# A design of an optimal shape of domain described by NURBS curves using the topological derivative and boundary element method

### Katarzyna Freus

,### Sebastian Freus

Journal of Applied Mathematics and Computational Mechanics |
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A DESIGN OF AN OPTIMAL SHAPE OF DOMAIN DESCRIBED BY NURBS CURVES USING THE TOPOLOGICAL DERIVATIVE AND BOUNDARY ELEMENT METHOD

Katarzyna Freus^{1},
Sebastian Freus^{2}

^{1}Institute
of Mathematics, Czestochowa University of Technology, Poland

^{2} Institute of Computer and Information Science, Czestochowa
University of Technology

Czestochowa, Poland

katarzyna.freus@im.pcz.pl, sebastian.freus@icis.pcz.pl

Received: 30 April 2017; accepted: 17 May 2017

**Abstract.** The aim of this paper is to create
an optimal shape of the 2D domain that is
described by the Non-Uniform Rational B-Splines (NURBS) curves. This work
presents
a method based on the topological derivative for the Laplace equation that
determines the sensitivity of a given cost function to the change of its
topology. As a numerical approach, the boundary element method is considered. To check the effectiveness of the
proposed
approach, the example of computations was carried out.

*MSC 2010:** 35J25, 65N38, 49K10, 49Q10, 53A04*

*Keywords:** Laplace equation,
boundary element method, topological derivative, NURBS curves*

1. Introduction

Topological optimization is a mathematical
method that allows one to find an
optimal material layout of a domain, such that a cost function gives its
optimum
value after optimization under given constraints. Material is removed by
creating
a small hole that appears in the optimization process. The topological
derivative (*D _{T}*) indicates the position in the domain of
interest where a hole should be formed. Wherever

*D*

_{T}_{ }is low enough, a hole is created. In the opening, the Neumann condition is taken into account. The topological-shape sensitivity method is utilized as a procedure to calculate the topological derivative taking the total potential energy as a cost function [1-4]. The boundary of the domain is described by the NURBS curves which are commonly used for representing and designing a shape in numerical implementation. In order to determine the temperature field of the domain considered, the boundary element method (BEM) is used in its direct version. In this paper, firstly a review of the BEM and the NURBS curves are presented. After that, the topological derivative for the Laplace equation is introduced and the numerical results are shown. Finally, conclusions are expressed.

2. Boundary element method

The domain W bounded by contour
Γ_{ }is taken into account. The Laplace equation
supplemented by the boundary condition is the following [4, 5]

(1)

where *x *= (*x*_{1}, *x*_{2}) are the
spatial coordinates, λ [W/mK] is the
thermal conductivity, *T** _{ }*(

*x*) denotes the temperature,

*∂*

*T*

*/∂*

_{ }*n*is the normal derivative,

*n*= [cosa

_{1}, cosa

_{2}] is the normal outward vector.

*T*

_{b}_{ }and

*q*are known as the boundary temperature and heat flux, respectively.

_{b}*T*

_{¥}is the ambient temperature and

*α*[W/m

^{2}K] is the heat transfer coefficient. The boundary integral equation for problem (1) is the following

(2)

where is the coefficient connected with the local shape of a boundary, x is the observation point and is the heat flux. and are the following

, (3)

where *r* indicates the distance between x = (x_{1},
x_{2}) and *x *=
(*x*_{1}, *x*_{2})

(4)

while

(5)

*n _{x}*,

*n*are the directional cosines of the normal outward vector

_{y}**n**.

Using the linear boundary elements, Equation (2) can be given in the form

| (6) |

Considering the known boundary condition, Eq. (6) can be written

(7)

where **A** is the
main matrix, **X** is the unknown vector and **B** is the free terms
vector. Equation (7) ensures the determination of the missing boundary
conditions.

3. NURBS curves

In the numerical example, we take into
account the domain Ω where the segments of the boundary are described by
the NURBS curves so, in this part of paper, we present the main information
about these curves. A *n*-th degree NURBS curve is as follows

(8) |

where *w _{j}* are the weights,

**P**

*are the control points forming a control polygon and*

_{j}*N*(

_{j,n }*t*) are the B-spline basis functions

(9) |

prescribed for the set of nodes

(10) |

at the same time the values *a* and *b* appear *n *+1
times. It should be mentioned that the number of control points equals *r *+1
and corresponds to the number of nonzero basis functions. Details about the NURBS curves can be found
in [6].

4. Topological derivative

In this work, the topological derivative for
the Laplace equation is considered. In the inside of the original domain W, a small hole
of radius is created. The idea of the topological
derivative is based on determining the sensitivity of a given cost function (total
potential energy) when the size of this hole is changed. The local value of the
*D _{T}* is defined as follows [1-4]

(11)

where and are
the cost functions calculated for the original W and the new domain , respectively, and *f* is a
regularizing function. There
are several papers in the area of the *D _{T}* for the steady state heat transfer. In this work, we adopt the definition called the topological - shape
sensitivity method proposed in [1], which is based on the following formula

(12)

where is a small perturbation on the radius of the hole.

5. Problem formulation

Let be the domain with a small hole. The Laplace equation supplemented by the boundary conditions is taken into account [1-4]

(13)

where on the holes H* _{e}* created via

*D*, the Neumann boundary condition is prescribed. Problem (13) can be written in the variational form with a test function Find such that

_{T}(14)

For the perturbed configuration, expression (14) is the following

(15) |

Using the total potential energy [1], the cost function for problem (13) has the form

(16)

The optimization problem can be expressed as the minimization of equation (16) with the variational formulation (Eq. (14) and (15)) as constraints. All these equations are used to obtain topological derivative (12). After some mathematical manipulations, the final expression for the topological derivative is the following

(17)

It is important to mention that *T* is
the solution of the original problem (without
a hole). Details of the
calculation of *D _{T}* are described in [1-3].

In this work, the gradient is obtained by differentiating the integral equation

(18)

with respect to the internal points, so

(19)

where and are given by expressions (3).

In order to obtain an optimal shape of the
domain, we perform the iterative procedure that is
carried out in some steps. First, the initial domain described by the NURBS
curves and the stopping criterion are provided. Then, using the BEM, the
temperature field is obtained. Next, the topological derivative at the boundary nodes is calculated by means of expressions (17). The point with the lowest absolute value of
*D _{T}* is chosen. On the selected point a hexagonal hole is created
(Fig. 1).

It is assumed that the side length of a
regular hexagon _{ }is approximately equal to the length of the
boundary element (i.e. the arithmetic average of the
length of all the boundary elements). So, a radius *r* of the hole can be
changed by increasing or decreasing the numbers of the boundary and internal
nodes. Generally the radius is a fraction of a
dimension of the original domain (this means where
*d *= min(*height,width*)) [3].

Fig. 1. Hexagonal hole

On the hole the Neumann condition is prescribed. Finally, the boundary of the domain is rebuilt. All the previous steps are repeated until a given stop criterion is obtained. The material volume is checked and removed after each iteration until an expected value is obtained. The elimination process of material is halted when

** **(20)

where *b* presents a
determined percentage of material to be eliminated.

**6. Numerical
example and results**

This section presents the solution of a problem
defined by (13). It is assumed that l = 30 W/mK. Figure 2 illustrates dimensions of the domain considered
while Figure 3 shows the position of control points: **P**_{0 }= (0,
0), **P**_{1 }= (0.02, 0), **P**_{2 }= (0.02, 0.015), **P**_{3
}= (0.035, 0.015), **P**_{4 }= (0.05, 0.015), **P**_{5 }=
(0.05, 0), **P**_{6 }= (0.104, 0), **P**_{7 }= (0.104,
0.011), **P**_{8 }= (0.115, 0.011), **P**_{9 }= (0.126,
0.011), **P**_{10 }= (0.126, 0), **P**_{11 }= (0.15, 0),
**P**_{12 }= (0.15, 0.075), **P**_{13 }= (0.075, 0.075),**
P**_{14 }= (0, 0.075).

Fig. 2. Domain considered

Since the original domain is symmetrical, only half of the domain will be taken into account in further examination. The boundary of the region is represented by the NURBS curves as follows (see Figs. 3 and 4):

Fig. 3. Position of control points Fig. 4. Segments of boundary

The gray area (see Figures 2 and 4) will not
be perturbed (this is the structural part of the problem). On the top of the
domain (G_{6}), the Robin condition is
considered where the heat transfer coefficient is α = 10 W/ (m^{2}K) and the ambient
temperature is *T _{∞ }*= 20°C. On the boundary the temperature

*T*= 500°C is accepted while on ,

_{b}*T*= 200°C is given.

_{b}On the remaining parts of the boundary the Neumann condition *q _{b}* = 0 is prescribed. The initial boundary was divided into 90 linear boundary elements
and the grid of 298 internal nodes was used.

Figure 5 illustrates the temperature distribution
while Figure 6 shows the topological
derivative calculated in the first iteration (*k *= 1). Figure 7 presents the domain after the first iteration. In the opening, is assumed. Holes with
0.002 were
used and during each iteration 3% of
material was eliminated. The iterative procedure was stopped when 60% of
material from the initial domain was removed.
The final result is presented in Figure 8.

Fig. 5. Temperature distribution

Fig. 6.
Topological derivative at *k *= 1

Fig. 7. Domain after the first iteration

Fig. 8. Final result

**7. Conclusions**

In the present work, the boundary
element method and the topological derivative are used to obtain an optimal shape
of the domain described by NURBS curves. The topological-shape sensitivity
method gives information about the position where the opening can be inserted.
The appropriate criterion is used to stop creating holes. After applying the
iterative process, an optimal shape of the domain is found. To inspect the
correctness of the method, the example of computation was conducted. The proposed approach confirms an effective the BEM coupled
with the NURBS curves and the *D _{T}* implementation for the
design
of an optimal topology of the domain considered applied
in the heat transfer process modelling.

References

[1] Navotny A.A., Feijoo R.A., Taroco E., Padra C., Topological-shape sensitivity analysis, Comput. Methods Appl. Mech. Eng. 2003, 192, 803-829.

[2] Marczak R.J., Topology optimization and boundary elements - a preliminary implementation for linear heat transfer, Engineering Analysis with Boundary Elements 2007, 31, 793-802.

[3] Anflor C.T.M., Marczak R.J., Topological sensitivity analysis for two-dimensional heat transfer problems using the Boundary Element Method, Optimization of Structures and Components Advanced Structured Materials 2013, 43, 11-33.

[4] Anflor C., Marczak R.J., A boundary element approach for shape and topology design in orthotropic heat transfer problems, Mecanica Computacional vol. XXVII, 2473-2486, San Luis, Argentina, 10-13 Noviembre 2008.

[5] Brebbia C.A., Dominguez J., Boundary Elements, An Introductory Course, CMP, McGraw-Hill Book Company, London 1992.

[6] Majchrzak E., Boundary Element Method in Heat Transfer, Publ. of the Techn. Univ. of Czest., Czestochowa 2001 (in Polish).

[7] Piegl L., Tiller W., The NURBS Book, Springer, 1995.