A locally refined adaptive isogeometric analysis for steady-state heat conduction problems

https://doi.org/10.1016/j.enganabound.2020.05.005Get rights and content

Highlights

  • An adaptive isogeometric analysis is developed to model steady-state heat conduction in 2D solid.

  • A temperature gradient recovery-based posteriori error estimator is evaluated to drive the local mesh refinement.

  • Adaptive local mesh refinement technique increases the accuracy and reduces the computational time.

  • Simulated results manifest the efficiency and accuracy of the developed method.

Abstract

This paper presents an isogeometric analysis with adaptivity using locally refined B-splines (LR B-splines) for steady-state heat conduction simulations in solids. Within this framework, the LR B-splines, which have an efficient and simple local refinement algorithm, are used to represent the geometry, and are also employed for spatial discretization, thus providing a seamless interaction between the CAD models and the numerical analysis. A Zienkiewicz-Zhu a posteriori error estimator in terms of the temperature gradient recovery is used to identify the regions for local mesh refinement. The accuracy and convergence properties of the proposed framework are demonstrated through several two-dimensional isotropic examples. Numerical results indicate good performance of the present method as the adaptive refinement technique yields more accurate solution compared to the uniform global refinement with an improved convergence rate.

Introduction

Thermal conduction is an important problem in practical engineering, hence it has been a crucial research topic in scientific and industrial communities. Numerical simulation can be an effective tool for solving the general thermal conduction problems. In the past decades, numerous numerical methods have been introduced to solve such thermal conduction problems, such as finite element method (FEM) [1], boundary element method (BEM) [2], meshless method [3], Lattice-Boltzmann method [4], and singular boundary method (SBM) [5].

Among the aforementioned methods, the FEM is widely applied in scientific and industrial communities. It relies on discretizing the domain into non-overlapping regions and generally employing a polynomial representation for approximation of the unknown field. These polynomials are called ‘approximation functions’ or ‘basis functions’ and are typically, Co continuous, and the mesh generation process not only is time consuming, but also leads to discretization error for complex shape structures. Hence, there is still a growing interest to develop new advanced numerical methods. Moreover, in the numerical simulation, the accuracy of solving the problem depends largely on the size, shape and the location of the element. A more accurate solution can be obtained by reducing the element sizes, but it will increase the computation efforts. The efficient and economic solution of problems is not merely dependent on the number of elements but also on their location. In order to improve the computational efficiency and accuracy, the fine elements are required in the regions with large gradients (e.g. region of high heat flux), while comparatively coarser elements may be used in the quiescent regions. The region of fine elements is determined by using an error analysis. The adaptive procedure can yield an optimal mesh. Huang et al. [6] developed an error estimation and adaptive technique for linear steady heat transfer problems. Based on the Zienkiewicz-Zhu recovery strategy, the smoothed temperature gradients are obtained, and an error norm is defined with the heat dissipation. Lewis et al. [7] later extended the adaptive technique to the nonlinear transient heat transfer problems. A novel self-adaptive approach based on the nested grid method is reported by Wilson et al. [8]. They show that the adaptively can handle spatial and temporal scales for thermal modeling of high-performance integrated circuits.

The isogeometric analysis (IGA) [9] facilitates a seamless integration of the CAD models and analysis as its underlying idea is to employ the basis functions used to represent the geometry to describe the unknown fields. Some of the salient features of the IGA include, for instance, accurate representation of the geometry, higher-order continuity, and more importantly, it circumvents the traditional mesh generation procedure. A successful application of the IGA to various engineering problems has been reported in the literature, such as two/three dimensional elasticity, plates and shells, vibration, topology optimization, fluid flow, acoustics, to name a few, see Refs. [10], [11], [12], [13], [14], [15], [16], [17]. For diffusion problems, Vuong et al. [18] presented a tutorial 2D MATLAB computer code for elliptic diffusion-type problems using the IGA, and explained the basic steps of IGA. Anders et al. [19] used the IGA for numerically investigating the effect of thermal diffusion on microstructural evolution of the binary polymer blend consisting of poly(dimethylsiloxane) and poly (ethyl-methylsiloxane) under impact loading. An et al. [20] reported the results dealing with the 2D steady-state thermal conduction problems using the IGABEM (isogeometric boundary element method), which possesses the excellent features of the common boundary representation of both the BEM and IGA. Recently, Lai et al., [21], by exploiting the user defined elements in Abaqus, developed an elegant platform to link IGES files and the Abaqus software.

In IGA, the non-uniform rational B-splines (NURBS) are extensively applied. However, conventional NURBS lack the ability of local refinement because of the global tensor products. In other words, the adaptivity analysis could not be conducted using the NURBS-based IGA. To overcome such limitations of the NURBS in adaptivity analysis, some other splines with local refinement ability have been introduced including T-splines [22], truncated hierarchical B-splines [23], polynomial splines over hierarchical T-meshes  [24], rational splines over hierarchical T-meshes [25], analysis-suitable++ T-splines [26] and locally refined B-splines [27], [28]. Bekele et al. [29] first solved steady-state groundwater flow using the adaptive IGA with locally refined B-splines (LR B-splines). Recently, the present authors have successfully developed the adaptive IGA with LR B-splines and its applications to simulating several engineering problems, for instance, 2D elasticity [30], inclusions [31], holes in orthotropic media [32], and cracks [33], [34].

To the best knowledge of the authors, studies concerning the heat conduction using an adaptive IGA have not been reported in the literature. Our previous work focused on the mechanical behavior of continuous and discontinuous elasticities, and obtained results have shown the effectiveness and accuracy of the adaptive IGA based on LR B-splines. The objective of this work is to extend the approach to model the two dimensional steady-state heat conduction problems. The main difference between this study and our previous work lies in the posteriori error estimation. In addition, although we call the LR B-splines, the NURBS are used to describe the complicated structure such as the curved pipe in the last example. The present formulation utilizes the LR B-splines basis for the spatial discretization and the temperature field approximation. Based on the Zienkiewicz-Zhu recovery technique [35], the smoothed temperature gradient field is computed to evaluate a posteriori error estimator for local refinement. The salient features of the proposed framework are: (a) the geometry can be accurately reproduced; (b) only the required domain is locally refined; (c) a smooth temperature gradient distribution can be produced; (d) accuracy and computational efficiency can be considered at the same time; and (e) high accuracy and convergence rate can be obtained.

After the introduction, we present the fundamental equations for steady-state heat conduction in Section 2. Subsequently, we describe the adaptive IGA based on the LR B-splines for modeling 2D steady-state heat conduction in Section 3. Then, we demonstrate the accuracy and performance of the present method in Section 4 through several numerical examples. We finally end with some major concluding remarks in Section 5.

Section snippets

Governing equations

Consider an isotropic solid occupying ΩR2, bounded by a surface Γ with an outward normal n. Let the boundary accommodate the following decompositions: Γ=Γ1Γ2Γ3 with Γ1Γ2Γ3=, where Dirichlet, Newmann and convection boundary conditions are specified on Γ1, Γ2 and Γ3, respectively. Let T:ΩR2 be the temperature field at a point x of the body. The strong form of the equation is: find the temperature field TR2, such that·(kT)+qv=0where k, T and qv are the thermal conductivity parameter, the

A brief introduction to LR B-splines

In 2013, Dokken et al. [27] developed LR B-splines function, which is a weighted B-splines function. LR B-splines have the ability of local refinement and are suitable for analysis. Here, we give a brief introduction to LR B-splines. Through embedding some level and upright lines, a rectangular region is partitioned into some smaller rectangles which are called as a box mesh. An LR mesh is a box mesh, which results from a series of single line insertions from an initial tensor, and each

Numerical experiments and discussion

Five numerical examples with different configurations are particularly considered and their computed results are then given to show the accuracy and effectiveness of the developed method. Unless mentioned otherwise, cubic bivariate LR B-splines basis functions are adopted. In all examples, the refinement parameter is taken as β=20% [30], and the coefficient of heat conduction is taken as k=1 W/moC.

The first four examples whose analytical solutions are available are considered as benchmark,

Conclusions

In this study, we proposed an effective computational method, through the adaptive IGA with the LR B-splines basis functions, for modeling steady-state heat conduction problems. This class of methods has been found to be particularly advantageous when, for instance, problems with highly localized regions are considered. Owing to the versatile and flexible local refinement strategy, the LR B-splines are used in association with the adaptive IGA. The adaptive local refinement is here guided by

Declaration of Competing Interest

The authors declare that there are no conflict of interest.

Acknowledgment

This work is supported by the National Natural Science Foundation of China (Grant Nos.11932006 and 11972146).The financial supports are gratefully acknowledged.

References (37)

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    Based on the above advantages, IGABEM has been widely applied in recent years. Examples of such works are the acoustic field analysis [25,26], the fracture and crack growth [27,28], the potential problem [29,30], the shape optimization [31,32], the topology optimization [33] and the defect identification [34]. Nevertheless, due to the limitation of the fundamental solution, IGABEM mainly solves steady-state problems in homogeneous materials.

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