Multi-patch Isogeometric Analysis Research Articles

Multipatch IGA for stress-constrained minimum weight structural topology optimization

In this article, an isogeometric analysis (IGA) formulation with various patches (multipatch approach) is proposed to solve the structural topology optimization (STO) problem. The multipatch formulation enables the solution of STO problems with geometries not equivalent to rectangles by using IGA. For this purpose, a minimum weight with stress constraints (MWSC) problem is stated. A quadratic B-splines basis defined for each patch is used to model the material distribution. The structural analysis is computed with IGA. The introduction of all of the stress constraints in the formulation of the problem is made with an overweight constraint (OC). Finally, the multipatch approach is validated with the solution of some benchmark problems.

Multi-patch isogeometric analysis for smart plates with distributed piezoelectric patches

Multi-patch isogeometric analysis for smart plates with distributed piezoelectric patches

Stress-based topological shape optimization for thick shells using the level set method and trimmed non-conforming multi-patch isogeometric analysis

Stress-based topological shape optimization for thick shells using the level set method and trimmed non-conforming multi-patch isogeometric analysis

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.

Mixed-mode thermo-mechanical fracture: An adaptive multi-patch isogeometric phase-field cohesive zone model

This work presents an adaptive phase-field cohesive zone model (PF-CZM) for simulating mixed-mode crack nucleation and growth in isotropic rock-like materials subjected to thermo-mechanical interactions. The proposed approach combines an adaptive multi-patch isogeometric analysis (MP-IGA) and length-scale insensitive PF-CZM. The formulation captures the distinct critical energy release rates for Mode-I and Mode-II fractures, which is crucial for predicting mixed-mode thermo-mechanical fracture behavior in isotropic rock-like materials. The PF-CZM governing equations are solved with isogeometric analysis based on locally refined non-uniform rational B-splines (LR NURBS), and the complex structural geometry is exactly described with multiple LR NURBS patches. The field variables, such as displacement, phase-field, and temperature at the interface of adjacent patches, are coupled using Nitsche’s method. To enhance the computational efficiency while maintaining accuracy, a refinement-correction adaptive scheme combined with the structured mesh refinement strategy is developed. The proposed framework is validated against recent numerical and experimental results in the literature, particularly in the context of capturing complex behavior of mixed-mode crack propagation in isotropic rock-like materials subjected to thermo-mechanical loading.

Topology optimization of curved thick shells using level set method and non-conforming multi-patch isogeometric analysis

We present a novel framework for topological shape optimization of curved non-conforming multi-patch and trimmed thick-shells subjected to external loads. Our method integrates the level set method (LSM) with a diffuse interface, a Hadamard shape derivative, and multi-patch isogeometric analysis (IGA) into a gradient descent algorithm to systematically capture the evolution of the shape. This integration enables us to directly manipulate CAD-compatible geometries and analysis techniques and to obtain the results as a CAD surface. The novelty lies in the utilization of multi-patch IGA models based on NURBS functions, which allows us to simultaneously maximize the stiffness and minimize the volume of the shell by searching for an optimal material distribution within its middle surface. The material is modeled under a small strain assumption in linear elasticity using a Reissner–Mindlin kinematic shell model in plane stress. The effectiveness of our approach is demonstrated on several curved conforming and non-conforming multi-patch geometries in 3D.

Adaptive multi-patch isogeometric analysis with truncated hierarchical B-splines in isotropic/orthotropic media

We present an adaptive multi-patch isogeometric analysis on the basis of truncated hierarchical B-splines (THB-splines) for solving two-dimensional complex isotropic/orthotropic elasticity. The THB-splines have local refinement property and their basis functions exhibit linear independence, so these features are highly applicable for adaptive isogeometric analysis. Guided by a posterior error estimator based on stress recovery, the adaptive algorithm is utilized in isogeometric analysis. In order to further extend the proposed method to solve complex geometry problems, the multi-patch technique is adopted to achieve exact modeling with Nitsche’s method as a multi-patch coupling approach. An isotropic numerical example with exact analytical solutions and three orthotropic numerical examples are presented to verify the effectiveness and accuracy of the developed method. Numerical solutions show that the developed adaptive isogeometric analysis method has high computational efficiency.

A [formula omitted] continuous multi-patch framework for adaptive isogeometric topology optimization of plate structures

This study proposes a novel computationally efficient methodology to perform topology optimization (TO) of fourth-order plate structures within the framework of multi-patch isogeometric analysis. This is realized by taking the multifold benefits of isogeometric PHT-Splines to (1) discretize the C1 continuous weak form of plate structures, (2) develop a C0 continuous density field for the material distribution in TO and inherently remove the need for filters, and (3) provide a hierarchical tree structure for the structural mesh to effortlessly implement an adaptive mesh refinement (AMR) strategy. Moreover, to ensure continuity between isogeometric patches, we adopt a strong C1 coupling between the boundaries. This is established by constructing new basis functions, defined as a linear combination of existing C0 functions at the patch interfaces. The density field in TO is further enhanced with a first-neighbourhood smoothening algorithm based on the Shepard function to generate printable topologies and alleviate the post-processing stages after optimization. An element-centre density, based on the control point densities of the isogeometric mesh, is used as the marking scheme for the AMR to determine the subdomains to be refined. Utilizing the Geometry Independent Field approximaTion, the design and adaptive analysis-optimization stages were independently discretized respectively through NURBS and PHT-Splines, allowing easy transfer of multi-patch geometries from industry-standard packages. Multiple numerical examples illustrate the stability of the multi-patch algorithm in optimizing the geometries effectively. The results also show considerable advantages in terms of solution accuracy such as precise field, smooth topology and computational efficiency.

An adaptive multi-patch isogeometric phase-field model for fatigue fracture

Fatigue fracture is one of the main causes of structural failure under cyclic loading, so accurate evaluation of fatigue fracture is very important for the design of structures subjected to cyclic loading. In this work, we propose an adaptive fatigue phase-field model within the framework of multi-patch isogeometric analysis to simulate crack nucleation and propagation in brittle materials under cyclic loading. Multiple non-uniform rational B-splines (NURBS) patches are used to exactly represent complex structures, and the Nitsche’s method is adopted to enforce the compatibility of the displacements, stresses and phase-field variables between two patches. The coupled governing equations of fatigue phase-field model are established based on the Nitsche’s method, and the selection of Nitsche’s parameters are determined from numerical experiments. A refinement-correction adaptive approach based on locally refined non-uniform rational B-splines (LR NURBS) structured mesh refinement strategy is developed to alleviate the computational burden caused by the requirement of small element size around crack surface and the large number of fatigue loading. The proposed multi-patch isogeometric phase-field model can deal with the coupling between equal/unequal length interfaces, thus the fatigue crack propagation simulation in arbitrary complex structure can be implemented. Several numerical examples are investigated to verify the practicality of the multi-patch isogeometric phase-field model and the effectiveness of the adaptive scheme.

Geometrically nonlinear analysis of Reissner–Mindlin plates using multi-patch isogeometric analysis based on Nitsche’s method

Within the isogeometric analysis framework, industrial products or complex shapes are represented using multiple NURBS patches, resulting in non-matching interfaces and introducing additional numerical challenges, particularly in scenarios involving nonlinear behavior. This paper introduces the application of Nitsche’s method to address interface coupling challenges presented in non-matching multi-patch configurations. A detailed formulation addressing geometric non-linearity in multiple Reissner–Mindlin plates is developed, and the resulting nonlinear equations are solved using the Newton–Raphson approach. The proposed formulation’s effectiveness is demonstrated by a series of numerical examples involving complex geometries represented by multi-patches with non-matching interfaces. These examples are validated against the analytical solutions and results obtained using the commercial finite element package, Abaqus.

Isogeometric Tearing and Interconnecting solvers for linear elasticity in multi-patch Isogeometric Analysis with theory for two dimensional domains

We consider the linear elasticity equation, discretized with multi-patch Isogeometric Analysis. A standard discretization error analysis is based on Korn’s inequality, which degrades for certain geometries, such as long and thin cantilevers. This phenomenon is known as geometry locking. We observe that high-order methods, like Isogeometric Analysis is beneficial in such a setting. The main focus of this paper is the construction and analysis of a domain decomposition solver, namely an Isogeometric Tearing and Interconnecting (IETI) solver, where we prove that the convergence behavior does not depend on the constant of Korn’s inequality for the overall domain, but only on the corresponding constants for the individual patches. Moreover, our analysis is explicit in the choice of the spline degree. Numerical experiments are provided which demonstrates the efficiency of the proposed solver.

Multi-patch isogeometric material optimization of bi-directional functionally graded plates

This study investigates an effective design methodology for the optimal material distribution of bi-directional functionally graded plates (2D-FGPs) with complex shape. In mechanical analysis, a multi-patch isogeometric method is used to analyze the statics of 2D-FGP, which is based on the third-order shear deformation theory and Nitsche’s technology. This method can effectively avoid the use of duplicate nodes in the parameterization of IGA for complex shapes to obtain C1-continuity of 2D-FGPs. In the optimal design problem, we have constructed a rectangular material design mesh (RMDM) based on the shape of 2D-FGPs, which can map the material distribution on the surface of plate to achieve the implementation of the optimization process. This two-dimensional B-spline control points of RMDM are set as the design variable with mass reduction and the first-order natural frequency maximization as optimization objectives, and limited arbitrary deflection as constraint conditions. In addition, an improved multi-objective particle swarm optimization algorithm (IMOPSO) is used to obtain a series of Pareto optimization solutions that meet the needs of the designer. The validity and applicability of this innovative combination of multi-patch isogeometric analysis and IMOPSO are demonstrated through several numerical examples in integrated design. This approach further accomplishes the numerical unified CAD/CAE optimization design of 2D-FGPs across multiple non-smooth boundaries.

Multipatch isogeometric analysis for geometrically exact shell elements using B-bar method and Bézier extraction

In the present study, a multipatch isogeometric analysis method for geometrically exact shell problems is proposed. Generalized curvilinear coordinates are used for the first-order shear-deformable geometrically exact shell elements. This general tensor-based shell formulation can be directly linked to a computer-aided design system such as nonuniform rational basis spline (NURBS), which is represented by two parameters. Nitsche’s method for patch-to-patch connections in conforming or nonconforming meshes is adopted. The compatibility conditions between NURBS patches in Nitsche’s coupling are weakly imposed using the continuity of displacements, rotations, and stress resultants. The final algebraic equation for the solution fields has a simple symmetric form with previously derived strain–displacement relation matrices and material modulus for membrane, bending, and transverse shear terms. To alleviate the locking phenomenon, the B-bar method, which utilizes the strain projected onto a lower-order field, is used in the isogeometric shell element. Reduced-order B-splines for strain fields are constructed based on tying points, which eased the locking phenomenon compared with the classical B-bar method. To further alleviate additional shell locking by a higher-order regularity, the Bézier extraction method is adopted to reduce the element regularity. The proposed combination of Nitsche’s coupling and B-bar projection demonstrates the superior accuracy and robustness in the multipatch isogeometric shell examples with a high convergence rate.

IsoGeometric Analysis with non-conforming multi-patches for the hull structural mechanical analysis

IsoGeometric Analysis (IGA) has been applied in hull structural analysis and proved to be efficient and accurate. Nevertheless, the advantages of IGA have not been fully utilized. In this paper, IGA technics for hull structural analysis are further developed. A 3 Degrees-Of-Freedoms C1-continuous Kirchhoff–Love shell element is used to analyze hull structures, and the weighted non-symmetric Nitsche’s method is used to couple connected patches. With the proposed shell elements and coupling method, the modeling of hull structures for IGA is greatly simplified. A typical hull block structure with complex interfaces is used to verify the correctness and accuracy of non-conforming multi-patch IGA. The commercial software ANSYS 17.0 is used to obtain reliable reference solutions. An extreme non-conforming IGA model with 231 non-conforming interfaces is adopted to test the robustness of IGA. When using the weighted non-symmetric Nitsche’s method, the maximum relative error of displacements is 1.7%, and the maximum relative error of von Mises stresses is 1.0%, which is acceptable in engineering. The correctness and robustness of non-conforming multi-patch IGA for the hull structural analysis are verified. The engineering practicability of IGA is proved.

Multi-patch isogeometric topology optimization for cellular structures with flexible designs using Nitsche’s method

Cellular structures have recently been received a tremendous growth in discussions and applications in the engineering due to their several fascinating structural properties, such as the ultra-lightweight, high stiffness and crashworthiness. However, the discussions on the design of cellular structures in the complex structural domain with the non-conforming mesh are still unavailable, resulting from the fact that the non-conforming mesh causes several difficulties in numerical analysis and design optimization. Hence, the main purpose of the work is to propose a Multi-Patch Isogeometric Topology Optimization (MP-ITO) method with powerful capabilities for periodic or graded cellular structures. Firstly, the Nitsche’s method is applied to couple non-conforming meshes in multiple NURBS patches and conduct multi-patch isogeometric analysis. Secondly, a multi-patch topology description model is developed, in which a local smoothing mechanism and the two-resolution scheme of discretization meshes are constructed to avoid terrible structural features and improve smoothness and continuity of boundaries at the interfaces within adjacent subdomains. The separation and independency of the Density Distribution Function (DDF) at each subdomain can offer high flexibility for cellular designs with the imposing of several kinds of periodic constraints. Thirdly, the MP-ITO method is proposed for complex structures and the mathematical formulation for cellular designs considering periodic constraints is developed. Finally, the effectiveness and indispensability of the local smoothing mechanism and the two-resolution scheme in the MP-ITO are discussed, and several numerical examples are addressed to present the compelling effectiveness and capabilities of the MP-ITO method with the high flexibility on the designs of periodic and graded cellular structures.

Stable discretizations and IETI-DP solvers for the Stokes system in multi-patch IgA

We are interested in a fast solver for the Stokes equations, discretized with multi-patch Isogeometric Analysis. In the last years, several inf-sup stable discretizations for the Stokes problem have been proposed, often the analysis was restricted to single-patch domains. We focus on one of the simplest approaches, the isogeometric Taylor–Hood element. We show how stability results for single-patch domains can be carried over to multi-patch domains. While this is possible, the stability strongly depends on the shape of the geometry. We construct a Dual-Primal Isogeometric Tearing and Interconnecting (IETI-DP) solver that does not suffer from that effect. We give a convergence analysis and provide numerical tests.

Practical unstructured splines: Algorithms, multi-patch spline spaces, and some applications to numerical analysis

In this work, we show how some recent advances on simplex spline spaces can be used to construct a polynomial-reproducing space of unstructured splines on multi-patch domains of arbitrary shape and topology. The traces of these functions on the subdomain boundaries reproduce the usual traces of standard polynomial bases used in discontinuous Galerkin (DG) approximations, allowing to borrow many theoretical and practical tools from these methods. Concurrently, we recast some theoretical results on the construction and evaluation of spaces of simplex splines into an explicit, algorithmic form. Together, these efforts allow to formulate a practical, efficient and fully unstructured multi-patch discontinuous Galerkin - isogeometric analysis (DG-IGA) scheme that bridges the gap between some current multi-patch isogeometric analysis (IGA) approaches and the more traditional mesh-based interior penalty discontinuous Galerkin (IPDG) method. We briefly discuss the advantages of this unified framework for time-explicit hyperbolic problems, and we present some interesting numerical examples using the acoustic wave equation.

Influence of aperture shape on dynamic behaviors of sandwich plates by means of improved isogeometric analysis and experimental tests

Due to assembly and lightweight design, various apertures are involved in composite and sandwich plates, which will influence the natural frequencies and vibration modes of these structures. If the influence mechanism of aperture shapes on dynamic characteristics of composite and sandwich plates is not clarified, structural safety will be significantly impacted as external loads are close to the natural frequencies of composite plates with apertures. Nevertheless, experimental tests in conjunction with accurate analysis on the dynamic response of composite and sandwich plates with different aperture shapes are less reported in published literature, so the influence mechanism of aperture shapes on dynamic characteristics of composite and sandwich plates is required to be clarified precisely. Thus, a novel refined sinusoidal higher-order theory (RSHT) combined with isogeometric analysis (IGA) is proposed, where the improved bending strip method (IBSM) has been reconstructed to satisfy approximately the C1-continuity in multi-patch IGA. Four sets of tests on free vibration of rectangular sandwich plates with different sizes of circular cutouts are carried out to verify the reliability of the proposed method. Numerical results and experimental tests show that the proposed RSHT can predict more accurately the influence of aperture shapes on dynamic behaviors of sandwich plates, and the influence mechanism of aperture shapes on dynamic characteristics of sandwich plates has been illustrated, which can provide a reference for other investigators.

IETI-DP methods for discontinuous Galerkin multi-patch Isogeometric Analysis with T-junctions

We study Dual–Primal Isogeometric Tearing and Interconnecting (IETI-DP) solvers for non-conforming multi-patch discretizations of a generalized Poisson problem. We realize the coupling between the patches using a symmetric interior penalty discontinuous Galerkin (SIPG) approach. Previously, we have assumed that the interfaces between patches always consist of whole edges. In this paper, we drop this requirement and allow T-junctions. This extension is vital for the consideration of sliding interfaces, for example between the rotor and the stator of an electrical motor. One critical part for the handling of T-junctions in IETI-DP solvers is the choice of the primal degrees of freedom. We propose to add all basis functions that are non-zero at any of the vertices to the primal space. Since there are several such basis functions at any T-junction, we call this concept “fat vertices”. For this choice, we show a condition number bound that coincides with the bound for the conforming case.

Initial solution estimation of one-step inverse isogeometric analysis for sheet metal forming with complex topologies

The one-step inverse isogeometric analysis method has been successfully applied in sheet metal forming with simple geometries. Generally, actual stamping parts frequently contain numerous trimmed NURBS-based CAD surfaces. The key step in sheet metal forming is to unfold the undevelopable CAD model onto a planar domain and obtain a good initial solution. In this study, we estimate the initial solution of stamping parts with complex topologies using an energy-based algorithm. The trimmed surface analysis technique and Nitsche’s method are adopted for trimmed and multi-patch isogeometric analysis. A new coordinate transformation system is used to avoid special treatment for negative angle and vertical wall problems, which are common in metal sheet forming. We demonstrate our algorithm with three examples and compare the results with one-step inverse isogeometric analysis of simple geometry or the traditional finite element method. These examples illustrate the performance of the new method and its applicability for the integration of design and analysis in sheet metal forming with complex topologies.