Application of isogeometric analysis in structural shape optimization

Iterative solution of Helmholtz problem with high-order isogeometric analysis and finite element method at mid-range frequencies

Isogeometric size optimization of bi-directional functionally graded beams under static loads

Phase-field fracture analysis for implicit geometric model via adaptive extended isogeometric analysis

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

Compact dc high-voltage photoelectron guns are able to meet the sophisticated demands of high-current applications such as energy recovery linacs. A main design parameter for such sources is the electric field strength, which depends on the electrode geometry and is limited by the field emission threshold of the electrode material. In order to minimize the maximum field strength for optimal gun operation, isogeometric analysis (IGA) can be used to exploit the axisymmetric geometry and describe its cross section by nonuniform rational B-splines, the control points of which are the parameters to be optimized. This computationally efficient method is capable of describing CAD-generated geometries using open source software (geopdes, nlopt, octave) and it can simplify the step from design to simulation. We will present the mathematical formulation, the software workflow, and the results of an IGA-based shape optimization for a planned high-voltage upgrade of the dc photogun teststand Photo-CATCH at TU Darmstadt. The software builds on a general framework for isogeometric analysis and allows for easy adaptations to other geometries or quantities of interest. Simulations assuming a bias voltage of $\ensuremath{-}300\text{ }\text{ }\mathrm{kV}$ yielded maximum field gradients of $9.06\text{ }\mathrm{MV}\text{ }{\mathrm{m}}^{\ensuremath{-}1}$ on the surface of an inverted insulator electrode and below $3\text{ }\mathrm{MV}\text{ }{\mathrm{m}}^{\ensuremath{-}1}$ on the surface of the photocathode.

Isogeometric analysis of functionally graded plates using a new quasi-3D shear deformation theory based on physical neutral surface

Dynamic analysis of two-dimensional elasticity problems using the Isogeometric analysis method and its comparison with the finite element method is the subject of this research. For this purpose, formulation of the governing differential equation is obtained by using the B-spline basis functions. Some numerical examples employing consistent and lumped mass matrices are presented in order to compare the results of Isogeometric analysis and finite elements. The effect of increasing the number of control points and the degree of B-spline basis functions on the accuracy of the solution are also investigated. Furthermore, since the Newmark method is employed for transient dynamic analysis, the effects of parameters α and β on the solution accuracy are investigated. The obtained results demonstrate higher accuracy of the isogeometric analysis method in comparison with the FEM, despite employing fewer numbers of degrees of freedom and consequently reduction in the dimensions of system of equations and computational effort.

Shape optimization directly from CAD: An isogeometric boundary element approach using T-splines

We introduce a manufacturing constraint for controlling the minimum member size in structural shape optimization problems, which is for example of interest for components fabricated in a molding process. In a parameter-free approach, whereby the coordinates of the FE boundary nodes are used as design variables, the challenging task is to find a generally valid definition for the thickness of non-parametric geometries in terms of their boundary nodes. Therefore we use the medial axis, which is the union of all points with at least two closest points on the boundary of the domain. Since the effort for the exact computation of the medial axis of geometries given by their FE discretization highly increases with the number of surface elements we use the distance function instead to approximate the medial axis by a cloud of points. The approximation is demonstrated on three 2D examples. Moreover, the formulation of a minimum thickness constraint is applied to a sensitivity-based shape optimization problem of one 2D and one 3D model.

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.

Isogeometric Analysis (IGA) has emerged as a powerful computational method that seamlessly integrates computer-aided design (CAD) and finite element analysis (FEA). It provides an innovative approach for solving complex engineering problems by employing the same basis functions used in CAD representations, such as Non-Uniform Rational B-Splines (NURBS), to discretize the computational domain. However, one of the challenges in IGA is the occurrence of crosspoints, which are regions where multiple NURBS patches meet. This presents a novel development of a crosspoint modification technique for Isogeometric Analysis, leveraging the capabilities of Artificial Intelligence (AI) algorithms. The aim is to overcome the limitations posed by crosspoints and enhance the accuracy and efficiency of IGA simulations. It utilizes AI-based methods, such as deep learning and pattern recognition, to identify and modify the crosspoint regions in the NURBS-based representation. The AI model is trained on a large dataset of known crosspoint configurations and their corresponding modifications, enabling it to learn the underlying patterns and relationships between the geometry and the required modifications. This trained model is then employed to automatically identify and modify crosspoints in new IGA problems, streamlining the pre-processing stage and reducing human intervention. The crosspoint modification technique incorporates various strategies to ensure geometric continuity and smooth transition across the modified crosspoint regions. These strategies include local refinement of the NURBS control points, adaptively adjusting the degrees of the adjacent patches, and optimizing the blending functions in the crosspoint vicinity. Several numerical experiments and comparative studies are conducted. The results demonstrate that the crosspoint modification using AI significantly improves the accuracy of the IGA simulations, particularly in regions affected by crosspoints. Moreover, the computational efficiency is enhanced as the automated modification reduces the manual effort required for crosspoint handling.
 This research contributes to the advancement of Isogeometric Analysis by introducing a novel AI-based crosspoint modification technique. By effectively addressing the challenges associated with crosspoints, the proposed method enhances the accuracy and efficiency of IGA simulations. The integration of AI algorithms with IGA holds great potential for future applications in various engineering disciplines, including structural analysis, fluid dynamics, and electromagnetics.

In structural optimization, static loads are generally utilized although real external forces are dynamic. Dynamic loads have been considered only in small-scale problems. Recently, an algorithm for dynamic response optimization using transformation of dynamic loads into equivalent static loads has been proposed. The transformation is conducted to match the displacement fields from dynamic and static analyses. This algorithm can be applied to large-scale problems. However, the application has been limited to size optimization. The present study applies the algorithm to shape optimization. Because the number of degrees of freedom of finite element models is usually very large in shape optimization, it is difficult to conduct dynamic response optimization with conventional methods that directly treat dynamic response in the time domain. The optimization process is carried out by interfacing an optimization system and an analysis system for structural dynamics. Various examples are solved to verify the algorithm. The results are compared to the results from static loads. It is found that the algorithm using static loads transformed from dynamic loads based on displacement is valid for very large-scale shape optimization problems. Copyright © 2005 John Wiley & Sons, Ltd.

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.

Isogeometric analysis of free-surface flow

Stern–Gerlach magnets are used to magnetically separate a beam of atoms or atom clusters. The design is difficult, since both the magnetic field gradient and its homogeneity should be maximized. This paper proposes the numerical optimization of the device’s pole-shoe shapes, starting from a reference geometry given in the literature. The main contributions of this paper are the generalization of a magnetic field–magnetic circuit coupling and the application of isogeometric analysis (IGA), which are shown to reduce the computational complexity and increase the accuracy. The field–circuit coupling significantly reduces the size of the field model part, in which the IGA’s spline-based framework enables a highly accurate evaluation of local field quantities, even across elements.