Three-dimensional Isogeometric Analysis Research Articles

Optimization of multi-directional functionally graded plates in thermal environment based on 3D isogeometric analysis and adaptive-hybrid evolutionary firefly algorithm

In this study, a numerical model based on three-dimensional isogeometric analysis and adaptive hybrid evolutionary firefly algorithm is developed to concurrently optimize the shape and material distribution of multi-directional functionally graded plates. The optimization problems focus on maximizing the fundamental frequency of the plates subjected to thermal effects, while the volume fraction of ceramic constituent material and volume of the plates are considered as side constraints. Isogeometric multi-mesh design approach is employed to generate design and analysis domain. The free vibration analysis of multi-directional functionally graded plates with variable thickness is conducted within the framework of three-dimensional elasticity theory and isogeometric analysis. The generalized numerical framework developed in this study could accurately capture the thermal effects on the behavior of nonhomogeneous plates with variable thickness. Various numerical examples are conducted to illustrate the optimal shapes and material distributions within square, rectangular, and circular plates. Influences of temperature field, boundary conditions, and geometries of the plates on the optimal results are also addressed.

Three-dimensional isogeometric analysis of functionally graded carbon nanotube-reinforced composite plates

Three-dimensional isogeometric analysis of functionally graded carbon nanotube-reinforced composite plates

Bending and free vibration analyses of multi-directional functionally graded plates in thermal environment: A three-dimensional Isogeometric Analysis approach

This study presents a three-dimensional numerical approach to analyze the bending and free vibration behavior of multi-directional functionally-graded plates under the effect of thermal environment. The effective material properties are temperature dependent and the temperature field within the plate’s volume are determined directly based on a generalized heat transfer equation. The governing equations for static bending and free vibration analyses are developed based on the three-dimensional elasticity theory. The Isogeometric Analysis approach is utilized to discretize the weak forms of heat transfer equation, static bending equation and free vibration equation. A iterative scheme is proposed to heat transfer problems with material being dependent on temperature. Rectangular and circular plates are investigated to validate the efficiency of Isogeometric Analysis approach on modeling and analyzing aspects. Various numerical examples are also presented to validate the accuracy of the proposed approach and investigate the influences of material variation profiles, thickness ratios, temperature rise, and boundary conditions on thermal responses of the plates.

Hellinger–Reissner principle based stress–displacement formulation for three-dimensional isogeometric analysis in linear elasticity

In case of Lagrangian finite element formulation, three-dimensional (3-D) stress-based hybrid solid elements have shown excellent coarse mesh accuracy for a wide range of applications. However, to the best of our knowledge, there is no work available towards the development of 3-D stress-based hybrid solid elements for isogeometric analysis (IGA). In this work, we propose stress-based hybrid solid elements to alleviate the issue of locking arising in non-uniform rational B-spline (NURBS)-based IGA. The present work primarily focuses on the linear elasticity, though the formulation can be extended to the non-linear regime. We believe that the excellent coarse mesh accuracy provided by the proposed elements will further enhance the IGA in various applications, especially involving structures with high aspect ratios and nearly incompressible materials. The proposed elements are constructed based on a two-field Hellinger–Reissner variational statement, where stress and displacement fields are interpolated separately. The stress interpolation functions have been derived systemically for various orders of NURBS displacement interpolation functions. Further, we show mathematically that the choice of the stress interpolation functions is free from spurious zero-energy modes. Lastly, the results of numerous 3-D linear-elastic benchmark problems are presented to demonstrate the efficacy and robustness of the proposed elements. The results confirm the superior coarse mesh accuracy for structures with high aspect ratios and almost incompressible materials.

A locking-free formulation for three-dimensional isogeometric analysis

Non-uniform rational B-splines (NURBS)-based isogeometric analysis (IGA) suffers from locking while analyzing highly slender geometries or problems dominated by incompressibility or near incompressibility constraint. The authors have recently proposed a class of reliable and efficient NURBS-based hybrid elements to alleviate locking in two-dimensional linear elasticity regime. Nevertheless, in several practical situations, the problem often necessitates the three-dimensional elements to analyze an accurate behavior of the domain. In the present work, novel stress-based elements are introduced to eliminate the adverse effects of locking in three-dimensional linear elastic NURBS-based IGA. The comprehensive assessment of several benchmark numerical examples shows the method’s capabilities.

Alternating directions parallel hybrid memory iGRM direct solver for non-stationary simulations

The three-dimensional isogeometric analysis (IGA-FEM) is a modern method for simulation. The idea is to utilize B-splines or NURBS basis functions for both computational domain descriptions and the engineering computations. Refined isogeometric analysis (rIGA) employs a mixture of patches of elements with B-spline basis functions, and $C^0$ separators between them. It enables a reduction of the computational cost of direct solvers. Both IGA and rIGA come with challenging sparse matrix structure, that is expensive to generate. In this paper, we show a hybrid parallelization method to reduce the computational cost of the integration phase using hybrid-memory parallel machines. The two-level parallelization includes the partitioning of the computational mesh into sub-domains on the first level (MPI), and loop parallelization on the second level (OpenMP). We show that hybrid parallelization of the integration reduces the contribution of this phase significantly. Thus, alternative algorithms for fast isogeometric integration are not necessary.

Isogeometric analysis for geometric modelling and acoustic attenuation performances of reactive mufflers

In the present study, the acoustic attenuation performances of mufflers are predicted by employing three-dimensional isogeometric analysis (IGA) and multi-patch technique for the first time. Circular, elliptical and conical reactive expansion chamber mufflers are exactly parameterized by non-uniform rational B-splines (NURBS), and the Helmholtz equation governing the interior acoustic field is also solved by NURBS. Compared with traditional finite element method (FEM) which is rooted in Lagrange and Hermite functions, IGA can effectively avoid the time-consuming meshing and totally preserve exact geometrical information during the modelling-analysis process. The computational effectiveness of multi-patch IGA for calculating transmission loss is verified by comparing with experimental approach, boundary element method and other numerical methods from available research articles. The effects of geometry, such as shapes of the expansion cavity and lengths of the extended inlet/outlet are discussed in detail.

Parallel Refined Isogeometric Analysis in 3D

We study three-dimensional isogeometric analysis (IGA) and the solution of the resulting system of linear equations via a direct solver. IGA uses highly continuous $C^{p-1}$Cp-1 basis functions, which provide multiple benefits in terms of stability and convergence properties. However, smooth basis significantly deteriorate the direct solver performance and its parallel scalability. As a partial remedy for this, refined Isogeometric Analysis (rIGA) method improves the sequential execution of direct solvers. The refinement strategy enriches traditional highly-continuous $C^{p-1}$Cp-1 IGA spaces by introducing low-continuity $C^0$C0-hyperplanes along the boundaries of certain pre-defined macro-elements. In this work, we propose a solution strategy for rIGA for parallel distributed memory machines and compare the computational costs of solving rIGA versus IGA discretizations. We verify our estimates with parallel numerical experiments. Results show that the weak parallel scalability of the direct solver improves approximately by a factor of $p^2$p2 when considering rIGA discretizations rather than highly-continuous IGA spaces.

Exact conversion from Bézier tetrahedra to Bézier hexahedra

Modeling and computing of trivariate parametric volumes is an important research topic in the field of three-dimensional isogeometric analysis. In this paper, we propose two kinds of exact conversion approaches from Bézier tetrahedra to Bézier hexahedra with the same degree by reparametrization technique. In the first method, a Bézier tetrahedron is converted into a degenerate Bézier hexahedron, and in the second approach, a non-degenerate Bézier tetrahedron is converted into four non-degenerate Bézier hexahedra. For the proposed methods, explicit formulas are given to compute the control points of the resulting tensor–product Bézier hexahedra. Furthermore, in the second method, we prove that tetrahedral spline solids with Ck-continuity can be converted into a set of tensor–product Bézier volumes with Gk-continuity. The proposed methods can be used for the volumetric data exchange problems between different trivariate spline representations in CAD/CAE. Several experimental results are presented to show the effectiveness of the proposed methods.

A wrapper around parallel MUMPS solver to reduce its memory usage and execution time for finite element method computations

In this paper, we present a wrapper around MUMPS solver, called Hierarchical Solver Wrapper (HSW), that is tailored to domain decomposition-based parallel finite element method computations on distributed memory systems. It offers the same interface as parallel MUMPS with matrix entries in coordinate format provided in a distributed fashion among multiple processors. The algorithm implemented by the wrapper utilizes multiple sequential instances of MUMPS solver to compute Schur complements over subdomains. Next, it deallocates sequential MUMPS solvers and LU factors, and it calls the parallel MUMPS solver feeded with the Schur complements, stored in distributed manner. In the backward substitution stage it recomputes the local LU factors before solving the local problems. The wrapper has been tested with three-dimensional isogeometric analysis computations, and we show it reduces both the memory usage and the execution time, in comparison with a single parallel MUMPS call.

An unsteady 3D Isogeometrical Boundary Element Analysis applied to nonlinear gravity waves

In this paper we describe a three-dimensional Isogeometric Analysis based on the Boundary Element Method (IGA–BEM) in the time domain. We show the capabilities and accuracy of the method for the simulation of non-linear gravity waves. The flow is assumed to be inviscid and irrotational and this leads to a mixed boundary value problem governed by the Laplace’s equation. The Boundary Integral Equation is solved at each time step and the time marching scheme is performed with a fourth-order Runge–Kutta method. The hydrodynamic force is calculated with an auxiliary boundary equation. In the simulations, the analysis suitable-T-spline and NURBS basis are used to approximate both the geometry and the BEM variables in the context of the Bézier extraction framework. The main advantages of this approach are: (1) the control of the continuity and smoothness of the T-spline and NURBS basis, which makes the model numerically stable without the need of artificial smooth techniques; (2) the high geometrical approximation of the non-rational splines; (3) the refinement capabilities without affecting the geometry and BEM variables and (4) and the direct integration with computer aided geometrical design tools. Some numerical benchmark examples are analysed to demonstrate the accuracy and the stability of the method. In addition, we report simulations of waves generated by the movement of submerged foils, which have implications in some wave generator systems installed in surf parks.

Efficient construction of multi-block volumetric spline parameterization by discrete mask method

Isogeometric analysis attempts to unify the mathematical languages in design and analysis to realize the seamless integration of CAD and CAE. In three-dimensional isogeometric analysis, parametric volume is employed as the computational domain for a given set of boundary information. In this paper, we propose a discrete mask method for the efficient construction of multi-block volumetric parameterization based on a set of given boundary spline surfaces. Given the block-partition information of a model, the interior control points can be obtained efficiently by solving a sparse linear system. The existence and uniqueness of the solution for the linear system are also proved. After performing a pre-process on the non-compatible boundary surfaces, the discrete Coons mask is generalized to a unified form, which provides more choices for the construction of inner control points. The proposed method is not only suitable for the multi-block case with C1 continuity, but can also be used for the case in which the boundary surfaces are of different degrees, and with different number of control points and knot vectors. Several examples are presented to illustrate the effectiveness of the proposed method.

High-quality construction of analysis-suitable trivariate NURBS solids by reparameterization methods

High-quality volumetric parameterization of computational domain plays an important role in three-dimensional isogeometric analysis. Reparameterization technique can improve the distribution of isoparametric curves/surfaces without changing the geometry. In this paper, using the reparameterization method, we investigate the high-quality construction of analysis-suitable NURBS volumetric parameterization. Firstly, we introduce the concept of volumetric reparameterization, and propose an optimal Mobius transformation to improve the quality of the isoparametric structure based on a new uniformity metric. Secondly, from given boundary NURBS surfaces, we present a two-stage scheme to construct the analysis-suitable volumetric parameterization: in the first step, uniformity-improved reparameterization is performed on the boundary surfaces to achieve high-quality isoparametric structure without changing the shape; in the second step, from a new variational harmonic metric and the reparameterized boundary surfaces, we construct the optimal inner control points and weights to achieve an analysis-suitable NURBS solid. Several examples with complicated geometry are presented to illustrate the effectiveness of proposed methods.

Isogeometric analysis suitable trivariate NURBS representation of composite panels with a new offset algorithm

Trivariate NURBS (non-uniform rational B-splines) representation of composite panels which is suitable for three-dimensional isogeometric analysis (IGA) is constructed with a new curve/surface offset algorithm. The proposed offset algorithm, which is required by IGA, is non-existent in the CAD literature. Using the presented approach, finite element analysis of composite panels can be performed with the only input being the geometry representation of the composite surface. The method proposed provides a bi-directional system in which one can go forward from CAD to analysis and backwards from analysis to CAD. This is believed to facilitate the design of composite structures. Different parts (patches) can be parametrized independently of each other and glued together, in the finite element solver, by a discontinuous Galerkin method. A stress analysis of curved composite panel with stiffeners is provided to demonstrate the proposed framework.