Geometric Multigrid Method for Isogeometric Analysis

The isogeometric analysis method(IGA)is a new type of numerical method solving partial differential equations.Compared with the traditional finite element method,IGA based on geometric spline can keep the model consistency between geometry and analysis,and provide higher precision with less freedom.However,huge stiffness matrix fromthe subdivision progress still leads to the solution efficiency problems.This paper presents amultigrid method based on geometric multigrid(GMG)to solve the matrix system of IGA.This method extracts the required computational data for multigrid method fromthe IGA process,which also can be used to improve the traditional algebraic multigrid method(AGM).Based on this,a full multigrid method(FMG)based on GMG is proposed.In order to verify the validity and reliability of these methods,this paper did some test on Poisson’s equation and Reynolds’equation and compared the methods on different subdivision methods,different grid degrees of freedom,different cyclic structure degrees,and studied the convergence rate under different subdivision strategies.The results show that the proposed method is superior to the conventional algebraic multigrid method,and for the standard relaxed V-cycle iteration,the method still has a convergence speed independent of the grid size at the same degrees.

Monte Carlo Simulation of Fractures Using Isogeometric Boundary Element Methods Based on POD-RBF

This paper presents a novel framework for stochastic analysis of linear elastic fracture problems.Monte Carlo simulation(MCs)is adopted to address the multi-dimensional uncertainties,whose computation cost is reduced by combination of Proper Orthogonal Decomposition(POD)and the Radial Basis Function(RBF).In order to avoid re-meshing and retain the geometric exactness,isogeometric boundary element method(IGABEM)is employed for simulation,in which the Non-Uniform Rational B-splines(NURBS)are employed for representing the crack surfaces and discretizing dual boundary integral equations.The stress intensity factors(SIFs)are extracted by M integral method.The numerical examples simulate several cracked structures with various uncertain parameters such as load effects,materials,geometric dimensions,and the results are verified by comparison with the analytical solutions.

A Combined Shape and Topology Optimization Based on Isogeometric Boundary Element Method for 3D Acoustics

A combined shape and topology optimization algorithm based on isogeometric boundary element method for 3D acoustics is developed in this study.The key treatment involves using adjoint variable method in shape sensitivity analysis with respect to non-uniform rational basis splines control points,and in topology sensitivity analysis with respect to the artificial densities of sound absorption material.OpenMP tool in Fortran code is adopted to improve the efficiency of analysis.To consider the features and efficiencies of the two types of optimization methods,this study adopts a combined iteration scheme for the optimization process to investigate the simultaneous change of geometry shape and distribution of material to achieve better noise control.Numerical examples,such as sound barrier,simple tank,and BeTSSi submarine,are performed to validate the advantage of combined optimization in noise reduction,and to demonstrate the potential of the proposed method for engineering problems.

Interpolating Isogeometric Boundary Node Method and Isogeometric Boundary Element Method Based on Parameter Space

In this paper,general interpolating isogeometric boundary node method(IIBNM)and isogeometric boundary element method(IBEM)based on parameter space are proposed for 2D elasticity problems.In both methods,the integral cells and elements are defined in parameter space,which can reproduce the geometry exactly at all the stages.In IIBNM,the improved interpolating moving leastsquare method(IIMLS)is applied for field approximation and the shape functions have the delta function property.The Lagrangian basis functions are used for field approximation in IBEM.Thus,the boundary conditions can be imposed directly in both methods.The shape functions are defined in 1D parameter space and no curve length needs to be computed.Besides,most methods for the treatment of the singular integrals in the boundary element method can be applied in IIBNM and IBEM directly.Numerical examples have demonstrated the accuracy of the proposed methods.

Subdivision Surface-Based Isogeometric Boundary Element Method for Steady Heat Conduction Problems with Variable Coefficient

The present work couples isogeometric analysis(IGA)and boundary element methods(BEM)for three dimensional steady heat conduction problems with variable coefficients.The Computer-Aided Design(CAD)geometries are built by subdivision surfaces,and meantime the basis functions of subdivision surfaces are employed to discretize the boundary integral equations for heat conduction analysis.Moreover,the radial integration method is adopted to transform the additional domain integrals caused by variable coefficients to the boundary integrals.Several numerical examples are provided to demonstrate the correctness and advantages of the proposed algorithm in the integration of CAD and numerical analysis.

Quantifying Uncertainty in Dielectric Solids’ Mechanical Properties Using Isogeometric Analysis and Conditional Generative Adversarial Networks

Accurate quantification of the uncertainty in the mechanical characteristics of dielectric solids is crucial for advancing their application in high-precision technological domains,necessitating the development of robust com-putational methods.This paper introduces a Conditional Generation Adversarial Network Isogeometric Analysis(CGAN-IGA)to assess the uncertainty of dielectric solids’mechanical characteristics.IGA is utilized for the precise computation of electric potentials in dielectric,piezoelectric,and flexoelectric materials,leveraging its advantage of integrating seamlessly with Computer-Aided Design(CAD)models to maintain exact geometrical fidelity.The CGAN method is highly efficient in generating models for piezoelectric and flexoelectric materials,specifically adapting to targeted design requirements and constraints.Then,the CGAN-IGA is adopted to calculate the electric potential of optimum models with different parameters to accelerate uncertainty quantification processes.The accuracy and feasibility of this method are verified through numerical experiments presented herein.

Generalized nth-Order Perturbation Method Based on Loop Subdivision Surface Boundary Element Method for Three-Dimensional Broadband Structural Acoustic Uncertainty Analysis

In this paper,a generalized nth-order perturbation method based on the isogeometric boundary element method is proposed for the uncertainty analysis of broadband structural acoustic scattering problems.The Burton-Miller method is employed to solve the problem of non-unique solutions that may be encountered in the external acoustic field,and the nth-order discretization formulation of the boundary integral equation is derived.In addition,the computation of loop subdivision surfaces and the subdivision rules are introduced.In order to confirm the effectiveness of the algorithm,the computed results are contrasted and analyzed with the results under Monte Carlo simulations(MCs)through several numerical examples.