This paper presents an isogeometric boundary element method (IGABEM) for transient heat conduction analysis. The Non-Uniform Rational B-spline (NURBS) basis functions, which are used to construct the geometry of the structures, are employed to discretize the physical unknowns in the boundary integral formulations of the governing equations. Bézier extraction technique is employed to accelerate the evaluation of NURBS basis functions. We adopt a radial integration method to address the additional domain integrals. The numerical examples demonstrate the advantage of IGABEM in dimension reduction and the seamless connection between CAD and numerical analysis.

Isogeometric analysis (IGA) has drawn extensive attentions in engineering science community since the seminal works of Hughes et al. [

Although the concept of IGA was originally proposed in the context of finite element methods, its application in engineering practice is restricted because FEM relies on volume parameterization which conflicts with the boundary representation in CAD field. Mierzwiczak et al. [

In this paper, we are aimed to extend the application of IGABEM to heat conduction problems which are an increasingly important topic in many engineering fields. The radial integration method proposed by Gao [

For the sake of completeness, the fundamentals of NURBS are briefed in this section. NURBS are generalized from B-splines, whose basis functions are defined over a knot vector which is a monotonic increasing real number sequence, denoted by _{i}

where

The derivative expression of the basis function can be derived as follows:

The B-spline curve is described by the linear combination of NURBS basis functions and the corresponding control points,

where

Non-uniform rational B-spline (NURBS) basis functions are obtained by introducing the weight coefficient

where

Similar to B-splines, the NURBS curve is expressed by:

Since the NURBS basis function adopts the rational form composed of the B-spline basis function and the weight factor

The computation of IGABEM can be accelerated using Bézier extraction. The idea is to extract linear operators and map the Bernstein polynomial basis functions of Bézier elements to global B-spline basis functions. The linear transformation is defined by a matrix called an extraction operator. The transpose of the extraction operator maps the control points of the global B-spline to that of Bézier curves. The process of Bézier extraction operation is proposed in this work, also see [

To perform Bézier decomposition, all internal knots of a knot vector are repeated until their multiplicity equals

where

After the knot is inserted, it brings about a change in the control point. Let

Then, the corresponding control points can be calculated by:

We rewrite

To decompose a B-spline curve defined in the knot vector

Using

Since

It should be noted that only the knot vector determines the C matrix. In other words, the extraction operator is a product of parameterization and does not depend on control points or basis functions. Therefore, we can apply the extraction operator directly to NURBS. Define weights as diagonal matrices

Converting

Using

The NURBS curve is expressed as:

In

The expression of the Bézier control points can refer to the method of homogeneous coordinates:

Both sides are multiplied by

After solving the weight factor ^{b} after the Bézier operation, we arrive at the final Bézier representation of the NURBS:

Thus, we can express NURBS equivalently with a series of ^{0} Bézier elements.

The governing partial differential equation of transient heat conduction in isotropic medium with internal heat source is expressed by:

where

The following three kinds of boundary conditions are considered:

where

The transient problem is time-dependent, so the initial conditions need to be prescribed:
_{0} indicates initial temperature on the domain.

Through weighted residual operation and partial integration operation,

where the coefficient

where

In this work, we do not consider the source. Thus,

By observing

We firstly approximate unknown temperature gradient

where _{i}

It is worth noting that collocation points can be boundary points or internal points. Generally, combining boundary points and internal points to form collocation points can effectively improve the approximation accuracy. In addition, the accuracy of approximation can be further improved by combining radial basis functions and polynomials, which can be expressed in combination with quadratic polynomials,

where

It is particularly important to determine these coefficients

where

and

where ^{ij} = _{j}

When the collocation point is not repeated, the matrix

By substituting

where

and

It can be found that the domain integral in

It is worth noting that these coefficients in _{0}, _{1}, and _{2} directly through the boundary integral calculation. In fact, these coefficients do not need to be calculated directly.

When the source point _{0}, _{1}, and _{2} can be rewritten as a formulas of vector and vector multiplication, respectively. By combining the three terms, we can get the following formulas

where the vector

where

First, we discretize the boundary into _{e}

NURBS basis functions are used for approximation. In any NURBS element

where ^{l} are normalized temperature and its flux at the

By substituting

where

After collecting the equations for all collocation points and expressing them in matrix forms, one can obtain the following system of linear algebraic equations:

In this work, the finite difference method is used for solution of the above time domain equations.

It is worth noting that weakly singular integrals exist in

_{0} to _{m}

Herein, _{i}_{i+1}. When the time marching method is used to solve the equations, we can obtain the derivative of temperature with respect to time, as follows:

where the time step

where

Thus,

After merging the similar terms, we can get the following expression:

where

When solving the unknown temperature _{i+1}, the temperature _{i}

where

Consider a two-dimensional flat plate, the initial temperature is 100 K, the upper and lower boundaries are adiabatic, the left boundary temperature is maintained at 100 K. Convective boundary condition is applied to the right boundary, and the ambient temperature is 400 K, as shown in the ^{3}, specific heat capacity _{x}

Consider an elliptical plate with a long radius of 2 m and a short radius of 1 m. The initial temperature is 100 K, and the ambient temperature of the plate boundary is 400 K. Material density ^{3}, specific heat capacity _{x}

The shape of ellipse and the distribution of internal points are shown in

In this work, the finite difference method is used to solve the time-domain equation. Therefore, we first investigate the influence of time step on the calculation results, as shown in

NSTEP | NODE 1 | NODE 2 | NODE 3 | NODE 4 | NODE 5 | NODE 6 | NODE 7 | NODE 8 | NODE 9 |
---|---|---|---|---|---|---|---|---|---|

( −1.6, 0) | ( −1.2, 0) | ( −0.8, 0) | ( −0.4, 0) | (0, 0) | (0.4, 0) | (0.8, 0) | (1.2, 0) | (1.6, 0) | |

5 | 136.156 | 117.783 | 109.520 | 106.530 | 105.821 | 106.535 | 109.522 | 117.784 | 136.159 |

6 | 135.715 | 117.454 | 109.308 | 106.386 | 105.691 | 106.386 | 109.309 | 117.456 | 135.717 |

7 | 135.273 | 117.127 | 109.096 | 106.238 | 105.562 | 106.238 | 109.097 | 117.128 | 135.275 |

8 | 136.043 | 117.776 | 109.588 | 106.621 | 105.908 | 106.621 | 109.588 | 117.778 | 136.045 |

9 | 135.603 | 117.450 | 109.376 | 106.472 | 105.778 | 106.472 | 109.377 | 117.452 | 135.605 |

10 | 135.969 | 117.773 | 109.632 | 106.678 | 105.966 | 106.678 | 109.633 | 117.775 | 135.971 |

NSTEP | NODE 1 | NODE 2 | NODE 3 | NODE 4 | NODE 5 | NODE 6 | NODE 7 | NODE 8 | NODE 9 |
---|---|---|---|---|---|---|---|---|---|

( −1.6, 0) | ( −1.2, 0) | ( −0.8, 0) | ( −0.4, 0) | (0, 0) | (0.4, 0) | (0.8, 0) | (1.2, 0) | (1.6, 0) | |

5 | 170.298 | 148.477 | 134.829 | 127.678 | 125.473 | 127.678 | 134.830 | 148.479 | 170.300 |

6 | 169.674 | 147.887 | 134.314 | 127.229 | 125.049 | 127.229 | 134.316 | 147.889 | 169.678 |

7 | 170.076 | 148.287 | 134.688 | 127.575 | 125.384 | 127.575 | 134.690 | 148.289 | 170.080 |

8 | 170.185 | 148.404 | 134.808 | 127.694 | 125.502 | 127.694 | 134.809 | 148.407 | 170.187 |

9 | 169.559 | 147.815 | 134.294 | 127.246 | 125.079 | 127.247 | 134.295 | 147.817 | 169.563 |

10 | 170.108 | 148.356 | 134.794 | 127.705 | 125.521 | 127.705 | 134.795 | 148.358 | 170.112 |

In this work, we use the radial basis method to approximate the temperature gradient. Therefore, it is significant to investigate the influence of different types of radial basis functions on the numerical results. Herein, eight different types of radial basis functions are given, as shown in

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

(1+^{−1} |
^{−R} |
^{−R2} |

NTYP | NODE 1 | NODE 2 | NODE 3 | NODE 4 | NODE 5 | NODE 6 | NODE 7 | NODE 8 | NODE 9 |
---|---|---|---|---|---|---|---|---|---|

( −1.6, 0) | ( −1.2, 0) | ( −0.8, 0) | ( −0.4, 0) | (0, 0) | (0.4, 0) | (0.8, 0) | (1.2, 0) | (1.6, 0) | |

1 | 136.417 | 117.892 | 109.492 | 106.436 | 105.70 | 106.436 | 109.491 | 117.892 | 136.416 |

2 | 136.541 | 118.114 | 109.663 | 106.374 | 105.545 | 106.370 | 109.648 | 118.109 | 136.559 |

3 | 136.156 | 117.783 | 109.521 | 106.535 | 105.82 | 106.535 | 109.522 | 117.784 | 136.159 |

4 | 135.574 | 117.337 | 109.233 | 106.515 | 105.737 | 106.515 | 109.232 | 117.338 | 135.575 |

5 | 135.595 | 117.413 | 109.232 | 106.419 | 105.539 | 106.419 | 109.232 | 117.414 | 135.595 |

6 | 135.991 | 117.575 | 109.300 | 106.550 | 105.903 | 106.549 | 109.300 | 117.575 | 135.992 |

7 | 135.878 | 117.538 | 109.341 | 106.606 | 105.948 | 106.607 | 109.341 | 117.537 | 135.877 |

8 | 135.530 | 117.347 | 109.292 | 106.615 | 105.954 | 106.615 | 109.291 | 117.347 | 135.529 |

NTYP | NODE 1 | NODE 2 | NODE 3 | NODE 4 | NODE 5 | NODE 6 | NODE 7 | NODE 8 | NODE 9 |
---|---|---|---|---|---|---|---|---|---|

( −1.6, 0) | ( −1.2, 0) | ( −0.8, 0) | ( −0.4, 0) | (0, 0) | (0.4, 0) | (0.8, 0) | (1.2, 0) | (1.6, 0) | |

1 | 170.288 | 148.415 | 134.783 | 127.719 | 125.568 | 127.719 | 134.783 | 148.415 | 170.288 |

2 | 170.101 | 148.329 | 134.821 | 127.816 | 125.700 | 127.813 | 134.812 | 148.319 | 170.107 |

3 | 170.298 | 148.477 | 134.829 | 127.678 | 125.473 | 127.678 | 134.830 | 148.479 | 170.301 |

4 | 170.851 | 148.993 | 134.882 | 127.143 | 124.591 | 127.143 | 134.882 | 148.993 | 170.851 |

5 | 171.047 | 149.173 | 134.912 | 126.998 | 124.356 | 126.998 | 134.913 | 149.173 | 171.050 |

6 | 170.625 | 148.741 | 134.815 | 127.396 | 125.048 | 127.395 | 134.816 | 148.741 | 170.625 |

7 | 170.633 | 148.781 | 134.852 | 127.383 | 125.004 | 127.384 | 134.852 | 148.781 | 170.632 |

8 | 170.524 | 148.743 | 134.853 | 127.370 | 124.965 | 127.370 | 134.853 | 148.744 | 170.524 |

In this section, we consider an example of octagonal leaf, as shown in

Similar to the previous example, we investigate the influence of different radial basis functions on the calculation results, as shown in

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

( −0.6, 0) | ( −0.2, 0) | (0, 0) | (0.4, 0) | (0, 0.5) | (0, −0.3) | (0, −0.7) | |

1 | 135.010 | 111.721 | 109.027 | 119.731 | 126.151 | 115.106 | 145.490 |

2 | 135.254 | 112.784 | 110.301 | 120.490 | 126.821 | 115.988 | 145.524 |

3 | 135.570 | 112.847 | 110.281 | 120.632 | 126.975 | 116.130 | 145.897 |

4 | 135.335 | 111.355 | 108.550 | 119.775 | 126.252 | 114.895 | 146.002 |

5 | 135.087 | 109.657 | 106.608 | 118.714 | 125.596 | 113.462 | 146.163 |

6 | 134.604 | 108.323 | 105.146 | 117.715 | 124.901 | 112.269 | 145.893 |

7 | 134.215 | 107.510 | 104.270 | 117.068 | 124.414 | 111.519 | 145.591 |

8 | 134.725 | 109.683 | 106.691 | 118.533 | 125.347 | 113.404 | 145.643 |

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

( −0.6, 0) | ( −0.2, 0) | (0, 0) | (0.4, 0) | (0, 0.5) | (0, −0.3) | (0, −0.7) | |

1 | 197.535 | 168.822 | 165.221 | 179.613 | 187.604 | 173.319 | 209.127 |

2 | 197.461 | 169.004 | 165.495 | 179.734 | 187.701 | 173.441 | 209.002 |

3 | 197.869 | 169.333 | 165.777 | 180.052 | 188.015 | 173.806 | 209.420 |

4 | 198.077 | 169.287 | 165.685 | 180.174 | 188.180 | 173.813 | 209.715 |

5 | 198.025 | 169.040 | 165.398 | 179.983 | 188.084 | 173.606 | 209.729 |

6 | 197.647 | 168.557 | 164.890 | 179.494 | 187.650 | 173.136 | 209.365 |

7 | 197.342 | 168.207 | 164.522 | 179.133 | 187.315 | 172.786 | 209.064 |

8 | 197.590 | 168.590 | 164.942 | 179.512 | 187.604 | 173.151 | 209.280 |

The influence of time step parameters on the calculation accuracy can be found in

NSTEP | NODE 1 | NODE 2 | NODE 3 | NODE 4 | NODE 5 | NODE 6 | NODE 7 |
---|---|---|---|---|---|---|---|

( −0.6, 0) | ( −0.2, 0) | (0, 0) | (0.4, 0) | (0, 0.5) | (0, −0.3) | (0, −0.7) | |

5 | 135.335 | 111.355 | 108.550 | 119.775 | 126.252 | 114.895 | 146.002 |

6 | 134.721 | 111.012 | 108.241 | 119.303 | 125.704 | 114.502 | 145.315 |

7 | 134.112 | 110.675 | 107.943 | 118.834 | 125.159 | 114.114 | 144.629 |

8 | 135.381 | 111.702 | 108.934 | 119.978 | 126.376 | 115.187 | 145.960 |

9 | 134.771 | 111.361 | 108.630 | 119.507 | 125.831 | 114.795 | 145.276 |

10 | 135.413 | 111.930 | 109.187 | 120.112 | 126.459 | 115.379 | 145.935 |

NSTEP | NODE 1 | NODE 2 | NODE 3 | NODE 4 | NODE 5 | NODE 6 | NODE 7 |
---|---|---|---|---|---|---|---|

( −0.6, 0) | ( −0.2, 0) | (0, 0) | (0.4, 0) | (0, 0.5) | (0, −0.3) | (0, −0.7) | |

5 | 198.077 | 169.287 | 165.685 | 180.174 | 188.180 | 173.813 | 209.715 |

6 | 196.833 | 168.046 | 164.445 | 178.926 | 186.929 | 172.568 | 208.479 |

7 | 197.713 | 168.968 | 165.370 | 179.833 | 187.825 | 173.484 | 209.341 |

8 | 197.985 | 169.272 | 165.677 | 180.124 | 188.108 | 173.783 | 209.602 |

9 | 196.744 | 168.037 | 164.444 | 178.881 | 186.859 | 172.544 | 208.367 |

10 | 197.925 | 169.263 | 165.676 | 180.093 | 188.060 | 173.764 | 209.527 |

This paper applied IGABEM to the heat conduction problem of temperature field. IGABEM can accurately represent the geometric model. In addition to the advantages of dimensionality reduction, IGABEM can also realize the seamless integration of CAD modeling and numerical analysis. The use of Bézier extraction technique further simplifies the implementation of IGA. This method transformed the basis functions of NURBS into Bernstein polynomials and maintained the consistency of geometric models. The main feature of this method was that the iterative calculation of NURBS basis function was avoided in the process of element physical interpolation calculation of BEM, which can effectively improve the computational efficiency. The domain integral term was successfully converted to the boundary integral in IGABEM with radial integration method. The present method provides a powerful tool for fast and high fidelity simulation of transient heat conduction problems commonly encountered in numerous industrial sectors.