Isogeometric Analysis Formulation Research Articles

Isogeometric topology optimization of structures using the overweight approach

In this paper, a 2D isogeometric formulation of the material distribution for structural topology optimization considering minimum weight and local stress constraints using the overweight approach is proposed. The aim of this isogeometric formulation is to provide solutions with high spatial definition using a lower number of design variables in comparison with the formulations previously developed to define the material layout. Despite of this, an important number of local stress constraints has to be considered in the solution of the problem. For this purpose, an Overweight Constraint is used to consider all of them. The structural analysis is performed by means of the Isogeometric Analysis (IGA) and the distribution of material is modeled by means of quadratic B-splines. Moreover, the optimization is addressed by means of the Sequential Linear Programming algorithm (SLP) that is driven by the information provided by a full first-order sensitivity analysis extension of the IGA formulation. Finally, the proposed formulation is tested by means of some benchmark problems, and the results show that the isogeometric formulation provides solutions with high spatial definition. A comparison with a Finite Element Method (FEM) topology optimization formulation is included.

IGA-based topology optimization in the design of stress-constrained compliant mechanisms

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.

The isogeometric collocated contact surface approach

We propose a frictionless contact formulation for isogeometric analysis, which combines a collocated formulation for the contact surfaces with a standard Galerkin treatment of the bulk. We denote it as isogeometric Collocated Contact Surface (CCS) formulation. The approach is based on a simple pointwise enforcement of the contact constraints, performed in this study with the penalty method. Unlike pointwise (node-to-surface or point-to-surface) contact algorithms in the Galerkin framework, the CCS formulation passes the contact patch test to machine precision by naturally exploiting the favorable properties of isogeometric collocation. Compared with approaches where the discretization of both bulk and contact surfaces is based on collocation, the CCS approach does not need enhancements to remove oscillations for highly non-uniform meshes. With respect to integral contact approaches, the CCS algorithm is less computationally expensive, due to the reduced amount of contact evaluation points. In addition, the CCS approach is easy to code and can be added to a pre-existing isogeometric analysis code with minimal effort. Numerical examples in both small and large deformations are investigated to compare the CCS approach with some available contact formulations and to demonstrate its accuracy.

Spline-based space-time finite element approach for fluid-structure interaction problems with a focus on fully enclosed domains

Non-Uniform Rational B-Spline (NURBS) surfaces are commonly used within Computer-Aided Design (CAD) tools to represent geometric objects. When using isogeometric analysis (IGA), it is possible to use such NURBS geometries for numerical analysis directly. Analyzing fluid flows, however, requires complex three-dimensional geometries to represent flow domains. Defining a parametrization of such volumetric domains using NURBS can be challenging and is still an ongoing topic in the IGA community.With the recently developed NURBS-enhanced finite element method (NEFEM), the favorable geometric characteristics of NURBS are used within a standard finite element method. This is achieved by enhancing the elements touching the boundary by using the NURBS geometry itself. In the current work, a new variation of NEFEM is introduced, which is suitable for three-dimensional space-time finite element formulations. The proposed method makes use of a new mapping which results in a non-Cartesian formulation suitable for fluid-structure interaction (FSI).This is demonstrated by combining the method with an IGA formulation in a strongly-coupled partitioned framework for solving FSI problems. The framework yields a fully spline-based representation of the fluid-structure interface through a single NURBS.The coupling conditions at the fluid-structure interface are enforced through a Robin-Neumann type coupling scheme. This scheme is particularly useful when considering incompressible fluids in fully Dirichlet-bounded and curved problems, as it satisfies the incompressibility constraint on the fluid for each step within the coupling procedure.The accuracy and performance of the introduced spline-based space-time finite element approach and its use within the proposed coupled FSI framework are demonstrated using a series of two- and three-dimensional benchmark problems.

An isogeometric scaled boundary plate formulation for the analysis of ionic electroactive paper

In recent years, electroactive paper emerged as a new alternative in the field of smart actuators. It is based on a cellulose material which is able to bend under the influence of an external electric field similarly as ionic polymer metal composites. The bending mechanism is mainly attributed to the migration of ionic charges over the thickness of a thin sheet of paper. The present contribution proposes a numerical framework for the simulation of electroactive paper. It is based on a scaled boundary plate formulation for isogeometric analysis. In contrast to the standard scaled boundary plate approach, the scaling direction is solved numerically by a B-Spline approximation. This allows to render nonlinear effects over the plate thickness as well as displacement fields of higher continuity. The model is applicable to very thin structures such as electroactive paper, and it also captures the nonlinear ionic charge distribution which is coupled to the bending mechanism of the actuator.

Panel flutter analysis of perforated plate repaired by VSCL bonded patch using the multi-patch IGA approach

The crashed structure fixing procedures using the bonded repair patches seems highly frequent in the aeronautical vehicle’s maintenance. The resulting composite repaired perforated thin-walled plate is generally a complex geometry with a relatively high computational cost. The present study focuses on the linear panel flutter analysis of the perforated flat panels repaired on one side by an external bonded tow-steered variable stiffness composite patch. A variable stiffness laminate repair design can increase the composite structural performance. It also provides high flexibility toward tailoring the structure’s mechanical properties. The potent multi-patch NURBS-based isogeometric analysis formulation is implemented based on the first-order shear deformation theory of plates. In the context of the multi-patch geometry modeling approach, the repaired perforated panel is divided into several separate but connected adjacent geometries. Assuming sufficiently thin repair laminates, an aerodynamic loading according to the first-order piston theory is applied. Consequently, the performance and accuracy of the implemented formulation are approved. The panel flutter behavior of various repaired perforated plates is inspected through some parametric studies on geometry, layup, boundary conditions, and flow direction. It is found that the critical flutter pressure parameter is decreased for wider cutouts with the changes dependent on the hole layout. It is found that the optimal position of the VSCL repair and cutout zone depends strongly on the type of base plate boundary conditions.

Panel flutter analysis of cracked functionally graded plates in yawed supersonic flow with thermal effects

In the present paper, an isogeometric analysis formulation along with the Nitsche technique is implemented based on the first order shear deformation plate theory to investigate the aero-thermo-elastic panel flutter behavior of a functionally graded (FG) plate containing cracks in the supersonic flow field. A FG material is considered with temperature-dependent mechanical properties distributed in the plate thickness according to the power law. The temperature distribution in the plate thickness is calculated through solution of the one-dimensional steady-state conductive heat transfer equation while the surface temperatures are kept fixed. The physical contact and the crack growth phenomena are overlooked in the present research. The panel flutter behavior is investigated under combination of crack orientation, crack length, thermal conditions, flow direction, FG volume fraction index and boundary conditions. The accuracy and quality of the present isogeometric formulation is shown in comparison with those available in literature.

An isogeometric formulation with a three-variable high order shear deformation theory for free vibration analysis of FG porous plates reinforced by graphene platelets

We present a generalized three-variable high order shear deformation theory (THSDT) using isogeometric analysis (IGA) to analyze free vibration of functionally graded porous (FGP) plates reinforced by graphene platelets (GPLs) in this work. It is named as FGP-GPLs for a short. The proposed theory only has got three degrees of freedom (DOFs) per node as the same way of numerically solutions in three-dimensional (3D) solids. THSDT fulfills the classical plate theory (CPT), the first-order shear deformation theory (FSDT) and even the higher-order shear deformation theory (HSDT). IGA is chosen to analyze because of its noteworthy advantages in numerical computational sides of plate problems. In addition, the displacement field of THSDT needs the high continuity in approximated formulation with high-order derivatives for a weak form of fourth order equation. According to IGA formulation based on the generalized THSDT, the shear locking phenomenon is free. The variables of THSDT are less than HSDTs which contain five DOFs per node. The influences of weight fractions, the coefficient porosity, dispersion patterns of GPLs and distribution types of porosity on structure’s natural frequencies are studied through some numerical examples. In order to prove the reliability and accuracy of present method, the numerical results are compared to available published works.
 Keywords:
 FG-porous plate; graphene platelet reinforcements; three-variable high order shear deformation theory (THSDT); isogeometric analysis; free vibration.

A nonlinear geometric couple stress based strain gradient Kirchhoff–Love shell formulation for microscale thin-wall structures

We present a nonlinear Kirchhoff–Love micro-shell element based on isogeometric analysis (IGA) and couple stress theory. Higher-order NURBS functions are exploited for analyzing the strain gradient effect which automatically fulfill the higher-order continuity requirements. We express the strain gradient elastic formulation in natural curvilinear coordinates, which leads to an efficient numerical tool to examine geometric nonlinearities of thin micro-shell structures. The presented IGA formulation is verified through comparisons to analytical solution, experimental data as well as other popular benchmark problems of nonlinear geometric shells. We believe that the presented formulation is particularly suitable for analyzing two-dimensional materials at larger length scales, which are commonly studied at nanoscale.

Free vibration analysis of functionally graded plates containing embedded curved cracks

In the present paper, the free vibration behavior of functionally graded plates containing straight and curved embedded crack is investigated. A NURBS-based multi-patch isogeometric analysis formulation is utilized based on the firstorder shear deformation plate theory. The Nitsche technique is implemented to meet the inter-patch connection constraints. The crack line is assumed as a narrow cut along a straight or free-shape curve path within the plate. The crack growth phenomena are overlooked. The accuracy and quality of the obtained results are compared to those available in the literature. Subsequently, the effect of various material and geometry parameters on the free vibration characteristics of cracked FG plate including the volume-fraction index, crack shape, crack length, crack orientation and, crack location are examined. It learned that the straight crack inclination angle mainly influences the fifth and fourth natural modes. Moreover, the FGM mixture index doesn't noticeably affect the frequency trends. No considerable impact is noted between the edge constraint setups on the fundamental frequency.

Isogeometric analysis for in-plane free vibration of centrifugally stiffened beams including Coriolis effects

In this paper, in-plane free vibration of a rotating Euler-Bernoulli beam with uniform cross section is studied using a rotation-free isogeometric analysis formulation. Using Hamilton’s principle, the derived equations governing the in-plane bending and axial deformations are coupled by means the Coriolis terms. These two equations are transformed into a one linear gyroscopic eigenvalue problem via a weak formulation. The boundary conditions are enforced by means of Lagrange multipliers. Numerical tests of convergence are conducted and the obtained results are compared to the reference solutions to show the effectiveness and accuracy of the proposed method. Effects on natural frequencies, and mode shapes originated from Coriolis terms, angular speed, slenderness ration, hub radius and boundary conditions are numerically investigated.

A physically and geometrically nonlinear formulation for isogeometric analysis of solids in boundary representation

AbstractThis contribution concerns a physically and geometrically nonlinear formulation for isogeometric analysis of solids in boundary representation. The parametrization of the proposed approach is inspired by the scaled boundary finite element method, where the geometry of the boundary is sufficient to define the entire solid surface. This suits perfectly the boundary representation modeling technique, which is commonly employed for the design of solids in CAD. In order to provide a boundary‐oriented approach, the solid surface is parametrized by a radial scaling parameter emanating from a scaling center and a parameter in the circumferential direction of the boundary. According to the idea of isogeometric analysis, NURBS basis functions define the geometry of the boundary and approximate the solution on the boundary. We also employ NURBS to approximate the solution in the scaling direction in the interior of the domain. This enables a straightforward treatment of nonlinear problems while preserving the exact geometry of the boundary. The approximation in scaling direction is in principle flexible and requires for coupling of adjacent patches inside the domain. The Galerkin method is employed for the solution in both parametric directions and a linearization is derived which is used within an iterative Newton‐Raphson scheme. We consider nonlinear problems in solid mechanics including elasto‐plastic material behavior and large deformations as well as complex geometries commonly employed in engineering applications. In this paper we study a numerical example of a complex geometry in order to evaluate the performance of the proposed approach.

Anisotropic solid-like shells modeled with NURBS-based isogeometric approach: Vibration, buckling, and divergence analyses

We proposed an IGA formulation for free vibration, buckling and divergence analyses of generally anisotropic solid-like composite shells. Recently developed Rayleigh–Ritz based methods are not accurate enough for curved shells since they are not able to capture twisting mode shapes. Here, we use the advantages of isogeometric analysis (IGA) and develop a three-dimensional higher-order continuous solid-like shell (SLS) model. Since the continuity of material orientation is crucial in numerical analysis of anisotropic shells, non-uniform rational B-splines (NURBS) are used to establish a smooth interpolation through the thickness and on the middle surface of the shell for both the geometry and displacement fields. The developed isogeometric SLS is coupled with Bézier extraction operators to represent continuity between multi-patch anisotropic domains. Numerical solutions for composite laminated shells with different boundary conditions are compared with analytical solutions or other solutions from the literature. In this context, we study the influence of span-to-thickness ratios, different shapes, modulus ratios, member aspect ratios and various layouts on the mechanical response. With the presented model, we were able to obtain the twisting mode shapes, which are not frequently observed in orthotropic and isotropic ones. We also study the influence of the polynomial order of the IGA formulation on the results.

Effects of embedded perforation geometry on the free vibration of tow-steered variable stiffness composite laminated panels

The free vibration characteristics of the variable stiffness composite laminated plates containing embedded cutout of desired shapes is investigated. The circular, elliptical and quadrilateral perforation geometries are taken into account. In order to provide a strongly flexible calculation tool, an isogeometric analysis formulation based on the non-uniform rational B-splines (NURBS) along with the Nitsche technique is developed based on the first order shear deformation plate theory. The effects of change in the mechanical properties of tow steered laminates throughout the geometry due to fibers following prescribed curvilinear paths are taken into account in the integration procedures. The accuracy and reliability of the calculations is shown through some representative comparisons. The effects of change in the cutout geometry, location and orientation in conjunction with the curvilinear fiber placement are studied.

Isogeometric analysis of functionally graded plates with a logarithmic higher order shear deformation theory

This paper presents a new logarithmic higher order shear deformation theory (LHSDT) based on isogeometric analysis (IGA) to study the static bending, free vibration, and buckling behaviors of functionally graded plates. The temperature change conditions are considered. In the LHSDT fashion, shear stresses disappear at the top and bottom surfaces of the plates and shear correction factor vanishes. The requirement for C1-continuity in terms of the LHSDT is straightforwardly possessed with the aid of inherent high order continuity of non-uniform rational B-spline (NURBS), which serves as basis functions in our IGA formulation. The superior performance and accuracy of the proposed method is demonstrated through extensive numerical examples. The computed results of static bending, vibration and buckling from the proposed theory are in a very good agreement with reference solutions available in literature obtained by various plate theories and different solving method.

An isogeometric analysis to identify the full flexoelectric complex material properties based on electrical impedance curve

In this paper, we present a new approach to identify all material parameters of flexoelectric materials based on electrical impedance curves. This approach combines an Isogeometric Analysis (IGA) formulation with a gradient-based optimization algorithm using the Method of Moving Asymptotes (MMA). The IGA formulation allows for efficient modeling of flexoelectricity taking advantage of the higher order continuity of IGA. The proposed methodology starts with determining preliminary real parts based on resonant modes in order to avoid local minima which gives the numerical impedance curves close to the experimental impedance curve. The results in the preliminary step are used as initial parameters of the refinement step to simultaneously determine both real and imaginary part by minimizing the difference between pseudo-experimental and numerical impedance curve. Some numerical examples are illustrated to show the good agreement between the numerical and pseudo-experimental impedance curves.

A Weak Formulation for Isogeometric Analysis of Vibro-Acoustic Systems with Non-Conforming Interfaces

This paper develops a weak formulation for steady-state dynamic analysis of vibro-acoustic systems in the isogeometric framework. The Non-Uniform Rational B-Splines (NURBS) polynomials are used as basis functions for both geometrical transformation and numerical approximation. Starting from a NURBS-based geometrical projection that transforms a general, complex domain to a regular, spectral domain, an energy-based numerical model is then built, in which NURBS are again used as wholly admissible functions to approximate the structural displacement and the acoustic sound pressure. This process eventually leads to a set of discretized equations, based on which the system’s dynamic properties can be obtained. Besides better accuracy and excellent geometrical adaptivity, the main merit of the proposed method lies in its element-free feature of system modeling, which provides a natural solution for structural-acoustic coupling with non-conforming interfaces. This non-conformity results from either non-matching control points along the boundary or distinct polynomial orders used by adjacent patches, which is inevitable for a vibro-acoustic system given its multi-physics nature. The reliability of the proposed method is validated by a benchmark example. Its accuracy and efficiency are further investigated through a more general, car-like vibro-acoustic system.

Non-overlapping domain decomposition solution schemes for structural mechanics isogeometric analysis

Isogeometric analysis (IGA) is a novel computer aided engineering technique that addresses diverse problems in computational mechanics [1–4], all under the exact geometric representation. Apart from the exact geometric representation, the high continuity of IGA shape functions enhances the accuracy and robustness of the method. However, the price to pay is that the resulting matrices are denser, with increased bandwidth and overlapping which make the solution of large-scale problems more computationally intensive. As a result, effective solution techniques are still considered an open issue for further research. In this paper, an innovative family of solution schemes based on domain decomposition methods (DDM) is proposed, that significantly reduces the computational cost. Specifically, the solution of the global system is performed with the preconditioned conjugate gradient algorithm (PCG) whose preconditioning step is evaluated with a dual DDM where special care is taken to avoid the overlapped subdomains which are inherent in decomposed IGA formulation due to the increased continuity of the shape functions.

An isogeometric analysis formulation for red blood cell electro-deformation modeling

An isogeometric analysis formulation for simulating red blood cell (RBC) electro-deformationis presented. Electrically-induced cell deformation experiments are receiving increasing attention as an attractive strategy for single-cell mechanical phenotyping. As the RBC structure consists in a very thin biological membrane enclosing a nearly-incompressible fluid, (i) a surface shell kinematic model and (ii) the imposition of the shell enclosed-volume conservation constraint are proposed within the isogeometric analysis framework. With regard to the electro-deformation, an accurate evaluation of the electric-field induced forces is achieved by the Maxwell stress tensor approach. A staggered fixed-point iteration scheme is then proposed for performing the electro-mechanical coupling, in order to use reliable mechanical and electrical problem solvers sequentially. Supported by the comparison with experimental results and reference solutions, numerical simulations concerning the large deformation of a RBC by optical tweezers and an in silico electro-deformation experiment prove the accuracy and the effectiveness of the proposed formulation.

A reconstructed local [formula omitted] formulation for isogeometric Kirchhoff–Love shells

An efficient assumed strain formulation for avoiding the membrane locking of non polar shells is developed in the context of B-spline interpolation. Assumed membrane strains are introduced locally in each element with a local L2-projection, and then a spline reconstruction algorithm, developed by Thomas et al. (2015), is employed for reconstructing the membrane strain at the global patch level. In this way, a banded stiffness matrix is obtained that strongly reduces the computational cost with respect to the standard B̄ formulation for isogeometric analysis. The main advantages of the proposed method are: the reproduction of regular membrane strain fields, an accurate membrane locking-free solution and an high computational efficiency with respect to standard B̄ formulation. The method can be applied for any polynomial degree of the B-spline interpolation. The effectiveness of the proposed method is demonstrated analyzing some bending and membrane dominated problem, commonly employed as benchmark in the shell literature.