XIGA: An eXtended IsoGeometric analysis approach for multi-material problems

Earth System Models utilized to predict the ocean physical and biogeochemical responses under anthropogenic climate change do not yet routinely resolve mesoscale eddies due to computational resource constraints, and ocean mesoscale eddies are often parameterized. Mesoscale eddy parameterizations are known to affect the large-scale ocean circulation through impacts on the stratification, leading to for example changes of the nutrient stream, which is known to have large-scale impacts for the nutrient supply. Here we examine numerical ocean models with different representations of the mesoscale eddies and their combined physical and biogeochemical response in a hierarchy of models differing in the horizontal resolution, ranging from a non-eddying model with eddy parameterization, eddy permitting models without and with eddy parameterization, and eddy resolving models that serves as the model truth. In the case of non-eddying models, we find that existing prescriptions can result in a "better" bulk biogeochemical response but for the wrong physical reasons, relative to the model truth. On the other hand, more recent parameterization schemes can improve the physical response in terms of the sensitivity to changes in forcing, but the biogeochemical response is more subtle. It is confirmed here that the biggest change in the biophysical response stems from a model becoming eddy permitting. However, the eddy permitting model without parameterization is found to overshoot and be "too good" relative to the model truth in the biogeochemical response, attributed to the explicit eddies being too weak. Results are presented from a new approach that supplements the action of explicit eddies in eddy permitting model via eddy parameterization, but without adversely damping the explicit eddies. Combined with more recent parameterizaton schemes, the approach can lead to a biogeochemical response that compares very favorably to the model truth, attributed entirely to the improvement in the physical response in the eddy permitting model, so that we are getting the right answer for the right reasons.

The isogeometric analysis (IGA), which employs non-uniform rational B-splines (NURBS) basis as shape functions for both representation of geometry and approximation of the field variables, owns several inherent advantages to become an effective numerical method such as an exact geometry description with fewer control points, high-order continuity, and high accuracy. Unlike the C0-continuity shape functions in the conventional finite element method (FEM), the high-order basis functions in IGA are not confined to one element, but span on several elements instead. This property makes the programming task difficult, and more importantly they cannot be straightforwardly embedded into the existing FEM framework. In this paper, we provide an effective numerical scheme by further extending the IGA based on the Bezier extraction of NURBS to study mechanical behavior of elasto-plastic problems in three-dimension (3D). The Bezier extraction operator decomposes NURBS functions into a set of Bernstein polynomials and provides C0-continuity Bezier elements for the IGA, which are similar to Lagrange elements structure. Consequently, the implementation of IGA is now similar to that of conventional FEM, and can easily be embedded in most existing FEM codes. The merits of the proposed formulation are also demonstrated by its convergence and validation studies against reference solutions considering both simple and complicated configurations.

Boccherini's Body: An Essay in Carnal Musicology. By Elisabeth Le Guin. Berkeley: University of California Press, 2006. [374 p. ISBN 0-520-240170. $39.95.] Illustrations, music examples, compact disc. Instrumental music of the second half of the eighteenth century is, by nature, an abstract art form. Barring a specific reference by a composer linking his or her work to a text, image, or idea, the analyst has traditionally turned to an objective approach in studying a work. One reason Jan LaRue's Guidelines for Stylistic Analysis (New York: W. W. Norton, 1970) has worked so well for so many is that it provides a common and straightforward vocabulary which can be used to discuss a composition. Small-, medium-, and large-scale levels with regard to texture, harmony, melody, surface rhythm, and the like can be distinguished, and comparisons with other compositions by the same composer as well as others are imminently possible. In her Boccherini's Body: An Essay in Carnal Musicology, Elisabeth Le Guin offers a different methodology. Drawing from cultural ideas of the eighteenth century, and incorporating theories and concepts from the fields of art, dance, and theater, the author proposes an approach which relies heavily on the subjective and physical nature of performance, one which she refers to as a performance- and body-oriented musicology. Boccherini's Body is a personal statement of the author's approach to Boccherini, informed as much by the physicality of the act of as the intellectual thought of the day. To complement the book, Le Guin has included a compact disc, filled with beautifully performed selections, some entire movements, some just snippets, so that the reader can hear all the written music examples. Le Guin's basic premise, as outlined in her introduction, is that one can attach meaning, in the form of a physical response, to an instrumental composition by applying images, movements, and texts from art, dance, and theater, as well as infusing these with theories, either from the eighteenth century or the present day. The problem, of course, is in convincing one's reader of the viability of this approach and in making those connections work. From the very beginning, Le Guin assert[s] the centrality of performance (p. 2) and in fact, one finds that nearly every discussion relies on the physicality of performance. The author argues that the concept of kinesthesia, or fundamental feeling, is essential when examining Boccherini's works. Physical comfort and discomfort become primary means of explaining passages and even entire movements. Individual embodied experiences take precedence over communal music making or considering what can communicate to the listener? By the end of the introduction, one wonders whether the intent of the book is a musicological study or a discussion of informed by historical and theoretical information. Chapter 1 offers us a glimpse as to how Le Guin approaches a Boccherini sonata, in this case, the Cello Sonata in E-flat Major, Fuori catalogo. She argues that by fully identifying with Boccherini, the man, she can render a better of the work. Le Guin provides a lengthy discussion of the physical aspects of rendering the piece: comfort/discomfort, movement of arms and fingers, use of muscles, weight, gravity, and other aspects of cello performance. Ideas and motives are related to the body; physical sensations are equated with topos and affect. Through this approach, the author claims a form of communication with Boccherini himself: I become aware of a poignance of presence, the unmistakable sensation of someone hereand not only here, but inhabiting my body (p. 25). Here, as elsewhere in the book, Le Guin argues that her own physical and emotional responses allow her to understand Boccherini himself and his intent. She fails to address the notion that each of us will have varied physical responses to this particular sonata, none being more accurate or real than another. …

ABSTRACTThis paper aims to optimize both the topology of microstructure and macrostructure, by using the Moving Iso‐ Surface Threshold (MIST) method under the framework of Isogeometric Analysis (IGA). To achieve this, the physical response and the equivalent properties of the microstructure are solved by IGA. The physical response functions are defined at the control points. The NURBS fitting method is used to fit the physical response function, generating a NURBS surface for describing the physical response function, that is, the explicitly described physical response function surface. An iso‐surface cut the NURBS physical response surfaces to obtain the explicitly described structure topology. In addition, the standard IGS files of the NURBS physical response surface and iso‐surface can be transferred to Computer‐Aided‐Design (CAD) software without post‐processing. The multiscale structure can be easily assembled in CAD software using the proposed method. Finally, several numerical examples are performed to demonstrate the effectiveness and efficiency of the proposed method. Obtained results show good agreement with examples in literature in terms of both topology and final value of objective function.

Isogeometric analysis (IGA) has been a fundamental step forward in the computational mechanics for the past few years, which maintains the accuracy of the description of computational domain geometry throughout the analysis process. However, the research on IGA in the area of flexible multibody dynamics is little and mainly concentrates on the univariate or bivariate NURBS geometry. This paper applies the trivariate IGA to the flexible multibody dynamics and proposes a continuum mechanics-based method to construct the system dynamic equations within the framework of IGA. A significant feature of this method is that it only employs the position coordinates of the control points as the system variables. To solve the large rotation and deformation coupled problems without introducing any rotation angles, the Green-Lagrange strain tensor is adopted. The evaluation of the elastic force and its Jacobian is easy and accurate by exploiting the appropriate mathematical transformation. In addition, the mass matrix and the generalized gravity force remain constant, and the centrifugal and Coriolis inertia forces equal zero. A numerical experiment is conducted using a thin-plate pendulum, which proves the feasibility and effectiveness of this method.

Isogeometric analysis (IGA) is a computational technique that integrates computer-aided design (CAD) with finite element analysis (FEA) by employing the same basis functions for both geometry representation and solution approximation. While IGA offers numerous advantages, such as improved accuracy and efficiency, it also presents several challenges related to geometric modeling. Some of these challenges include accurately representing complex geometries with NURBS (Non-Uniform Rational B-Splines) or other basis functions used in IGA and generating high-quality meshes that conform to the complex geometry represented by NURBS curves/surfaces. This paper introduces an analytical framework to provide a more efficient and theoretically grounded method for generating curvilinear configurations and its analytical solution in IGA, bridging the gap between generated data and its physical representations. This innovative approach is distinguished by integrating the NURBS parameterization in curve generation and providing a corresponding framework to achieve a broader and more accurate explanation of meshes and properties, especially constructing new coordinates and calculating the physical displacements under these conditions. Our model enables the analytical understanding of complex curves from the UIUC airfoil and superformula datasets, demonstrating a deeper dive into simulations. This study signifies a pivotal juncture wherein machine-learning-based complex geometrical formulations are synergistically combined with actual isogeometric analysis.

Computational cardiac models are progressively being used to understand, predict, and improve the treatment of cardiac diseases. These models commonly rely on the traditional finite element analysis (FEA), where the geometry description and consequent mesh generation are separate preprocessing steps that are required before conducting numerical analyses. The recent isogeometric analysis (IGA) paradigm eliminates the separate meshing step and integrates geometry construction and solution approximation using higher-order splines. In this study, we first investigate whether IGA can be efficiently used to model post-infarction left ventricular mechanics. Mechanics results from an established FEA model with a fine homogeneous mesh are used to investigate to what extent similar results could be obtained using hierarchical mesh refinement in IGA. The IGA-mechanics results show a good agreement while providing a mesh-independent geometry, but deviations are noticed close to the base, apex, and partially the endocardium. Second, both the FEA- and IGA-model results are used as input for an FEA-mechanics-driven electrophysiology model which is used for a ventricular tachycardia (VT)-inducibility study, as abnormal mechanics is believed to be a potential driving factor for the development of VT. The resulting VT propagation patterns agree visually for both mechanical inputs and differences in VT-exit points are within 1–7 mm for simulations where VT occurred. Furthermore, an agreement of 85% in the binary VT results is observed, where the 15% difference displays the electrophysiology-model sensitivity to deviations in mechanical input.

Analysis in computer aided design: Nonlinear isogeometric B-Rep analysis of shell structures

Isogeometric analysis (IGA) has recently become popular in computational science during the past decade or so. IGA tries to to unify both geometric and field representation; in other words, both the geometry and the fields are represented using the same underlying basis set. However, while the concept of IGA for differential equations is more common, extension to an integral equation framework is significantly more challenging. In this work, we present for the first time, the IGA as applied to integral equations encountered in electromagnetics. The presented approach relies on the subdivision scheme for both geometry and function representation. Results presented attest to the viability of the method.

The classic finite difference method (FDM) has been successfully adopted in the simulation of dendritic solidification, which is based on phase-field theory. Nevertheless, special strategies of boundary integral and projection are required for applying a supercooling rate to a droplet surface. In the present study, isogeometric analysis (IGA) is employed to discretize the phase-field equation due to the two advantages of Non-Uniform Rational B-Splines (NURBS) basis functions, namely an arbitrary order of derivatives and exact description of complex geometry. In addition, an improved, easy way to apply the supercooling rate on a melt droplet surface is proposed to avoid the integral and projection of the cellular boundary required in FDM. Firstly, dendrite growth in a square computational domain is simulated to verify the performance of IGA. Then, the influences of latent heat, anisotropic mode and initial angle on the dendrite shapes are studied by the presented IGA, FDM and finite element method (FEM). Finally, dendritic solidification in a droplet under different cooling rates along irregular boundaries is performed by the proposed IGA.

The isogeometric analysis (IGA) that integrates Computer Aided Design(CAD) and Computer Aided Engineering (CAE) is found so far the effectively numerical tool for the analysis of a variety of practical problems. In this paper, we develop further the NURBS based isogeometric analysis framework for piezoelectric structures.The method employs the NURBS basis functions in both geometry representation and analysis. The main advantages of the present method are capable of handling the exactgeometry of conic sections and making the flexibility of refinement and degree elevationwith an arbitrary continuity of basic functions. These features results in high accuracy of approximate solutions for practical applications, especially piezoelectric problems. Three numerical examples are provided to validate excellent performance of the present method.

The past few years have witnessed rapid advances in the integration of Computer-Aided Design (CAD) and Finite Element Analysis (FEA), such as the Isogeometric Analysis (IGA) method and the Integration of Computer Aided Design and Analysis (I-CAD-A) technique. Recent studies indicated that the geometry description used in the Absolute Nodal Coordinate Formulation (ANCF) is consistent with the geometry description in CAD systems, making it an attractive approach for I-CAD-A. ANCF as an accurate, non-incremental finite element method to deal with the dynamics of those systems subject to large motions and large deformations has developed towards practical application in the field of flexible multibody systems. The objective of this study is to enrich the family of ANCF elements with three new triangular shell elements: the thin triangular element, the low order thick triangular element and the high order thick triangular element, which are represented by Bezier triangles. The linear mapping relationship between the geometrical representation of the Bezier triangle and that of the proposed thin triangular element is established. The invertible property of the linear mapping relationship clearly indicates that the proposed thin triangular element can be also used to perform the I-CAD-A. The conditions of the approximate $G^{1}$ continuity between the proposed adjacent elements are also deduced in order to construct a smooth surface. Finally, several static and dynamic numerical examples are employed to validate the correctness and effectiveness of the proposed triangular shell elements.

We present a new isogeometric analysis (IGA) approach based on extended Loop subdivision scheme for solving various geometric flows defined on subdivision surfaces. The studied flows include the second‐order, fourth‐order, and sixth‐order geometric flows, such as averaged mean curvature flow, constant mean curvature flow, and minimal mean‐curvature‐variation flow, which are generally derived by minimizing the associate energy functionals with ‐gradient flow respectively. The geometric flows are discretized by means of subdivision based IGA, where the finite element space is formulated by the limit form of the extended Loop subdivision for different initial control meshes. The basis functions, consisting of quartic box‐splines corresponding to each subdivided control mesh, are utilized to represent the geometry exactly. For the cases of the evolution of open surfaces with any shape boundary, high‐order continuous boundary conditions derived from the mixed variational forms of the geometric flows should be implemented to be consistent with the isogeometric concept. For time discretization, we adopt an adaptive semi‐implicit Euler scheme. By several numerical experiments, we study the convergence behaviors of the proposed approach for solving the geometric flows with high‐order boundary conditions. Moreover, the numerical results also show the accuracy and efficiency of the proposed method.

This study proposes a comprehensive optimization framework that integrates single- and multi-objective algorithms for solving complex problems in structural mechanics. A genetic algorithm (GA) and non-dominated sorting genetic algorithm II (NSGA-II) are developed and validated through multiple mechanical case studies, including truss structures, beam vibration and fluid-structure interactions. An isogeometric analysis (IGA) approach is incorporated to improve the accuracy of boundary representation. The numerical results demonstrate the robustness and precision of the proposed methods compared to existing literature, providing an efficient and reliable optimization strategy for practical engineering applications.

A framework for conducting heat transfer analysis of nonuniformly heated plates using isogeometric analysis (IGA) is proposed. The modeling approach represents a method that is capable of capturing the full 3D thermal states of plate structures subjected to nonuniform heating. The formulation can be deemed a mixed IGA and control-volume scheme that describes the governing equations using a set of 2D lumped temperature layers. The non-matching grids between the fire domain and the solid domain are weakly coupled. Numerical results demonstrate that the approach is a suitable first step towards coupled CFD-IGA-based fire-structure simulation.