch and inherently adaptive. This concept is easily extended for bifurcation problems. The performance of the method is demonstrated using isogeometric Kirchhoff-Love shells on problems with snap-through and pitch-fork instabilities and applied to the problem of a snapping meta-material. These results show that parallel corrections are performed in a fraction of the time of the serial initialization, achievable on desktop scale., Ship and Offshore Structures, Numerical Analysis

The present study aims to develop an original solid-like shell element for large deformation analysis of hyperelastic shell structures in the context of isogeometric analysis (IGA). The presented model includes a new variable to describe the thickness change of the shell and allows for the application of unmodified three-dimensional constitutive laws defined in curvilinear coordinate systems and the analysis of variable thickness shells. In this way, the thickness locking affecting standard solid-shell-like models is cured by enhancing the thickness strain by exploiting a hierarchical approach, allowing linear transversal strains. Furthermore, a patch-wise reduced integration scheme is adopted for computational efficiency reasons and to annihilate shear and membrane locking. In addition, the Mixed-Integration Point (MIP) format is extended to hyperelastic materials to improve the convergence behaviour, hence the efficiency, in Newton iterations. Using benchmark problems, it is shown that the proposed model is reliable and resolves locking issues that were present in the previously published isogeometric solid-shell formulations., Ship and Offshore Structures, Numerical Analysis

Wrinkles are ubiquitous in the world around us. In our daily lives, we encounter wrinkles in various forms, whether in our clothes or on our skin. Wrinkles emerge as a result of a delicate interplay between bending, membrane, and foundation stiffness contributions within membranes. While experimental investigations provide insights into the physics underlying wrinkling, numerical investigations find their purpose in the design, analysis, and optimisation of membranes subjected to wrinkling. Nevertheless, the numerical simulation of membrane wrinkling presents several challenges. Firstly, wrinkling constitutes a buckling phenomenon in membranes with low bending stiffness. Wrinkles have the potential to evolve into folds, creases, or other wrinkling patterns as loads or displacements increase. Secondly, the wavelengths of wrinkling can be orders of magnitude smaller than the overall geometry, requiring a small resolution of the numerical simulation and hence increasing computational costs. Overall, the question arises of how to design robust and accurate numerical models for the analysis of wrinkled membranes. This dissertation is subdivided into four parts and aims to provide answers to this question. The first theme considers hyperelastic material modelling, with a focus on developing wrinkling models under large strains. The shell model employed in this dissertation is based on the isogeometric analysis paradigm. Specifically, the Kirchhoff--Love shell model is used, which leverages the higher-order continuity of underlying spline spaces. Chapter 3 extends hyperelastic material formulations to stretch-based materials, enabling the use of the isogeometric analysis paradigm for rubber-like shells. Since the modelling of wrinkling patterns imposes physical scales limiting element mesh sizes, chapter 4 introduces a hyperelastic isogeometric membrane element that incorporates an implicit wrinkling model, thus avoiding explicit modelling of wrinkling amplitudes., Ship and Offshore Structures

In order to perform isogeometric analysis with increased smoothness on complex domains, trimming, variational coupling or unstructured spline methods can be used. The latter two classes of methods require a multi-patch segmentation of the domain, and provide continuous bases along patch interfaces. In the context of shell modelling, variational methods are widely used, whereas the application of unstructured spline methods on shell problems is rather scarce. In this paper, we therefore provide a qualitative and a quantitative comparison of a selection of unstructured spline constructions, in particular the D-Patch, Almost-C1, Analysis-Suitable G1 and the Approximate C1 constructions. Using this comparison, we aim to provide insight into the selection of methods for practical problems, as well as directions for future research. In the qualitative comparison, the properties of each method are evaluated and compared. In the quantitative comparison, a selection of numerical examples is used to highlight different advantages and disadvantages of each method. In the latter, comparison with weak coupling methods such as Nitsche's method or penalty methods is made as well. In brief, it is concluded that the Approximate C1 and Analysis-Suitable G1 converge optimally in the analysis of a bi-harmonic problem, without the need of special refinement procedures. Furthermore, these methods provide accurate stress fields. On the other hand, the Almost-C1 and D-Patch provide relatively easy construction on complex geometries. The Almost-C1 method does not have limitations on the valence of boundary vertices, unlike the D-Patch, but is only applicable to biquadratic local bases. Following from these conclusions, future research directions are proposed, for example towards making the Approximate C1 and Analysis-Suitable G1 applicable to more complex geometries., Numerical Analysis, Ship and Offshore Structures

Mesh adaptivity is a technique to provide detail in numerical solutions without the need to refine the mesh over the whole domain. Mesh adaptivity in isogeometric analysis can be driven by Truncated Hierarchical B-splines (THB-splines) which add degrees of freedom locally based on finer B-spline bases. Labeling of elements for refinement is typically done using residual-based error estimators. In this paper, an adaptive meshing workflow for isogeometric Kirchhoff–Love shell analysis is developed. This framework includes THB-splines, mesh admissibility for combined refinement and coarsening and the Dual-Weighted Residual (DWR) method for computing element-wise error contributions. The DWR can be used in several structural analysis problems, allowing the user to specify a goal quantity of interest which is used to mark elements and refine the mesh. This goal functional can involve, for example, displacements, stresses, eigenfrequencies etc. The proposed framework is evaluated through a set of different benchmark problems, including modal analysis, buckling analysis and non-linear snap-through and bifurcation problems, showing high accuracy of the DWR estimator and efficient allocation of degrees of freedom for advanced shell computations., Ship and Offshore Structures, Numerical Analysis

Monitoring the cardiac function often relies on non-invasive vital measurements, electrocardiograms, and low-resolution echocardiograms. The monitoring data can be augmented with numerical modeling results to support treatment-risk assessment. Often, medical images are not suitable for high-fidelity modeling due to their spatial sparsity and low resolution. In this work, we present a workflow that converts a limited number of two-dimensional (2D) echocardiogram images into analysis-suitable left-ventricle geometries for isogeometric analysis (IGA). The image data is fitted by reshaping a multi-patch NURBS template. This fitting procedure results in a smooth interpolation with a limited number of degrees of freedom. In addition, a coupled 3D-0D cardiac model is extended with a pre-stressing method to perform ventricular-cardiac-mechanic analyses based on in vivo images. The workflow is benchmarked using population-averaged imaging data, which demonstrates that both the shape of the left ventricle and its mechanical response are well predicted in sparse-data scenarios. The case of a clinical patient is considered to demonstrate the suitability of the workflow in a real-data setting.

Consistent treatment of large rotations in common Reissner–Mindlin formulations is a complicated task. Reissner–Mindlin formulations that use a hierarchic parametrization provide an elegant way to facilitate large rotation shell analyses. This can be achieved by the assumption of linearized transverse shear strains, resulting in an additive split of strain components, which technically simplifies implementation of corresponding shell finite elements. The present study aims at validating this assumption by systematically comparing numerical solutions with those of a newly implemented hierarchic and fully nonlinear Reissner–Mindlin shell element., Deutsche Forschungsgemeinschaft (DFG)

[Abstract:] Topology design of compliant mechanisms has gained wide popularity among the scientific community, and their use in the mechanical engineering field is being of upmost importance. In this paper, an isogeometric analysis (IGA) formulation is used to solve the topology optimization problem of compliant mechanisms. Stress constraints are introduced in the problem to guarantee the attainment of realistic solutions. For this purpose, an overweight constraint is considered for the design process, replacing the use of local stress constraints. The material distribution in the domain is modeled with quadratic B-splines and with a uniform relative density within each element of the mesh. These strategies to define the material layout are used to compare the IGA-based formulation with the finite element (FEM) formulation. The IGA formulation provides several advantages with respect to the classical FEM-based approaches that are shown and analyzed with an input-parameters sensitivity analysis. The sensitivity analysis and the assessment of the importance of introducing of stress constraints in the problem are developed by solving two benchmark problems. Regarding the sensitivity analysis of input parameters, the results show that the ratio between the material and the springs stiffnesses is the parameter with the largest influence on the solutions of the problem. Moreover, the advantages of the IGA formulations over FEM formulations are related with the computational time, the smoothness of the structural borders, and the non-appearance of the checkerboard patterns. With respect to the stress constraints, the results show that they have to be considered in order to avoid instability and structural integrity problems.

This work develops, discretizes, and validates a continuum model of a molybdenum disulfide (MoS2) monolayer interacting with a periodic holey silicon nitride (Si3N4) substrate via van der Waals (vdW) forces. The MoS2 layer is modeled as a geometrically nonlinear Kirchhoff–Love shell, and vdW forces are modeled by a Lennard-Jones (LJ) potential, simplified using approximations for a smooth substrate topography. Both the shell model and LJ interactions include novel extensions informed by close comparison with fully-atomistic calculations. The material parameters of the shell model are calibrated by comparing small-strain tensile and bending tests with atomistic simulations. This model is efficiently discretized using isogeometric analysis (IGA) for the shell structure and a pseudo-time continuation method for energy minimization. The IGA shell model is validated against fully-atomistic calculations for several benchmark problems with different substrate geometries. Agreement with atomistic results depends on geometric nonlinearity in some cases, but a simple isotropic St.Venant–Kirchhoff model is found to be sufficient to represent material behavior. We find that the IGA discretization of the continuum model has a much lower computational cost than atomistic simulations, and expect that it will enable efficient design space exploration in strain engineering applications. This is demonstrated by studying the dependence of strain and curvature in MoS2 over a holey substrate as a function of the hole spacing on scales inaccessible to atomistic calculations. The results show an unexpected qualitative change in the deformation pattern below a critical hole separation.

This work develops, discretizes, and validates a continuum model of a molybdenum disulfide (MoS2) monolayer interacting with a periodic holey silicon nitride (Si3N4) substrate via van der Waals (vdW) forces. The MoS2 layer is modeled as a geometrically nonlinear Kirchhoff–Love shell, and vdW forces are modeled by a Lennard-Jones (LJ) potential, simplified using approximations for a smooth substrate topography. Both the shell model and LJ interactions include novel extensions informed by close comparison with fully-atomistic calculations. The material parameters of the shell model are calibrated by comparing small-strain tensile and bending tests with atomistic simulations. This model is efficiently discretized using isogeometric analysis (IGA) for the shell structure and a pseudo-time continuation method for energy minimization. The IGA shell model is validated against fully-atomistic calculations for several benchmark problems with different substrate geometries. Agreement with atomistic results depends on geometric nonlinearity in some cases, but a simple isotropic St.Venant–Kirchhoff model is found to be sufficient to represent material behavior. We find that the IGA discretization of the continuum model has a much lower computational cost than atomistic simulations, and expect that it will enable efficient design space exploration in strain engineering applications. This is demonstrated by studying the dependence of strain and curvature in MoS2 over a holey substrate as a function of the hole spacing on scales inaccessible to atomistic calculations. The results show an unexpected qualitative change in the deformation pattern below a critical hole separation.

We proposed a convolutional neural network (CNN)-based surrogate model to predict the nonlocal response for flexoelectric structures with complex topologies. The input, i.e. the binary images, for the CNN is obtained by converting geometries into pixels, while the output comes from simulations of an isogeometric (IGA) flexoelectric model, which in turn exploits the higher-order continuity of the underlying non-uniform rational B-splines (NURBS) basis functions to fast computing of flexoelectric parameters, e.g., electric gradient, mechanical displacement, strain, and strain gradient. To generate the dataset of porous flexoelectric cantilevers, we developed a NURBS trimming technique based on the IGA model. As for CNN construction, the key factors were optimized based on the IGA dataset, including activation functions, dropout layers, and optimizers. Then the cross-validation was conducted to test the CNN’s generalization ability. Last but not least, the potential of the CNN performance has been explored under different model output sizes and the corresponding possible optimal model layout is proposed. The results can be instructive for studies on deep learning of other nonlocal mech-physical simulations.

We explore how recent advances in Isogeometric analysis, Galerkin Least-Squares methods, and Augmented Lagrangian techniques can be applied to solve nonstandard problems, for which there is no classical stability theory, such as that provided by the Lax–Milgram lemma or the Banach-Necas-Babuska theorem. In particular, we consider continuation problems where a second-order partial differential equation with incomplete boundary data is solved given measurements of the solution on a subdomain of the computational domain. The use of higher regularity spline spaces leads to simplified formulations and potentially minimal multiplier space. We show that our formulation is inf-sup stable, and given appropriate a priori assumptions, we establish optimal order convergence.

Binary-fluid flows can be modeled using the Navier–Stokes–Cahn–Hilliard equations, which represent the boundary between the fluid constituents by a diffuse interface. The diffuse-interface model allows for complex geometries and topological changes of the binary-fluid interface. In this work, we propose an immersed isogeometric analysis framework to solve the Navier–Stokes–Cahn–Hilliard equations on domains with geometrically complex external binary-fluid boundaries. The use of optimal-regularity B-splines results in a computationally efficient higher-order method. The key features of the proposed framework are a generalized Navier-slip boundary condition for the tangential velocity components, Nitsche's method for the convective impermeability boundary condition, and skeleton- and ghost-penalties to guarantee stability. A binary-fluid Taylor–Couette flow is considered for benchmarking. Porous medium simulations demonstrate the ability of the immersed isogeometric analysis framework to model complex binary-fluid flow phenomena such as break-up and coalescence in complex geometries.

Constructing an analysis-suitable parameterization for the computational domain from its boundary representation plays a crucial role in the isogeometric design-through-analysis pipeline. PDE-based elliptic grid generation is an effective method for generating high-quality parameterizations with rapid convergence properties for the planar case. However, it may generate non-uniform grid lines, especially near the concave/convex parts of the boundary. In the present work, we introduce a novel scaled discretization of harmonic mappings in the Sobolev space H1 to remit it. Analytical Jacobian matrices for the involved nonlinear equations are derived to accelerate the computation. To enhance the numerical stability and the speed of convergence, we propose a simple and yet effective preconditioned Anderson acceleration framework instead of using computationally expensive Newton-type iteration. Three preconditioning strategies are suggested, namely diagonal Jacobian, block-diagonal Jacobian, and full Jacobian. Furthermore, we discuss a delayed update strategy of the preconditioner, i.e., the preconditioner is updated every few steps to reduce the computational cost per iteration. Numerical experiments demonstrate the effectiveness and efficiency of our improved parameterization approach and the computational efficiency of our preconditioned Anderson acceleration scheme., Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public., Numerical Analysis

We develop two new stabilized methods for the steady advection-diffusion-reaction equation, referred to as the Streamline GSC Method and the Directional GSC Method. Both are globally conservative and perform well in numerical studies utilizing linear, quadratic, cubic, and quartic Lagrange finite elements and maximally smooth B-spline elements. For the streamline GSC method we can prove coercivity, convergence, and optimal-order error estimates in a strong norm that are robust in the advective and reactive limits. The directional GSC method is designed to accurately resolve boundary layers for flows that impinge upon the boundary at an angle, a long-standing problem. The directional GSC method performs better than the streamline GSC method in the numerical studies, but it is not coercive. We conjecture it is inf-sup stable but we are unable to prove it at this time. However, calculations of the inf-sup constant support the conjecture. In the numerical studies, B-spline finite elements consistently perform better than Lagrange finite elements of the same order and number of unknowns.

In the search for more efficient and less environmentally harmful cooling technologies, the field of magnetocalorics is considered a promising alternative. To generate cooling spans, rotating permanent magnet assemblies are used to cyclically magnetize and demagnetize magnetocaloric materials, which change their temperature under the application of a magnetic field. In this work, an axial rotary permanent magnet assembly, aimed for commercialization, is computationally designed using topology and shape optimization. This is efficiently facilitated in an isogeometric analysis framework, where harmonic mortaring is applied to couple the rotating rotor-stator system of the multipatch model. Inner, outer and co-rotating assemblies are compared and optimized designs for different magnet masses are determined. These simulations are used to homogenize the magnetic flux density in the magnetocaloric material. The resulting torque is analyzed for different geometric parameters. Additionally, the influence of anisotropy in the active magnetic regenerators is studied in order to guide the magnetic flux. Different examples are analyzed and classified to find an optimal magnet assembly for magnetocaloric cooling. Keywords Harmonic mortaringIsogeometric analysisMagnet assemblyMagnetocaloricsShape optimizationTopology optimization

Pneumatic-driven robots are one of the most widely-used soft robots in the industry, such as manipulation tasks in factories and production lines. The structural design for the soft pneumatic actuator is a hotspot in practical applications. This paper proposes a novel optimal design framework based on isogeometric analysis (IGA) for soft pneumatic actuators, which integrates the design, analysis, and optimization in a NURBS-based framework. Firstly, the positions of the control points are defined to construct a NURBS-based geometry model. To solve the design-dependent analysis problem, a pressure-loading method based on IGA is introduced. An optimization model is built to generate the optimal structures based on the selected design variables related to the NURBS curve. After the optimization, the optimized configurations are printed using an extrusion-based additive manufacturing machine with flexible material, and a pressure-loading experiment is conducted. The experiment reveals the deformation pattern of different structures, demonstrates the proposed method's accuracy and reports the optimized structure performs better than the conventional design. The proposed method provides a high-efficiency optimization way for the structural design of soft pneumatic actuators, advancing in the diverse design space, programming design, and concise optimization model. With isogeometric analysis, the proposed method integrates the deign-analysis-manufacturing system in the NURBS framework, with significant adaptability for a fast-iterating manufacturing system. © 2023 The Author(s)

In this paper we use the Isogeometric Analysis (IgA) to solve the Helmholtz equation with Dirichlet boundary condition over a bounded physical 2D domain. Starting from the variational formulation of the problem, we show how to apply IgA to obtain an approximated solution based on biquadratic B-spline functions. We focus the attention on problems where the physical domain has very irregular boundary. To solve these problems successfully a high quality parametrization of the domain must be constructed. This parametrization is also a biquadratic tensor product B-spline function, with control points computed as the vertices of a quadrilateral mesh with optimal geometric properties. We study experimentally the influence of the wave number and the parametrization of the physical domain in the accuracy of the approximated solution. A comparison with classical Finite Element Method is also included. The power of IgA is shown solving several difficult model problems, which are particular cases of the Helmholtz equation and where the solution has discontinuous gradient in some points, or it is highly oscillatory. For all model problems we explain how to select the knots of B-spline quadratic functions and how to insert knew knots in order to obtain good approximations. The results obtained with our imple-mentation of the method prove that IgA approach is successful, even on regions with irregular boundary, since it is able to offer smooth solutions having at the same time some singular points and high number of oscillations., Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public., Mathematical Physics

We explore how recent advances in Isogeometric analysis, Galerkin Least-Squares methods, and Augmented Lagrangian techniques can be applied to solve nonstandard problems, for which there is no classical stability theory, such as that provided by the Lax–Milgram lemma or the Banach-Necas-Babuska theorem. In particular, we consider continuation problems where a second-order partial differential equation with incomplete boundary data is solved given measurements of the solution on a subdomain of the computational domain. The use of higher regularity spline spaces leads to simplified formulations and potentially minimal multiplier space. We show that our formulation is inf-sup stable, and given appropriate a priori assumptions, we establish optimal order convergence.

This paper proposes a novel isogeometric topology optimization method to obtain the hierarchical structures, where a special porous microstructures of identical topologies but with different length-width ratios are considered. First, the floating projection technique without penalization is adopted to find the topological configurations of hierachical structures, which make the optimized hierachical structures not easy to fall into the local optimums. Second, the isogeometric analysis based on the non-uniform rational B-spline basis functions, is utilized to calculate the structural compliances and sensitivity information. Third, this paper presents a limited number of microstructures with the changing aspect ratios, to satisfy arbitrary boundary conditions of macrostructures. The macroscopic equivalent properties are evaluated by the isogeometric analysis-based homogenization method. Finally, four numerical examples are provided to illustrate the effectiveness of the proposed method in designing hierarchical structures, and the influence of micro-structural properties on the macro-structural performances. © 2023

To accurately analyze the fracture behavior of piezoceramics at small length scales, flexoelectricity must be considered along with piezoelectricity. Due to its dependence on size, flexoelectricity predominates at the micro- and nano-length scales. Additionally, crack tips having the largest strain gradient state cause large flexoelectricity around them. Different approaches are employed in the past to model cracks computationally. However, extended isogeometric analysis (XIGA) is proven to be an accurate and efficient method. C1 continuity requirements for modeling gradients in flexoelectricity are met by non-uniform rational B-splines (NURBS) basis functions used in XIGA. In this work, XIGA-based computational model is developed and implemented to study the fracture behavior of the piezoelectric-flexoelectric domain. An in-house MATLAB code is developed for the same. Several numerical examples are studied to ensure the efficacy and efficiency of the implemented model, and crack behavior is presented in the form of an electro-mechanical J-integral. The analysis is carried out to investigate how cracks behave for different flexoelectric coefficients under different electrical and mechanical loading combinations. J-integral is also analyzed against crack parameters such as crack orientation and length. It is observed that boundary loads and flexoelectric material constants significantly influence J-integral. Results also show a considerable amount of fracture toughening in the presence of flexoelectricity. The peak value of J-integral is found to be reduced with an increase in the flexoelectric coefficient. A significant reduction in J-integral, as much as 45%, is observed when the flexoelectric constant varied from 0.5 to 2 µCm−1., Green Open Access added to TU Delft Institutional Repository 'You share, we take care!' - Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public., Transport Engineering and Logistics

We present an isogeometric method for Kirchhoff–Love shell analysis of shell structures with geometries composed of multiple patches and which possibly possess extraordinary vertices, i.e. vertices with a valency different to four. The proposed isogeometric shell discretisation is based on the one hand on the approximation of the mid-surface by a particular class of multi-patch surfaces, called analysis-suitable G1 (Collin et al., 2016), and on the other hand on the use of the globally C1-smooth isogeometric multi-patch spline space (Farahat et al., 2023). We use our developed technique within an isogeometric Kirchhoff–Love shell formulation (Kiendl et al., 2009) to study linear and non-linear shell problems on multi-patch structures. Thereby, the numerical results show the great potential of our method for efficient shell analysis of geometrically complex multi-patch structures which cannot be modelled without the use of extraordinary vertices., Funding Information: The authors wish to thank the anonymous reviewers for their comments that helped to improve the paper. A. Farahat and M. Kapl have been supported by the Austrian Science Fund (FWF) through the project P 33023-N. H.M. Verhelst is grateful for the funding from Delft University of Technology. J. Kiendl has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No 864482). Additionally, the authors are grateful for the support from the developers of the Geometry + Simulation Modules, in particular from A. Mantzaflaris (Inria Sophia Antipolis-Méditerranée). Funding Information: The authors wish to thank the anonymous reviewers for their comments that helped to improve the paper. A. Farahat and M. Kapl have been supported by the Austrian Science Fund (FWF) through the project P 33023-N . H.M. Verhelst is grateful for the funding from Delft University of Technology . J. Kiendl has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No 864482 ). Additionally, the authors are grateful for the support from the developers of the Geometry + Simulation Modules, in particular from A. Mantzaflaris (Inria Sophia Antipolis-Méditerranée)., Numerical Analysis, Ship Hydromechanics and Structures

Finite element analysis and multibody dynamics are popular mathematical modeling techniques for sev- eral mechanical applications. The finite element method is a costly method from a runtime standpoint. Moreover, both multibody dynamics and finite element methods require approximating the geometry of the application either by meshing or by analytical surfaces. Therefore, IsoGeometric Analysis (IGA) has been proposed as an alternate method for parameterizing geometric surfaces. IGA enables a uni- fied mathematical representation of both geometries and solution fields using B-Splines or NURBS. NURBS are the de-facto standard in modern Computer-Aided Design (CAD) workflows, but their use for analysis in finite elements is rare. By directly using the geometric information for the analysis, IGA reduces the cost of mesh generation, therefore enabling a more efficient design process. The IGA-based rigid body contact algorithm offers several advantages over the traditional FEM- based approach. First, it provides an accurate representation of the geometry. This improves the efficiency of the design process and reduces the need for mesh refinement. Second, by looking for the local orthogonal projection in the parameter space rather than the coordinate space, the algorithm simplifies the definition of boundary conditions. This makes the process of defining boundary conditions easier and more intuitive. Third, the algorithm approximates the contact area by mapping it from the parameter space to the physical space. This allows for the specification of the level of accuracy of the contact area, which can permit more precise solutions. The paper compares the IGA-based rigid body contact algorithm with the existing rigid FE mesh- based contact method in Simcenter Madymo. The results show that for the same level of accuracy, the IGA-based method can largely reduce execution time. This suggests that IGA could be a valuable alternative for use in the field of contact simulatio, Applied Mathematics

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

[Abstract:] The first formulation of topology optimization was proposed in the 1980s. Since then, many contributions have been presented with the purpose of improving its efficiency and expanding its field of application. The aim of this research is to develop a structural topology optimization algorithm considering minimum weight and stress constraints. Structural topology optimization with stress constraints has been previously formulated with several different approaches, mainly: local stress constraints, global stress constraints or block aggregation of stress constraints. In this research the overweight approach, an improvement of the so-called damage approach, is used. In this method, a virtual relative density (VRD) is defined as a function of the violation of the local stress constraints. VRD is increased as the stresses exceed the maximum allowable value, with the exception of the areas with the minimum value of the relative density, since full-void solutions are intended. The distribution of the material in the domain is modelled using two different approaches: a uniform relative density within each element of the mesh and a relative density defined by means of quadratic B-splines. For this reason, the structural analysis is performed by means of the finite element method (FEM) and the isogeometric analysis (IGA) respectively. The optimization is addressed by means of the sequential linear programming algorithm (SLP), which is driven by the information provided by a full first order sensitivity analysis extension of both FEM and IGA formulations. Finally, the overweight approach is tested by means of some two dimensional problems. The domain has been divided in an elevated number of elements to attain high spatial definition solutions. The results show that the overweight approach is a feasible alternative for the damage approach and the stress constraints aggregation techniques to solve the topology optimization problem. A comparison between both formulat

In this work, we study numerical solutions of a gradient-elastic Kirchhoff plate model on convex and concave geometries. For a convex plate, we first show the well-posedness of the model. Then, we split the sixth-order partial differential equation (PDE) into a system of three second-order PDEs. The solution of the resulting system coincides with that of the original PDE. This is verified with convergence studies performed by solving the sixth-order PDE directly (direct method) using isogeometric analysis (IGA) and the system of second-order PDEs (split method) using both IGA and C0 finite elements. Next, we study a concave pie-shaped plate, which has one re-entrant point. The well-posedness of the model on the concave domain is proved. Numerical solutions obtained using the split method differ significantly from that of the direct method. The split method may even lead to nonphysical solutions. We conclude that for gradient-elastic Kirchhoff plates with concave corners, it is necessary to use the direct method with IGA.

Implementation of an Explicit Algebraic Reynolds Stress Model in Isogeometric Analysis for seperating flows using a two-equation k-omega model. The model is verified for simple cases such as homogeneous isotropic decaying turbulence, and the turbulent backward facing step, Marine Technology | Hydromechanics

In order to model prominent failure modes experienced by multi-layered composites, a fine through-thickness discretisation is needed. If the structure also has large in-plane dimensions, the computational cost of the model becomes large. In light of this, we propose an adaptive isogeometric continuum shell element for the analysis of multi-layered structures. The key is a flexible and efficient method for controlling the continuity of the out-of-plane approximation, such that fine detail is only applied in areas of the structure where it is required. We demonstrate how so-called knot insertion can be utilised to automatise an adaptive refinement of the shell model at arbitrary interfaces, thereby making it possible to model multiple initiation and growth of delaminations. Furthermore, we also demonstrate that the higher-order continuity of the spline-based approximations allows for an accurate recovery of transverse stresses on the element level, even for doubly-curved laminates under general load. With this stress recovery method, critical areas of the simulated structures can be identified, and new refinements (cracks) can be introduced accordingly. In a concluding numerical example of a cantilever beam with two initial cracks, we demonstrate that the results obtained with the adaptive isogeometric shell element show good correlation with experimental data.

An isogeometric finite element method for incompressible fluid film equations is presented. The method can be applied to numerically model the behaviour of thin cellular membranes, such as lipid bilayers. The membranes are represented by infinitely thin closed surfaces. Both the surface parametrization and analysis are based on state-of-the-art polar spline spaces. These spaces are defined such that a C1 continuous genus 0 surface can be constructed. At the discrete setting, point-wise conservation of mass is attained, using the framework of discrete exterior calculus. Therefore, the polar spline spaces are called divergence conforming. Time discretization of the highly non-linear system is done via the fixed-point iterations. It is found that for certain non-uniformly curved domains, the iterations converge and time stepping can be performed. However, for surfaces that are closely resembling a perfect sphere, the iterations are not stable for any ∆t. The solution is parameter-dependent and this indicates a possible bug in the Matlab code., Animation of the basic domain: https://youtu.be/YBrJwtOAqZ8 Gitlab project, containing the Matlab code that was used for this research: https://gitlab.tudelft.nl/hmverhelst/lipid-bilayers, Applied Mathematics

The use of sequential time integration schemes becomes more and more the bottleneck within large-scale computations due to a stagnation of processor’s clock speeds. In this study, we combine the parallel-in-time Multigrid Reduction in Time method with a p-multigrid method to obtain a scalable solver specifically designed for Isogeometric Analysis. Numerical results obtained for two- and three-dimensional benchmark problems show the overall scalability of the proposed method on modern computer architectures and a significant improvement in terms of CPU timings compared to the use of standard spatial solvers., Numerical Analysis, Delft Institute of Applied Mathematics

Isogeometric Analysis (IgA) is an extension of the more well known Finite Element Method (FEM). It allows for more accurate descriptions of boundary value problems on irregular domains. However, many of the traditional iterative solution strategies that are known to work well in FEM do not show the same behavior in IgA, especially for increasing order of basis functions p. A method shown to have fast convergence in this situation is a p-multigrid method with a smoother based on a Block ILUT factorization. Most of the blocks of this factorization are efficiently calculated. The same holds for the smoothing steps. It is therefore our objective to make changes to the Block ILUT smoother. Inspiration is taken from methods where ILU factorizations are constructed using a fixed-point iteration. We combine these with the existing Block ILUT smoother. This ultimately leads to two new proposed methods we will call Block Fixed-point ILU and Block ParILUT. The existing methods as well as the newly suggested methods are tested and compared on computational costs of the factorization, the number of nonzero entries in this factorization and the number of multigrid iterations needed to reach convergence, if these factorizations are to be used as smoother. The benchmark used for these tests is a convection diffusion reaction (CDR) equation on a multipatch geometry with 4, 16 or 64 patches.

Significant research is ongoing in the world to meet the needs of social and environmental crisis by harnessing wind and solar energy at high altitudes. One such approach is the use of an inflatable High Altitude Aerial Platform (HAAP). In the presented work, such periodically supported beam-like structures are analyzed using various mathematical models primarily modeling them as an equivalent beam using one-dimensional theories. The Euler-Bernoulli Theory (EBT) has been widely used for high aspect ratio beams, whereas the First Order Shear Deformation Theory (FSDT), or the Timoshenko beam theory, considers transverse shear effects and hence is superior in modeling low aspect ratio beams. First, an Isogeometric Analysis (IGA) is conducted using both FSDT and EBT to predict thermal buckling of periodically supported composite beams. Isogeometric analysis overcomes the limitations of the Gibbs phenomenon at discontinuities for a periodically supported beam using a higher order textit{k}-refinement. Next, an Integral Equation Approach (IEA) is implemented using EBT to obtain natural frequencies and buckling loads of periodically supported non-prismatic beams. Ill-conditioning errors were alleviated using admissible orthogonal Chebychev polynomials to obtain higher modes. We also present the prediction of the onset of flutter instability for metal plate and inflatable wing shaped foam test articles analyzed using finite element analysis (FEA). FEA updating based on modal testing and by conducting a geometrically nonlinear analysis resulted in a good agreement against the experiment tests. Furthermore, a nonlinear co-rotational large displacement/rotation FEA including the effects of the pressure as a follower forces was implemented to predict deformations of an inflatable structures. The developed FEA based tool namely Structural Analysis of Inflatables using FEA (SAIF) was compared with the experiments and available literature. It is concluded that the validity of the

We propose Floating Isogeometric Analysis (FLIGA), which extends IGA to extreme deformation analysis. The method is based on a novel tensor-product construction of B-Splines for the update of the basis functions in one direction of the parametric space. With basis functions “floating” deformation-dependently in this direction, mesh distortion is overcome for problems in which extreme deformations occur predominantly along the associated (possibly curved) physical axis. In doing so, we preserve the numerical advantages of splines over many meshless basis functions, while avoiding remeshing. We employ material point integration for numerical quadrature, thus attributing a Lagrangian character to our technique. The paper introduces the method and reviews the fundamental properties of the FLIGA basis functions, including a numerical patch test. The performance of FLIGA is then numerically investigated on the benchmark of Newtonian and viscoelastic Taylor–Couette flow. Finally, we simulate a viscoelastic extrusion-based additive manufacturing process, which served as the original motivation for the new approach., Team Sid Kumar

Given an unstructured mesh consisting of quadrilaterals and triangles (we allow both planar and non-planar meshes of arbitrary topology), we present the construction of quadratic splines of mixed smoothness — C1 smooth away from the unstructured regions of T and C0 smooth otherwise. The splines have several useful B-spline-like properties – partition of unity, non-negativity, local support and linear independence – and allow for straightforward imposition of boundary conditions. We propose a non-nested refinement process for the splines with multiple advantages — a simple computer implementation, reduction in the footprint of C0 smoothness, boundary preservation, and excellent approximation behaviour in simulations. Furthermore, the refinement process leaves the splines invariant on the mesh boundary. Numerical tests indicate that the spline spaces demonstrate optimal approximation behaviour in the L2 and H1 norms under mesh refinement, and provide a viable approach to simulations on watertight reconstructions of trimmed surfaces., Numerical Analysis

The bottleneck in designing iterative solvers for the Helmholtz equation lies in balancing the trade-off between accuracy and scalability. Both the accuracy of the numerical solution and the number of iterations to reach convergence deteriorate in higher dimensions and increase with the wavenumber. To address these issues in this dissertation, we formulated three research pillars: accuracy, wavenumber independent convergence and linear complexity. Below, we summarize the core findings of this dissertation: WAVENUMBER INDEPENDENT CONVERGENCE We develop the first preconditioning technique which leads to close to wavenumber independent convergence for very large wavenumbers in 1D, 2D and 3D. Building on a two-level deflation projection method, we incorporated Quadratic Rational Bezier curves to construct the deflation space and vectors (Chapter 7). As a result, the near-zero eigenvalues of the coarse grid operator remain aligned with the fine-grid operator, keeping the spectrum of the preconditioned system clustered, leading to superior convergence properties compared to previous methods. LINEAR COMPLEXITY For over 30 years, applied mathematicians have tried to make convergent (standard) multigrid solvers for the Helmholtz equation. Multigrid solvers use sequences of smaller problem sizes and are computationally cheap and easy to implement. Unfortunately, multigrid methods diverge for Helmholtz and solving this issue remained an open problem. Using standard smoothing techniques, combined with similar higher-order coarse spaces, we constructed a fully convergent V- and W-cycle algorithm (Chapter 9). The key features of the algorithm are the use of higher-order transfer operators (instead of deflation vectors in the previous application) and a complex shift in the smoothing operator. While the method converges and the preliminary results have been proven, much research can still be conducted in this area, as this could support a paradigm shift, Numerical Analysis

We propose an adaptive mesh refinement strategy for immersed isogeometric analysis, with application to steady heat conduction and viscous flow problems. The proposed strategy is based on residual-based error estimation, which has been tailored to the immersed setting by the incorporation of appropriately scaled stabilization and boundary terms. Element-wise error indicators are elaborated for the Laplace and Stokes problems, and a THB-spline-based local mesh refinement strategy is proposed. The error estimation and adaptivity procedure are applied to a series of benchmark problems, demonstrating the suitability of the technique for a range of smooth and non-smooth problems. The adaptivity strategy is also integrated into a scan-based analysis workflow, capable of generating error-controlled results from scan data without the need for extensive user interactions or interventions.

Partial differential equations (PDE) on surfaces appear in a variety of applications, such as image processing, modeling of lubrication, fluid flows, diffusion, and transport of surfactants. In some applications, surfaces are drawn and modeled by using CAD software, giving a very precise patchwise parametric description of the surface. This thesis deals with the development of methods for finding numerical solutions to PDE posed on such parametrically described multipatch surfaces. The thesis consists of an introduction and five papers. In the first paper, we develop a general framework for the Laplace-Beltrami operator on a patchwise parametric surface. Each patch map induces a Riemannian metric, which we utilize to compute quantities in the simpler reference domain. We use the cut finite element method together with Nitsche’s method to enforce continuity over the interfaces between patches. In the second paper, we extend the framework to be able to handle geometries that consist of an arrangement of surfaces, i.e., more than two per interface. By using a Kirchhoff's condition this method avoids defining any co-normal to each surface and can deal with sharp edges. This approach is shown to be equivalent to standard Nitsche interface method for flat geometries. In the third paper, we developed a cut finite element method for elliptic problems with corner singularities. The main idea is to use an appropriate radial map that grades the finite element mesh towards the corner that counter-acts the solution's singularity. In the fourth paper, we present a new robust isogeometric method for surfaces described by CAD patches with gaps or overlaps. The main approach here is to cover all interfaces with a three-dimensional mesh and then use a hybrid variable in a Nitsche-type formulation to transfer data over the gaps. Using this hybridized approach leads to a convenient and easy to implement method with no restriction on the number of coupled patches per interface. In the

Partial differential equations (PDE) on surfaces appear in a variety of applications, such as image processing, modeling of lubrication, fluid flows, diffusion, and transport of surfactants. In some applications, surfaces are drawn and modeled by using CAD software, giving a very precise patchwise parametric description of the surface. This thesis deals with the development of methods for finding numerical solutions to PDE posed on such parametrically described multipatch surfaces. The thesis consists of an introduction and five papers. In the first paper, we develop a general framework for the Laplace-Beltrami operator on a patchwise parametric surface. Each patch map induces a Riemannian metric, which we utilize to compute quantities in the simpler reference domain. We use the cut finite element method together with Nitsche’s method to enforce continuity over the interfaces between patches. In the second paper, we extend the framework to be able to handle geometries that consist of an arrangement of surfaces, i.e., more than two per interface. By using a Kirchhoff's condition this method avoids defining any co-normal to each surface and can deal with sharp edges. This approach is shown to be equivalent to standard Nitsche interface method for flat geometries. In the third paper, we developed a cut finite element method for elliptic problems with corner singularities. The main idea is to use an appropriate radial map that grades the finite element mesh towards the corner that counter-acts the solution's singularity. In the fourth paper, we present a new robust isogeometric method for surfaces described by CAD patches with gaps or overlaps. The main approach here is to cover all interfaces with a three-dimensional mesh and then use a hybrid variable in a Nitsche-type formulation to transfer data over the gaps. Using this hybridized approach leads to a convenient and easy to implement method with no restriction on the number of coupled patches per interface. In the

Significant research is ongoing in the world to meet the needs of social and environmental crisis by harnessing wind and solar energy at high altitudes. One such approach is the use of an inflatable High Altitude Aerial Platform (HAAP). In the presented work, such periodically supported beam-like structures are analyzed using various mathematical models primarily modeling them as an equivalent beam using one-dimensional theories. The Euler-Bernoulli Theory (EBT) has been widely used for high aspect ratio beams, whereas the First Order Shear Deformation Theory (FSDT), or the Timoshenko beam theory, considers transverse shear effects and hence is superior in modeling low aspect ratio beams. First, an Isogeometric Analysis (IGA) is conducted using both FSDT and EBT to predict thermal buckling of periodically supported composite beams. Isogeometric analysis overcomes the limitations of the Gibbs phenomenon at discontinuities for a periodically supported beam using a higher order textit{k}-refinement. Next, an Integral Equation Approach (IEA) is implemented using EBT to obtain natural frequencies and buckling loads of periodically supported non-prismatic beams. Ill-conditioning errors were alleviated using admissible orthogonal Chebychev polynomials to obtain higher modes. We also present the prediction of the onset of flutter instability for metal plate and inflatable wing shaped foam test articles analyzed using finite element analysis (FEA). FEA updating based on modal testing and by conducting a geometrically nonlinear analysis resulted in a good agreement against the experiment tests. Furthermore, a nonlinear co-rotational large displacement/rotation FEA including the effects of the pressure as a follower forces was implemented to predict deformations of an inflatable structures. The developed FEA based tool namely Structural Analysis of Inflatables using FEA (SAIF) was compared with the experiments and available literature. It is concluded that the validity of the

This study constructs a linear buckling analysis procedure for planar curved beams by adopting a total Lagrangian Timoshenko-Ehrenfest beam formulation. The present approach is based on isogeometric analysis (IGA). Because of the ability to accurately describe complex geometry, non-uniform rational B-spline (NURBS) curves are employed in the IGA context. The system of equations from the adopted nonlinear formulation is linearized for linear buckling analysis. The geometric stiffness matrix in this study considers the influence of membrane, bending, and shear actions. The B-bar method is also adopted to ensure the proposed formulation is locking-free and applicable to curved beam structures with arbitrary geometry. The outputs of the proposed formulation are compared to the existing exact or numerical solutions.

This study constructs a linear buckling analysis procedure for planar curved beams by adopting a total Lagrangian Timoshenko-Ehrenfest beam formulation. The present approach is based on isogeometric analysis (IGA). Because of the ability to accurately describe complex geometry, non-uniform rational B-spline (NURBS) curves are employed in the IGA context. The system of equations from the adopted nonlinear formulation is linearized for linear buckling analysis. The geometric stiffness matrix in this study considers the influence of membrane, bending, and shear actions. The B-bar method is also adopted to ensure the proposed formulation is locking-free and applicable to curved beam structures with arbitrary geometry. The outputs of the proposed formulation are compared to the existing exact or numerical solutions.

In this work, we study numerical solutions of a gradient-elastic Kirchhoff plate model on convex and concave geometries. For a convex plate, we first show the well-posedness of the model. Then, we split the sixth-order partial differential equation (PDE) into a system of three second-order PDEs. The solution of the resulting system coincides with that of the original PDE. This is verified with convergence studies performed by solving the sixth-order PDE directly (direct method) using isogeometric analysis (IGA) and the system of second-order PDEs (split method) using both IGA and C0 finite elements. Next, we study a concave pie-shaped plate, which has one re-entrant point. The well-posedness of the model on the concave domain is proved. Numerical solutions obtained using the split method differ significantly from that of the direct method. The split method may even lead to nonphysical solutions. We conclude that for gradient-elastic Kirchhoff plates with concave corners, it is necessary to use the direct method with IGA.

In this paper, a deep learning framework combined with isogeometric analysis (IGA for short) called IGA-Reuse-Net is proposed for efficient reuse of numerical simulation on a set of topology-consistent models. Compared with previous data-driven numerical simulation methods only for simple computational domains, our method can predict high-accuracy PDE solutions over topology-consistent geometries with complex boundaries. UNet3+ architecture with interlaced sparse self-attention (ISSA) module is used to enhance the performance of the network. In addition, we propose a new loss function that combines a coefficients loss and a numerical solution loss. Several training datasets with topology-consistent models are constructed for the proposed framework. To verify the effectiveness of our approach, two different types of Poisson equations with different source functions are solved on three datasets with different topologies. Our framework can achieve a good trade-off between accuracy and efficiency. It outperforms the physics-informed neural network (PINN for short) model and yields promising results of prediction., Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public., Materials and Manufacturing

Easy to construct and optimally convergent generalisations of B-splines to unstructured meshes are essential for the application of isogeometric analysis to domains with non-trivial topologies. Nonetheless, especially for hexahedral meshes, the construction of smooth and optimally convergent isogeometric analysis basis functions is still an open question. We introduce a simple partition of unity construction that yields smooth blended B-splines, referred to as SB-splines, on semi-structured quadrilateral and hexahedral meshes, i.e. on mostly structured meshes with sufficiently separated unstructured regions. To this end, we first define the mixed smoothness B-splines that are C0 continuous in the unstructured regions of the mesh but have higher smoothness everywhere else. Subsequently, the SB-splines are obtained by smoothly blending in the physical space the mixed smoothness B-splines with Bernstein bases of equal degree. One of the key novelties of our approach is that the required smooth weight functions are assembled from the available smooth B-splines on the unstructured mesh. The SB-splines are globally smooth, non-negative, have no breakpoints within the elements and reduce to conventional B-splines away from the unstructured regions of the mesh. Although we consider only quadratic mixed smoothness B-splines in this paper, the construction generalises to arbitrary degrees. We demonstrate the excellent performance of SB-splines studying Poisson and biharmonic problems on semi-structured quadrilateral and hexahedral meshes, and numerically establishing their optimal convergence in one and two dimensions., Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public., Numerical Analysis