Isogeometric Research Articles

A simultaneous shape and topology optimization approach of shell structures based on isogeometric analysis and density distribution field

This paper presents a novel simultaneous shape and topology optimization approach of shell structures based on isogeometric analysis and density distribution field. In the optimization approach, Non-Uniform Rational B-Splines (NURBS) technology is utilized to describe the geometry and material distribution of the shell structures. The coordinates and densities of the NURBS control points are utilized as design variables to simultaneously optimize the shape and topology of shell structures. The proposed approach offers significant advantages, including ease of implementation, seamless integration with CAD models, high efficiency, and smooth, clear boundaries. Two representative examples are performed to demonstrate the effectiveness of the proposed approach. The optimized configurations are compared with other works and commercial software results.

A symmetric interior-penalty discontinuous Galerkin isogeometric analysis spatial discretization of the self-adjoint angular flux form of the neutron transport equation

This paper presents the first application of a symmetric interior-penalty discontinuous Galerkin isogeometric analysis (SIP-DG-IGA) spatial discretization to the self-adjoint angular flux (SAAF) form of the multi-group neutron transport equation. The penalty parameters are determined, for general element types, from a mathematically rigorous coercivity analysis of the bilinear form. The proposed scheme produces a compact spatial discretization stencil. It also yields symmetric positive-definite (SPD) matrices, which can be efficiently solved using pre-conditioned conjugate gradient (PCG) solution algorithms. The proposed discretization scheme is verified using the method of manufactured solutions (MMS) and several nuclear reactor physics benchmark verification test cases. For sufficiently smooth elliptic problems, the proposed spatial discretization can exploit higher-order continuity, or k-refinement, of the NURBS basis to consistently yield greater numerical accuracy per degree of freedom (DoF) than standard h-refinement. Since this is a discontinuous scheme, it can also accurately model significant changes in the neutron scalar flux that may occur near the material interfaces of heterogeneous problems.

Adaptive refinement in incompressible fluid flow simulation based on THB-splines-powered isogeometric analysis

In this paper, we deal with adaptive refinement in numerical simulation of incompressible flow solved by isogeometric analysis. We study various error estimators and compare them with respect to convergence to the exact solution. Further, we propose a new class of error estimators based on stabilization methods for numerical solving of incompressible flow and we show that they provide viable option to standard error estimators. Moreover, we comment on different choices of marking strategies and their suitability to the case of incompressible flow and provide comparison of error estimators also with respect to selected marking strategies and selected representative pairs of discretization spaces in isogeometric analysis.

From CAD to representations suitable for isogeometric analysis: a complete pipeline

AbstractNext generation CAD/CAE systems shall integrate design and analysis procedures as a means to simplify workflows, boost performance, and speed up time to market. In the present paper we demonstrate a proof of concept of such a unification using subdivision surfaces and the enabling technology of isogeometric analysis. In particular, we present a complete pipeline to convert CAD models into smooth $$G^1$$ G 1 spline representations, which are suitable for isogeometric analysis. Starting from a CAD boundary representation of a mechanical object, we perform an automatic control cage extraction by means of quadrangular faces, such that its limit Catmull-Clark subdivision surface approximates accurately the input model. We then compute a basis of the $$G^1$$ G 1 spline space over the quad mesh in order to carry out least squares fitting over a point cloud, acquired by sampling the original CAD geometry. The resulting surface is a collection of Bézier patches with $$G^1$$ G 1 regularity. Finally, we use the basis functions to perform isogeometric analysis simulations of realistic PDEs on the reconstructed $$G^1$$ G 1 model. The quality of the construction is demonstrated via several numerical examples performed on a collection of CAD objects presenting various challenging realistic shapes.

A comprehensive study on porosity modelling and its impact on fracture behavior of edge cracked FG structures using XIGA

In the present work, the fracture analysis of functionally graded (FG) porous structure containing an edge cracked is carried out in the presence of different types of porosity distributions using extended isogeometric analysis (XIGA). Firstly, the different types of porosity distribution functions are mathematically modeled in the porous FG structure across the length of the domain. The effective properties of the porous FG structure are computed using power law. Also, an additional term of porosity is incorporated in the power law to include the effect of porosity in the FG structure. The effective properties are computed across the length of the structure in the presence of different types of porosity distributions. Further, a pre-existing crack is modeled in the domain to study its influence on the fracture behaviour of porous FG structure using XIGA. To validate the accuracy, the results for the non-porous FG structure are compared with the available results in the literature (with the analytical and numerical solution), and they are found in good agreement (percentage error in the range of 0.04–––1.78%). Moreover, the comparative study is performed to investigate the influence of different types of porosity distributions on the fracture behaviour of FG structure.

Combined parameter and shape optimization of electric machines with isogeometric analysis

Combined parameter and shape optimization of electric machines with isogeometric analysis

A scaled boundary shell formulation in isogeometric analysis for static and dynamic analysis

AbstractIn modern applications of computer‐aided design (CAD) for the analysis of shell structures, isogeometric analysis is a powerful tool that integrates both design and analysis. An exact geometry description and a straightforward computation without loss of information are advantageous, especially for shell structures such as roofs, satellite hulls, or car bodies. In addition, the scaled boundary method provides a scale separation with a semi‐analytical solution procedure to consider a three‐dimensional linear elastic constitutive law. The presented approach deals with a scaled boundary solid shell formulation in the framework of isogeometric analysis. The formulation utilizes a normal scaling strategy which scales the shell along its normal vector at each point on the discretized bottom surface. This is fundamentally different from the well‐known radial scaling strategy, where each point on the problem domain is obtained from a fixed scaling center. This results in a separation of the analysis into an in‐plane direction and a scaling (normal) direction. By introducing the scaling, a scaled boundary differential equation is derived that is dependent on the scaling parameter only. Choosing a proper set of conditions, the differential equation can be solved by a Padé expansion. While the in‐plane direction is solved in a weak sense, the thickness direction along the normal vector can be solved analytically resulting in a semi‐analytical procedure. The isogeometric description of the CAD structure inherently yields the exact normal vector, its derivatives and a higher order continuity throughout the structure. Herein, the focus is on the solution technique in the thickness direction and the challenges are addressed. The power of the formulation is outlined in several numerical examples of static and dynamic analysis and a comparison to shell formulations in literature is provided.

Non-linear bending analysis and control of graphene-platelets-reinforced porous sandwich plates with piezoelectric layer subjected to electromechanical loading

Piezoelectric materials as the controlling element have been widely utilized to produce intelligent engineering structures, while these smart structures may fail to realize effective control of composite structures with large deformations. However, investigations on such issues are less reported in published literature, as an accurate and efficient model is required to well forecast the geometrically nonlinear behaviors of smart sandwich structures. As a result, a novel sinusoidal Legendre global-local higher-order shear deformation plate theory (SLHSDT) has been developed to accurately capture geometrically nonlinear behaviors of piezoelectric sandwich plates. The proposed model can fulfill the compatible conditions of transverse shear stresses and contain transverse normal strain, which can ensure precision in predicting electromechanical behaviors. The multi-patch isogeometric analysis (IGA) method for sandwich plates partially bonded with piezoelectric layers is proposed to overcome C1-continuity between patches for the first time. Moreover, the Newmark-β method and Newton-Raphson technique are attempted to solve the nonlinear equations. The present model has been utilized to investigate electromechanical behaviors of laminated structures with piezoelectric layers, which has been compared with the published results. In addition, experiments on macro fiber composite (MFC) integrated sandwich plates have been also carried out in the present work, which can effectively verify the performance of proposed model. Subsequently, the proposed model is employed to study electromechanical behaviors of the five-layer piezoelectric sandwich plates containing internal pores and graphene platelets. Then, influences of the porosity coefficient and GPLs weight fraction on the nonlinear electromechanical behaviors of sandwich plates are investigated. Eventually, the active control on nonlinear behaviors of piezoelectric porous sandwich plates with GPLs reinforcement is studied by using a closed-loop control system, and an effective approach slowing down large deformation has been proposed by selecting an appropriate distribution of GPLs along the thickness direction.

An efficient isogeometric buckling optimization framework for multi-patch laminated shells: Leveraging 3D solid elements, lamination parameters and penalty method

Buckling is a common failure mode of laminated shell structures during service, and isogeometric analysis methods are highly suitable for the buckling analysis due to their accurate geometric modeling capability and high-order continuity. However, the isogeometric buckling analysis for a multi-patch structure and its optimal design still face challenges in computational efficiency. To address this issue, this paper proposes an integrated framework for the buckling optimization of multi-patch laminated shells, which leverages 3D solid elements, lamination parameters, and the penalty coupling method. Solid-shell elements have advantages in shell analysis due to their simplicity, absence of rotational degrees of freedom, and adaptive layering in the thickness direction. Lamination parameters simplify the optimization of fiber orientations in laminated shells by condensing the design variable space and improving the convexity of the optimization problem. The penalty method can handle the coupling in multiple patches at a relatively low cost. Nevertheless, lamination parameters are rarely applied to the case of solid-shell elements. In addition, despite using lamination parameters, the number of design variables for multi-patch structures remains significantly large. To tackle these issues, this study derives the solid-shell formulation in lamination parameters and the sensitivity analysis formulae in the reverse mode. Experiment validation is conducted to assess the accuracy of the proposed method based on lamination parameters with solid elements, the feasibility of the multi-patch optimization model, and the computational efficiency of optimization sensitivity based on the reverse mode compared to the forward counterpart common in the literature.

Multiscale topology optimization of anisotropic multilayer periodic structures based on the isogeometric analysis method

Multiscale topology optimization of anisotropic multilayer periodic structures based on the isogeometric analysis method

Development of a robust and integrable pre-processing tool for isogeometric analysis

Development of a robust and integrable pre-processing tool for isogeometric analysis

Isogeometric topology optimization for maximizing band gap of two-dimensional phononic crystal structures

A smooth and efficient isogeometric topology optimization (ITO) method for maximizing band gaps in two-dimensional phononic crystals is developed in the paper. The band gaps of phononic crystals are computed by the NURBS-based isogeometric analysis, and the material distributions of two-dimensional phononic crystals are represented by the smooth, high-order continuity of the NURBS surface. The densities defined at each control point of the NURBS surface are employed as optimized design variables. The isogeometric analysis optimization using the same spline technique for both geometry and numerical analysis can benefit the optimization procedure and obtain smooth optimized structures, which can be easily used in 3D printing. The maximizing band gaps of phononic crystals for both out-of-plane and in-plane wave modes are obtained here using the present ITO approach. Numerical examples validated the effectiveness and reliability of the ITO method in broadening the band gaps and finding the optimal phononic crystals. And the numerical results show that the smooth optimized structures are obtained with fewer iterations.

Isotropic growth model for generalized scaled boundary isogeometric analysis on slender structures

AbstractComponents for timber constructions are usually based on simple geometrical shapes. In the production, bifurcated or curved components of the tree crown remain unused. Recently, building botany has evolved as a new discipline of living construction design. Therefore, trees are lead in the right shape to join as a load‐bearing structure interconnected with conventional parts. For the structural analysis, geometric features such as curvature or bifurcations exhibit unique static capabilities. It is crucial to transfer these geometries into the calculation model accurately. The structure can be depicted through three‐dimensional scanning techniques. A fundamental model (point cloud) is created in this process. This is the base for a Non‐uniform rational B‐spline (NURBS) model, existing as a 2D surface within a 3D space. Using the novel generalized scaled boundary isogeometric analysis (GSBIGA)‐approach, numerical disadvantages of conventional scaled boundary methods can be reduced. This method employs a scaling center line instead of a scaling center point. A growth simulation is used to describe the leading process. The present approach is based on an isotropic growth model applied to wooden growth applications. Therefore, the deformation gradient is split multiplicatively into an elastic and a growth part. The formulation for modeling volumetric growth is located on the kinematic level.

A CO-HSDT isogeometric analysis for free vibration of matrix cracked FG-GPLRC plates coupled with stationary fluid

This paper for the first time proposes an efficiently C0-higer order shear deformation theory (HSDT) and isogeometric analysis (IGA) for the free vibration analysis of matrix cracked functionally graphene nanoplatelets reinforced composite (FG-GPLRC) plate coupled with stationary fluid. This case represents essential components of sophisticated structures utilized in industries such as shipbuilding, nuclear, marine, and naval. The properties of the four GPLs distributions of FG-GPLRC are evaluated by using a combination of the modified Halpin-Tsai micromechanics model and the rule of mixtures, while the degraded stiffness of cracked layers is predicted by the self-consistent micromechanical (SCM) model. The fluid is assumed to be homogeneous, inviscid, incompressible and irrotational, so the free-surface waves and hydrostatic pressure effects on structures are neglected. The governing equation of motion is derived by the Hamilton's principle, where three different fluid-plate interaction systems are taken into consideration. After validating the proposed method against existing literature, the effect of various parameters such as crack density, interaction boundary conditions (IBC), fluid level, GPLs distribution pattern, total number of layers, and geometric parameter on the free vibration of matrix cracked FG-GPLRC plate are investigated. It is believed that the finding in this paper may be helpful for the accurate design and analysis of matrix cracked FG-GPLRC plate submerged in fluid.

Stochastic Dynamic Buckling Analysis of Cylindrical Shell Structures Based on Isogeometric Analysis

In this paper, we extend our previous work on the dynamic buckling analysis of isogeometric shell structures to the stochastic situation where an isogeometric deterministic dynamic buckling analysis method is combined with spectral-based stochastic modeling of geometric imperfections. To be specific, a modified Generalized-α time integration scheme combined with a nonlinear isogeometric Kirchhoff–Love shell element is used to simulate the buckling and post-buckling problems of cylindrical shell structures. Additionally, geometric imperfections are constructed based on NURBS surface fitting, which can be naturally incorporated into the isogeometric analysis framework due to its seamless CAD/CAE integration feature. For stochastic analysis, the method of separation is adopted to model the stochastic geometric imperfections of cylindrical shells based on a set of measurements. We tested the accuracy and convergence properties of the proposed method with a cylindrical shell example, where measured geometric imperfections were incorporated. The ABAQUS reference solutions are also presented to demonstrate the superiority of the inherited smooth and high-order continuous properties of the isogeometric approach. For stochastic dynamic buckling analysis, we evaluated the buckling load variability and reliability functions of the cylindrical shell with 500 samples generated based on seven nominally identical shells reported in the geometric imperfection data bank. It is noted that the buckling load variability in the cylindrical shell obtained with static nonlinear analysis is also presented to show the differences between dynamic and static buckling analysis.

Parallel isogeometric boundary element analysis with T-splines on CUDA

We present a framework for parallel isogeometric boundary element analysis (BEA) of elastic solids on CUDA. To deal with traction discontinuities, we propose a BEA model that supports multiple nodes and semi-discontinuous elements. The multiplicity of a node is defined by the number of regions containing any element influenced by the node. A region is a group of connected elements delimited by a closed crease curve. The default shape function of an element is determined by a linear operator applied to a set of basis functions. A BEA model is supposed to be generated from a watertight boundary representation of a solid. In this paper, we employ bicubic analysis-suitable T-splines. In this case, the shape of an element is defined by its Bézier extraction operator applied to the tensor product of Bernstein polynomials of degree 3. We describe the data structures of the BEA model and the main algorithms of the analysis pipeline on CUDA. In particular, we describe two strategies for parallel assembling of the linear system of equations. We also introduce a novel approach for inside integration based on the subdivision of the singular region in triangles with constant aspect ratio. In the T-splines context, we extend the Bézier extraction to handle unstructured T-meshes with crease edges. Moreover, we propose a scheme for embedding the influence of linked tangency handles on the shape of an element directly into the Bézier extraction operator. Such an embedding enables the removal of the corresponding nodes from the BEA model and the application of an alternative collocation method we discuss in the paper. We present several experiments to evaluate the accuracy and efficiency of the proposed framework. The results demonstrate that the GPU can be advantageously employed for parallelizing T-spline-based isogeometric analysis using boundary elements, achieving a speedup of up to 29x compared to the sequential code on a current laptop. We make the BEA code available in a prototype in MATLAB with a graphical interface that allows users to apply boundary conditions and visualize analysis results on the boundary.

Electric–Mechanical coupling analysis of two-dimensional piezoelectric heterogeneous materials in flexible electric devices with extended multiscale isogeometric analysis

Piezoelectric heterogeneous materials are widely used in flexible electronic device design, enhancing sensitivity to external stimuli like pressure and acceleration. Despite their usefulness, analyzing these inherently periodic structures poses significant computational challenges. In response, this paper presents a multiscale isogeometric analysis approach tailored for simulating piezoelectric materials. We introduce an electric–mechanical coupling model using isogeometric analysis (IGA) for two-dimensional piezoelectric membrane structures, assuming the plane stress hypothesis. Our proposed algorithm enables precise calculation of both displacement and electric potential solutions, demonstrating superior convergence properties compared to traditional finite element methods. Furthermore, we extend this approach to multiscale isogeometric analysis for computing numerical solutions in porous structures and heterogeneous composite piezoelectric materials under tensile and bending conditions. Through rigorous numerical testing, we evaluate the proposed extended multiscale isogeometric analysis method, showcasing its efficacy in achieving a balance between computational efficiency and simulation accuracy. This IGA-based electro-mechanical coupling model and numerical algorithm pave the way for more streamlined and precise simulations of piezoelectric materials within the context of flexible electronic devices.

First Failure Load of Rectangular Laminated and Sandwich Plates Using Isogeometric Analysis

This paper investigates the first failure load of simply supported and clamped laminate and sandwich plates loaded by a distributed normal traction on the top surface. A third-order shear and normal deformable plate theory, the isogeometric basis functions, and five failure criteria, namely, the maximum stress, the Tsai–Wu, the Tsai–Hill, the Hoffman, and the Hashin for laminated plates, and only the Tsai–Hill criterion for sandwich plates are used in this study. Of these, Hashin’s criteria distinguish between the fiber and the matrix failure. The in-plane stresses are found from the plate displacements and Hooke’s law, and the transverse stresses are recovered by integrating the equilibrium equations through the thickness. Effects of the plate aspect ratio, the fiber angle, the face sheet materials, and the core materials on the first failure load are identified. The computed results are found to agree well with those reported in the literature. Whereas the five failure criteria for laminated plates predict nearly the same value of the first failure load, they do not provide the same location of the failure initiation.

Isogeometric Analysis of Laminated and Carbon Nanotube Reinforced Composite Stiffened Plate Using Higher Order Shear Deformation Theory

Isogeometric Analysis of Laminated and Carbon Nanotube Reinforced Composite Stiffened Plate Using Higher Order Shear Deformation Theory

Free-Form Deformation as a non-invasive, discrete unfitted domain method: Application to the time-harmonic acoustic response of a saxophone

The Finite Element method, widely used for solving Partial Differential Equations, may result in suboptimal computational costs when computing smooth fields within complex geometries. In such situations, IsoGeometric Analysis often offers improved per degree-of-freedom accuracy but building analysis-suitable representation of complex shapes is generally not obvious. This paper introduces a non-invasive, spline-based fictitious domain method using Free-Form Deformation to efficiently solve the Helmholtz equation in complex domains, such as in musical instruments. By immersing a fine FE mesh into a simple B-spline box, the approximation subspace size is significantly reduced without compromising accuracy. Accompanied by specific conditioning treatment, the method not only proves to be efficient, but also robust and easy to implement in existing FE software. Applied to an alto saxophone, the method reduces the number of degrees of freedom by over two orders of magnitude and the computation time by more than one compared to standard FE methods with comparable accuracy when compared to experimental tests.