Isogeometric Finite Element Method Research Articles

Uncertainty Quantification of Vibroacoustics with Deep Neural Networks and Catmull–Clark Subdivision Surfaces

This study proposes an uncertainty quantification method based on deep neural networks and Catmull–Clark subdivision surfaces for vibroacoustic problems. The deep neural networks are utilized as a surrogate model to efficiently generate samples for stochastic analysis. The training data are obtained from numerical simulation by coupling the isogeometric finite element method and the isogeometric boundary element method. In the simulation, the geometric models are constructed with Catmull–Clark subdivision surfaces, and meantime, the physical fields are discretized with the same spline functions as used in geometric modelling. Multiple deep neural networks are trained to predict the sound pressure response for various parameters with different numbers and dimensions in vibroacoustic problems. Numerical examples are provided to demonstrate the effectiveness of the proposed method.

Weakly viscous two-dimensional incompressible fluid flows using efficient isogeometric finite element method

This paper presents a straightforward and efficient numerical simulation method for solving the Navier–Stokes equations for weakly viscous incompressible fluids describing steady flow. Our approach utilizes isogeometric finite elements to handle higher-order partial differential operators associated with weakly viscous incompressible flow problems. Specifically, our numerical formulation employs a principle of virtual power (PVP)-based weak formulation that utilizes a stream-function field, which distinguishes it from the more commonly used bi-harmonic type formulations. The usage of a stream-function field ensures a pointwise divergence-free velocity condition, making the present method suitable for low to moderately high Reynolds number flow problems. In contrast to the bi-harmonic formulation, which is typically used for describing internal flow and requires special treatment of outflow boundary conditions, the PVP-based formulation is more general and does not require special treatment at the outflow boundary. It is also demonstrated that both bi-harmonic and PVP-based weak formulations yield identical results for internal flow problems. Our method employs non-uniform rational B-spline basis functions, and we present a simple stitching technique for imposing no-slip Dirichlet boundary conditions. Finally, we solve Poisson's equation to recover the pressure field. Furthermore, we use an appropriate Gaussian quadrature that is exact for splines to speed up the computation of various element matrices, especially for high polynomial degrees. The proposed formulation is evaluated by solving a set of numerical problems, including internal flow and channel flow problems.

A Kronecker product linear-cost solver for the high-order generalized-α method for multi-dimensional hyperbolic systems

The main result of the work is the generalization of the concept of the variational splitting from [3] to the higher-order generalized-α schemes proposed in [5] in the context of hyperbolic and elastic wave propagation problems. It is possible to factorize the matrix of the linear system of equations (M+ηK) in the linear computational cost. Here M is the mass matrix, and K is the stiffness matrix. We discretize using the isogeometric finite element method. Following the idea proposed in [3], factorizing the matrix (M+ηK), we obtained an additional error of the order of η2. This paper introduces a technique to couple the possibility of fast factorization of (M+ηK) with higher-order generalized-α schemes proposed in [5] for hyperbolic and elastic wave propagation problems. Thus, we introduce a variationally separable splitting technique for the high-order accuracy generalized-α method. We use tensor-product meshes to develop the splitting method, which results in the linear cost for multi-dimensional problems. We consider standard C0 finite elements as well as smoother isogeometric analysis for spatial discretization. The direction splitting method requires the mass and the stiffness matrices to have the Kronecker product structure. For other problems, the corresponding Kronecker product solver can be employed as a preconditioner as long as a Kronecker product can sufficiently well approximate the problem matrix. We also study the spectrum of the amplification matrix to establish the unconditional stability of the method. We use various examples to demonstrate the performance of the method and the optimal approximation accuracy. In numerical tests, we compute the L2 and H1 norms to show the optimal convergence of the discrete method in space and high-order accuracy in time.

A Polynomial Chaos Expansion Method for Mechanical Properties of Flexoelectric Materials Based on the Isogeometric Finite Element Method

The paper proposes a method for analyzing the mechanical properties of flexoelectric materials based on the isogeometric finite element method (IGA-FEM) and polynomial chaos expansion (PCE). The method discretizes the flexoelectric governing equations utilizing the B-spline shape functions that satisfy the continuity requirement to obtain the mechanical properties (electric potential) of the material. To obtain a mechanical property with different input parameters, we choose the truncated pyramid model as the object of study, and use IGA-FEM and PCE to solve different single uncertain parameters, including independent Young’s modulus and uniformly distributed force, and two kinds of flexoelectric constants, respectively. Numerical examples are presented to bear out the accuracy and viability of the proposed methodology.

Aeroelastic Tailoring of Variable Stiffness Composite Laminated Quadrilateral and Circular Plates in Supersonic Flow Using Isogeometric Approach

In this study, a cubic NURBS-based isogeometric finite element method is developed to study the aeroelastic tailoring of variable stiffness composite laminated (VSCL) plates, including skew, skew-tapered, tapered, and circular shapes based on third-order shear deformation theory. Also, the first-order piston theory is employed to approximate aerodynamic loads with arbitrary yaw flow angles over the plate area to establish aeroelastic equations. The NURBS basis functions are then utilized to simulate exact geometries and estimate variable fields for discretizing governing equations by well-known isogeometric analysis. Finally, the free vibration and flutter analysis of a VSCL quadrilateral and circular plate will be presented via the obtained eigenvalue problem. The achieved results are compared with the available results in the literature to prove the accuracy and validity of the present method. The effect of different parameters such as the flow yaw angle, end and middle-length fiber orientation, longitudinal to transverse Young’s modulus ratio, geometric shape, boundary conditions, and turning symmetric layup to asymmetric layup is carried out on the critical flutter aerodynamic pressure and natural frequencies.

Sensitivity analysis of flexoelectric materials surrogate model based on the isogeometric finite element method

In this paper proposes a sensitivity analysis method based on a Polynomial Chaos Expansion (PCE) surrogate model for flexoelectric materials. The non-uniform rational B-splines (NURBS) basis functions to discretize the fourth-order partial differential equation for flexoelectricity and obtains a deterministic solution (electric potential). The mathematical expressions of surrogate model for the flexoelectric materials are established by considering uncertain parameters such as independent Young’s modulus, concentrated load and flexoelectric constants. The sensitivity expression is found by derivation the mathematical expression for the surrogate model. Moreover the finite difference method (FDM) are conducted in numerical examples to demonstrate the validity and correctness of the proposed algorithm.

Sensitivity analysis of structural-acoustic fully-coupled system using isogeometric boundary element method

In many engineering challenges, the whole interaction between the structural domain and the acoustic domain must be taken into account, particularly for the acoustic analysis of thin structures submerged in water. The fast multipole boundary element approach is used in this work to simulate the external acoustic domain and the finite element method is used to describe the structural components. To improve coupling analysis accuracy, discontinuous higher-order boundary components are created for the acoustic domain. The isogeometric boundary element method (IGABEM) discretizes unknown physical fields by using CAD spline functions as basis functions. IGABEM is inherently compatible with CAD and can perform numerical analysis on CAD models without having to go through the time-consuming meshing process required by traditional FEM/BEM and volume parameterization in isogeometric finite element methods. IGABEM’s power in tackling infinite domain issues and combining CAD and numerical analysis is fully used when it is applied to structural form optimization of three-dimensional external acoustic problems. The structural-acoustic design and optimization procedures benefit from the use of structural-acoustic design sensitivity analysis because it may provide information on how design factors affect radiated acoustic performance. This paper provides adjoint operator-based equations for sound power sensitivity on structural surfaces and direct differentiation-based equations for sound power sensitivity on arbitrary closed surfaces surrounding the radiator. Numerical illustrations are provided to show the precision and viability of the suggested approach.

Concurrent algorithm for integrating three-dimensional B-spline functions into machines with shared memory such as GPU

The aim of this paper is to analyze the integration for 3D isogeometric finite element method solvers and its effective scheduling on hierarchical computer architecture. Data necessary for concurrency over elements is independent, so computation on this level is trivially concurrent. However, constructing several layers of concurrency for the integration algorithm is challenging. In this work, we propose a multilevel concurrent integration algorithm associated with scheduling that brings one extra degree of possible speedup. Because of one extra degree of possible speedup, we analyze the concurrent integration inside elements. The scheduling algorithm is intended for strongly related hierarchical architectures of a GPU. Using trace theory and Foata Normal Form, we verify integrity of the proposed solution. Summing up, we propose a general method for analyzing concurrency of the integration algorithm. We instantiate this method on a classical element-based integration algorithm, however, this methodology is possible to apply for other integration algorithms, including sum factorization, fast numerical quadrature, or row-wise integration methods.

Isogeometric FE-BE method with non-conforming coupling interface for solving elasto-thermoviscoelastic problems

The isogeometric finite element (FE) and boundary element (BE) coupling method with non-conforming interface is applied to solve 3D elasto-thermoviscoelastic problems. The thermoviscoelastic constitutive relation is expressed by Stieltjes integral, and the equivalent integral weak form of the problem is derived according to the principle of virtual work, in which the time integral is discretized by the trapezoidal difference approximation formula. The parameters of viscoelastic material are described by Prony series, and the discrete formula can be simplified by recursive relationship to avoid storing all solutions in time steps. The equilibrium and compatibility conditions of FE and BE subdomains on the coupling interface are used to couple two different system matrices, and the coupling of non-conforming non-uniform rational B-splines (NURBS) surfaces in different subdomains is achieved by using virtual knot insertion technique. In addition, the time-temperature equivalent principle is applied to observe the mechanical behavior of viscoelastic materials over a wide temperature range in a short time. Through several numerical examples, the effects of different working conditions and geometrical parameters of combustion chamber on the structure of the solid rocket motor (SRM) are studied, and the correctness and robustness of this method are verified.

An isogeometric FE-BE method to investigate fluid–structure interaction effects for an elastic cylindrical shell vibrating near a free surface

In this study, an isogeometric finite element-boundary element (FE-BE) framework is proposed to investigate frequency-dependent hydrodynamic characteristics of elastic structures vibrating at free surface or in the vicinity of free surface. The overall numerical framework consists of two parts. In the first part, in vacuo dynamic characteristics are obtained by an isogeometric finite element method (IGAFEM) formulation. Then, in the second part, frequency-dependent generalized added mass and generalized hydrodynamic damping coefficients are obtained by an isogeometric boundary element method (IGABEM) formulation, assuming that the surrounding fluid is ideal, i.e., inviscid, incompressible and its motion is irrotational. To include free surface effects, the free surface boundary conditions are satisfied with an appropriate Green’s function in the mathematical formulation. To show the applicability of the presented framework, an elastic cylindrical shell which was studied previously in the literature is investigated considering that the structure is partially or fully submerged into fluid of infinite depth. It is shown that the results obtained in this study compare very well with the results available in literature.

A sample-efficient deep learning method for multivariate uncertainty qualification of acoustic–vibration interaction problems

We propose an efficient Monte Carlo simulation method to address the multivariate uncertainties in acoustic–vibration interaction systems. The deep neural network acts as a general surrogate model to enhance the sampling efficiency of Monte Carlo Simulation. Singular Value Decomposition - Radial Basis Functions (SVD-RBF) acts as a bridge between the original full model and the neural network, enabling the training datasets of the neural network to be evaluated rapidly from a reduced-order model. The snapshots of full order models are obtained with isogeometric analysis, in which we couple two numerical schemes for vibro–acoustic interaction problems: the isogeometric finite element method for simulating vibration of Kirchhoff–Love shells and isogeometric boundary element method for exterior acoustic waves. Numerical results show that the proposed algorithm can significantly improve the efficiency of uncertainty analysis.

Bi-material topology optimization for fully coupled structural-acoustic systems with isogeometric FEM–BEM

This paper presents a novel method for topology optimization of vibrating structures interacted with acoustic wave for the purpose of minimizing radiated sound power level. We consider exterior acoustic fields and bi-material shell models without damping in this work. Within the isogeometric analysis framework, we employ Catmull–Clark subdivision surfaces to construct geometries and discretize physical fields. The isogeometric finite element method with Kirchhoff–Love shell elements is coupled with the isogeometric boundary element method in acoustics. The topology optimization is performed through density-based approaches, in which the sensitivities are evaluated with adjoint variable methods. Numerical experiments demonstrate the validity and effectiveness of the algorithm for topology optimization of structural-acoustic interaction systems.

On the incorporation of curvature effects into the isogeometric analysis of fibre‐reinforced solids

AbstractIn the context of engineering on the micro‐ and nanoscale, size‐dependency is an important characteristic of material behaviour. In order to avoid complex experiments and predict size effects in simulations instead, classic continuum approaches are extended by the introduction of a length scale, e.g. through the consideration of gradient contributions. For the particular case of fibre‐reinforced materials, such a gradient‐enhanced approach can be achieved by introducing the fibre curvature as an additional kinematic quantity. This implies that basis functions with a global continuity higher than C0 are required for a finite element‐based approach which accounts for these fibre curvature effects. The present contribution shows that the isogeometric finite element method can provide a framework for the simulation of the respective higher‐gradient continua.

An isogeometric finite element approach to fibre-reinforced composites with fibre bending stiffness

In the modelling of fibre-reinforced composites, it is well established to consider the fibre direction in the stored energy in order to account for the transverse isotropy of the overall material, induced by a single family of fibres. However, this approach does not include any length scale and therefore lacks in the prediction of size effects that may occur from the fibre diameter or spacing. By making use of a generalised continuum model including non-symmetric stresses and couple-stresses, the gradient of the fibre direction vector can be taken into account as an additional parameter of the stored energy density function. As a consequence, the enhanced model considers the bending stiffness of the fibres and includes information on the material length scale. Along with additional material parameters, increased continuity requirements on the basis functions follow in the finite element analysis. The isogeometric finite element method provides a framework which can fulfil these requirements of the corresponding weak formulation. In the present contribution, the method is applied to two representative numerical examples. At first, the bending deformation of a cantilever beam is studied in order to analyse the influence of the fibre properties. An increasingly stiff response is observed as the fibre bending stiffness increases and as the fibre orientation aligns with the beam’s axis. Secondly, a fibre-reinforced cylindrical tube under a pure azimuthal shear deformation is considered. The corresponding simulation results are compared against a semi-analytical solution. It is shown that the isogeometric analysis yields highly accurate results for the boundary value problem under consideration.

A Non-Intrusive Stochastic Isogeometric Analysis of Functionally Graded Plates with Material Uncertainty

A non-intrusive approach coupled with non-uniform rational B-splines based isogeometric finite element method is proposed here. The developed methodology was employed to study the stochastic static bending and free vibration characteristics of functionally graded material plates with inhered material randomness. A first order shear deformation theory with an artificial shear correction factor was used for spatial discretization. The output randomness is represented by polynomial chaos expansion. The robustness and accuracy of the framework were demonstrated by comparing the results with Monte Carlo simulations. A systematic parametric study was carried out to bring out the sensitivity of the input randomness on the stochastic output response using Sobol’ indices. Functionally graded plates made up of Aluminium (Al) and Zirconium Oxide (ZrO2) were considered in all the numerical examples.

Shape optimization of heterogeneous materials based on isogeometric boundary element method

In this paper, the isogeometric boundary element method (IGABEM) is used to optimize the shape of heterogeneous materials. In contrast to the isogeometric finite element method (IGAFEM), the iterative optimization algorithm based on IGABEM can be implemented directly from Computer-Aided Design (CAD) without returning the optimization results to CAD designers. The discontinuous element method is extended to IGABEM which deals with corner point problems of inhomogeneous materials. After the singularity of sensitivity analysis of boundary integral equations is demonstrated, the power series expansion method (PSEM) is applied to IGABEM to evaluate various degrees of singularity about sensitivity analysis, which shows more accuracy and efficiency than the element sub-division method (ESDM). A set of control points on the geometric boundary are chosen as design variables, which can be passed from the design model to the analysis model, and the objective function is the elastic energy increment. Finally, several numerical examples in 2D and 3D problems are presented to demonstrate the validity and robustness of the present method.

An isogeometric FE-BE method and experimental investigation for the hydroelastic analysis of a horizontal circular cylindrical shell partially filled with fluid

Abstract In this study, the dynamic characteristics (i.e. natural frequencies and associated mode shapes) of a partially filled horizontal cylindrical shell are investigated experimentally and by an isogeometric finite element-boundary element method. The proposed numerical procedure is divided into two parts. In the first part, the dynamic characteristics of the cylindrical shell under in-vacuo conditions are obtained by the isogeometric finite element method (IGAFEM) based on a linear Kirchhoff-Love shell formulation. In the second part, the fluid-structure interaction effects are calculated in terms of generalized added mass coefficients by using the isogeometric boundary element method (IGABEM), assuming that the structure vibrates in its in-vacuo principle mode shapes. By adopting the linear hydroelasticity theory, it is assumed that the fluid flow is ideal, i.e., an incompressible flow and inviscid fluid. In order to show the versatility of the numerical method, the results are compared with those obtained by the conducted experiments. Relevant numerical challenges in the hydroelastic vibration analysis are highlighted and it is shown that the numerical predictions and experimental results are in good agreement.

Isogeometric free vibration of sector cylindrical shells with carbon nanotubes reinforced and functionally graded materials

An isogeometric numerical procedure based on non-uniform rational B-splines is applied to solve the free vibration of carbon nanotubes reinforced and functionally graded material sector cylindrical shells. A typical isogeometric analysis cylindrical shell element is introduced to transform curvilinear domain to parametric domain. The effects of the shear deformation, rotary inertia and axis extensibility are taken into consideration. The material properties are estimated by a mixture rule and effective carbon nanotube parameters. The weak form is derived in detail by applying virtual principle. The novelty of this paper is to use the isogeometric finite element method to analyze the dynamic behaviors of sector cylindrical shells with functionally graded material and four types of carbon nanotubes distributed material. By numerical examples, the convergence and accuracy of the current method are validated. Then, a series of natural frequencies and mode shapes are presented to serve as benchmark solutions in future researches. Moreover, the effects of boundary conditions, geometric proprieties as well as material parameters on the frequencies of the carbon nanotubes reinforced and functionally graded material sector cylindrical shells are examined.

Fast and green parallel isogeometric analysis computations for multi-objective optimization of liquid fossil fuel reserve exploitation with minimal groundwater contamination

We present a general optimization technique to minimize the execution time, energy consumption, and numerical error of a parallel isogeometric finite element method (IGA-FEM) solver for time-dependent problems. The IGA-FEM solver is called upon multiple times during inverse problem computations. It is used to evaluate the inverse problem parameters. As an exemplary challenging problem, we consider the IGA-FEM solver as a primal solver for the inverse optimization of hydraulic fracking. Liquid Fossil Fuel Extraction Problem (LFFEP) is defined to maximize extraction and minimize contamination for the hydraulic fracking process. Solving the LFFEP problem requires hundreds of computationally demanding calls to the IGA-FEM solver. Thus, we research the optimal configuration of the IGA-FEM solver to make computations more efficient; i.e., to reduce computation time and energy consumption where the input parameters are the number of utilized CPUs in a parallel environment, mesh size, and B-spline order. The algorithm presented in this paper can be used to optimize any similar time-dependent IGA-FEM simulation.

Structural shape optimization of three dimensional acoustic problems with isogeometric boundary element methods

The boundary element method (BEM) is a powerful tool in computational acoustics, because the analysis is conducted only on structural surfaces, compared to the finite element method (FEM) which resorts to special techniques to truncate infinite domains. The isogeometric boundary element method (IGABEM) is a recent progress in the category of boundary element approaches, which is inspired by the concept of isogeometric analysis (IGA) and employs the spline functions of CAD as basis functions to discretize unknown physical fields. As a boundary representation approach, IGABEM is naturally compatible with CAD and thus can directly perform numerical analysis on CAD models, avoiding the cumbersome meshing procedure in conventional FEM/BEM and eliminating the difficulty of volume parameterization in isogeometric finite element methods. The advantage of tight integration of CAD and numerical analysis in IGABEM renders it particularly attractive in the application of structural shape optimization because (1) the geometry and the analysis can be interacted, (2) remeshing with shape morphing can be avoided, and (3) an optimized solution returns a CAD geometry directly without postprocessing steps. In the present paper, we apply the IGABEM to structural shape optimization of three dimensional exterior acoustic problems, fully exploiting the strength of IGABEM in addressing infinite domain problems and integrating CAD and numerical analysis. We employ the Burton–Miller formulation to overcome fictitious frequency problems, in which hyper-singular integrals are evaluated explicitly. The gradient-based optimizer is adopted and shape sensitivity analysis is conducted with implicit differentiation methods. The design variables are set to be the positions of control points which directly determine the shape of structures. Finally, numerical examples are provided to verify the algorithm.