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.

Since the pioneering work of Hughes et al. [

The concept of IGA was first proposed in the context of finite element methods (FEM). Its application in engineering practice for complex geometries is limited due to the volume parameterization of FEM conflicts with the boundary representation of CAD. On the contrary, the boundary element method (BEM) only needs boundary parameterization and thus is naturally compatible with CAD model [

Heat conduction is of paramount importance in engineering. IGABEM based on NURBS was applied to 3D steady thermal conduction problems for a medium with non-homogenous inclusions [

The remainder of the paper is organized as follows. The Catmull–Clark subdivision surface method is introduced in Section 2. The IGABEM with subdivision surfaces is used for solution of heat conduction with variable coefficient in Section 3. Then, several numerical examples are discussed in Section 4, followed by the conclusion in Section 5.

Subdivision surface is defined as the limit of an infinite refinement or subdivision process. Each new subdivision step produces a new mesh with more polygonal elements and smoother geometries. By repeatedly refining the initial polygon mesh, a series of meshes that approach the final subdivision surface is generated. The subdivision surface inherits the reconfigurability of spline curve, so that all control meshes generated in subdivision process can accurately represent the same spline surface. An important technical advance in subdivision surfaces is to generate limiting surfaces by constructing an elementwise mapping from the subdivided meshes. In this way, the subdivided meshes play the same role as control grids of NURBS in geometry modeling, and the limiting surface resembles the physical surface in NURBS.

In this work we consider the quadrilateral subdivision surfaces proposed by Catmull–Clark. The Catmull–Clark subdivision scheme is an extension of bicubic uniform B-splines. Catmull–Clark subdivision algorithm subdivides each quadrilateral element into four sub-quadrilateral elements (1–4 splitting) as shown in

Let us set the original mesh subdivision level as _{v} of this vertex. If _{v} = 4, the vertex is called a regular vertex as shown in _{v} of Point 1.

A quadrilateral mesh can be divided into four sub-elements by inserting a new point in the middle of each edge and inside the element, which are called “E” and “F”, respectively. In addition, the position of original points called “V” is moved to construct smooth surface flexibly.

The coordinates of these vertices “V”, “E” and “F” on the

where

The idea of limit subdivision is used to subdivide the irregular elements locally and repeatedly through the following steps: (1) introducing subdivision matrices to establish the relationship between the hierarchical sub units and the parent units, (2) obtaining the eigenvector and eigenvalue of the matrix by Fourier transformation of the subdivision matrix, (3) storing the information of the hierarchical regular sub units by using the subdivision matrix, the picking matrix and the regular unit, and (4) building the base function of the original irregular parent unit 3. After constructing the basis function of the irregular element, the surface model corresponding to the whole mesh can be constructed by the fitting operation, and the model has _{2} continuity at the regular point and _{1} continuity at the irregular point.

In each regular quadrilateral there are sixteen quartic non-zero basis functions associated with its and
neighbouring quadrilateral vertices, as shown in ^{e} within the
quadrilateral

where _{i} denotes the coordinate of the _{1}, θ_{2}) represents a bicubic B-spline basis function, please see [

As shown in ^{ir} for quadrilateral patches that contain irregular vertices, as follows:

where ^{ir}(θ_{1}, θ_{2}) denotes the basis function for irregular elements, please see [

The governing differential equation of steady-state heat conduction in isotropic media with variable coefficient and internal heat source is expressed as follows:

where

where Γ_{T} represents a constant temperature boundary, Γ_{q} represents a constant temperature flux boundary, and

By introducing weight function

For the first term, by using the integral by parts method and Gauss divergence theorem, we can get the following formula:

The fundamental solution

and

By substituting

where the coefficient

where

By observing

The second domain integral in

where

where _{i}(^{i} is the coordinates of the

The commonly used radial basis functions are shown in

NTYP | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|

R | (1 + ^{−1} |
^{–R} |

How to determine the coefficients β and γ is particularly important. It is difficult to obtain these coefficients directly. However, through a series of transformation operations, we can get the implicit expression of these coefficients. When the point

where

and

where ^{ij} = _{j} (^{i} − ^{j}|, and

It is worth noting that the matrix

Upon substitution of

in which

By substituting

where

We can find that the domain integral in

The boundary of domain is defined through the union of discretization elements, as follows:

In this work, the set of discretization elements

where the coefficients

where _{b} is a known column vector formed by domain integral caused by heat source.

By substituting boundary conditions into

The collocation approach is used to generate the linear system of equations of BEM. In this work, the collocation points are chosen as a series of points on the physical surface mapped from the nodes of the control mesh, or see [

In this section, we consider a cube with cylindrical hole. Catmull–Clark subdivision surface scheme is used to construct a smooth model. The initial model is shown in _{f} = 293 K, and the convective heat transfer coefficient h = 1000 W/(m^{2}· K) between the fluid and structure. The right surface of the structure with the maximum coordinate value in positive y-axis is subjected to constant heat flux q = 50000 W/m^{2}. The other five surfaces are adiabatic. Material density ρ = 7800 kg/m^{3}, specific heat capacity c_{p} = 460 J/(kg · K), and thermal conductivity k = 500 W/(m · K).

Since there is no analytical solution to the problem, we compare the results of this algorithm proposed in this work with that of Fluent. In

We consider a problem with variable heat conduction coefficients. The thermal conductivity of the material is not a constant value, but satisfies

The value of the temperature at the computing points in

Consider a dumbbell model shown in

The top surface is subjected to constant heat flux ^{2}. The bottom of the structure contacts a heat source with _{f}^{2} · K). The other surfaces are adiabatic. Herein, material density is ρ = 7800 kg/m^{3}, heat capacity ratio is _{p}

As shown in

Z-axis coordinates | 0.01 | 0.04 | 0.07 | 0.10 | 0.13 | 0.16 | 0.19 | 0.22 | 0.25 |
---|---|---|---|---|---|---|---|---|---|

Constant | 402.29 | 406.27 | 411.49 | 420.47 | 432.11 | 444.17 | 456.23 | 467.87 | 476.83 |

r = 0.01 | 402.35 | 406.77 | 411.94 | 421.12 | 432.95 | 445.14 | 457.12 | 468.75 | 477.52 |

In this work, we present a novel framework for solving three-dimensional steady-state heat conduction problems with IGABEM. The Catmull–Clark subdivision surface method is used to construct a smooth geometric model and discretizing boundary integral equations. The radial integration method is incorporated to transform the domain integral caused by variable coefficient into the boundary integral. The numerical simulation can be conducted directly from CAD model without needing a meshing procedure. The numerical results validate the accuracy of the method and demonstrate its ability of integrating CAD and numerical analysis. In the future, we will extend the algorithm to transient heat conduction problems.