An isogeometric analysis approach for two-dimensional steady state heat transfer problems

th , 2015) ABSTRACT: The purpose of this article is studied the application of isogeometric analysis (IGA) to two-dimensional steady state heat transfer problems in a heat sink. By using high order basis functions, NURBS basis functions, IGA is a high rate convergence approach in comparison to a traditional Finite Element Method. Moreover, the development of this method decreased the gaps between CAD and mathematical model and increased the


INTRO DUCTIO N
Almost every technical operate process generate heat during activity duration; it can be active or inactive. In case of parts work in high temperature conditions, the size, material and other relative parameters must be optimized so that they can avoid destroying, and heat which generate unnecessary needed to effectively diffused [1,2]. It is an importance and necessity for heat transfer problems in techniques and industries, and it is interested in science and engineering communities. The heat transfer problems have solved by many different methods, like Finite Element Method [3], meshless method [4], or Finite Pointset Method for simulating heat transfer involving a moving source [5], even Analytical Solution. In this study, we focus on the introduction of the basic concept of isogeometric analysis using B-spline basis functions for heat transfer problems and discuss the accuracy of this method also mentioned other.
Isogeometric analysis (IGA) was introduced in [6] and has developed since 2005. Because of the existing gaps between Computer Aided Design (CAD) and the Finite Element Analysis (FEA), IGA was coined. The predominance that is using No-Uniform Rational B-spline to represent the complex geometries, while the geometry is replaced by finite element meshes approximated of the geometry in FEA. To obtain a high accuracy result, a refinement mesh is used with a coherence level. In traditional FEA, the refinement requires communication with the CAD geometry during a process of analysis, while simplify mesh refinement is a dominance of IGA. It is approximately 80% of overall analysis time to generate the mesh in FEM [7]. Therefore, we will save much time and cost with IGA. The IGA has been applied to several physical problems and will be clearly described in this paper.

BSPLINE AND NURBS
where 1 i n   and 0/0 is considered as zero.
Clearly, considering a B-spline basis function with p degree, the interior knot can be a multiplicity p in the knot vector. Furthermore, the first and last knots have multiplicity  There are several important features of NURBS geometry. The first is that the basis constitutes a partition of unity, that is , 1 The second, each basis function is pointwise nonnegative over the entire domain, that is The next feature is that each th p order function has 1 p  continuous derivatives across the element boundary. And an important note is the support of the B-spline functions of order p is always 1 p  knot spans.

B-spli ne c ur ve s
A B-spline curve for a given direction has the form of Basis function and knot vector as in Figure 1. Control point locations are denoted by ■ , and the knots, which define a mesh by partitioning the curve into elements, are denoted by ■

B-spli ne surface s
Given a control net

A TWO -DIM ENSIO NAL STATIC FO RM ULATIO N B ASED O N NURBS APPRO XIM ATIONS
Using NURBS basis function, the temperature variable can be interpolated as where i R are the NURBS basis functions, i T are the temperature at control point i and n is the number of control points.
The governing equation of static analysis for a linear structural system in the form as  KT f (10) where K is the global left hand side matrix expressing the properties of the overall system, f is the global load vector, which is the assemblage of individual load vectors.
In addition, K matrix is presented by [9] T where B is the derivative matrix, which relate the gradient of the field variable to the nodal values. And D matrix in form as where x k and y k is the thermal conductivity coefficients.

RESULTS AND DISCUSSIO N 4.1 Square pl ate with Dir ic hle t c ondi ti ons
To demonstrate the accuracy and performance of the isogeometric in heat conduction problems, we consider to a square plate of unit thickness, is shown in Figure 8, size 100 cm. At the top side, the plate is subjected to isothermal boundary conditions of 500 C  , and Isogeometric Analysis and compare that result with the analytical method and the Finite Element Method.
In order to obtain the exact solution of the steady state without heat generator, we shall use the Laplace equation and the analytical solution was calculated by Holman, 1989, [6], it is a Fourier sine series, and that solution is expressed in Equation (13).
where, 2 T is the temperature at the top side, and 1 T is at the other sides.
By apply the concrete boundary conditions and a range of n value, the temperature at the center of the plate is determined following the Equation (13). The value of temperature is static at 200.000 First, we consider to the basis is choosen as quadratic, cubic and quartic NURBS, is shown in Figure 9. The number of elements is constant while the number of control points and the order of basis functions simultaneously increase. To clearly observe the advantage of IGA, we consider to the temperature in the center of square plate with array of degree of freedoms.    . By increasing the order, p , of the basis functions, the obtained results converge to the best reference value determined by the expression (13), [7]. By using more fine discretization, there errors are reduces, as shown in Figure 11. As shown in the Figure 10, the convergence rate of the FEA is slower than IGA, it mean, at the certain number of DOFs, the IGA result closer the analytical result than FEA. The temperature distribution in the square plate is shown in Figure 12. .

Square plate wi th both Dir ic hle t and Ne umann c onditi ons
As a second example, a two-dimensional domain is prescribed with Dirichlet and Neumann boundary conditions applied along the boundaries is show in Figure 13. Heat enter at the bottom of the plate is 500 C  , and other sides entered  The Figure 15 shows the convergence of the temperature at the center of the square plate. As increasing a number of control points, the obtained results converge to a value.
Both of two method have similar temperature distribution in this problem, it is shown in Figure   16.

TAÏ P CHÍ PHAÙ T TRIEÅ N KH&CN, TAÄ P 18, SOÁ K4-2015
Trang 169   Figure 18, and h-Refinement technology was used to increase number of elements automatically, as show in Figure 19.   We can recognize the various of temperatures is closed to FEM value while FEM's number DOFs is much higher. Eventhough, the result with only coarse mesh also be better.
The temperature contour is shown in Figure   22 and Figure 23.

A practical pr oble m
This is a practice problem expressing the effect of IGA in heat transfer. The electrical technical advancement need to decrease the size of them, and a important problem is effect heat diffusion, or we can optimize the shape and dimension of heat sink. This section simply describe a few of problems listed above. That is the temperature distribution in fins of the heat sink to optimize it's profile. Consider to a heat sink is shown in Figure 24  The NURBS surfaces of the heat sink is also shown in Figure 24. With that number of elements, the number of degrees of freedom is 1327, is shown in Figure 25. FEM solution with about 57000 degrees of freedoms is shown in Figure 26.

CO NCLUSIO N
An isogeometric analysis approach for twodimensional static heat transfer problem is expressed above. Applying IGA to numerical problems lead significant effective results, as represent on above. More important that it can refine the mesh without the connection to the CAD geometry, it called h-refinement and prefinement and k-refinement, it is very convenient and makes the problem easier. Furthermore, IGA is base on high order basis functions, i.e., cubic basis functions are more often. Quartic basis functions have to take more time and the error decrease inappreciably, but they get a high accuracy in comparison with quadratic and cubic basis functions. Although, with industrial problems, where the accuracy is not necessary, FEM still gain advantages over.
Therefore, IGA should be applied to problems that have complex geometries. It will decrease the errors at the compound curve, surface, it contributes to the exact results. IGA also have some disadvantages because it is still be developing. To make up the accuracy results, IGA is with regards to computational time to achieve convergence. A particular reason is high order basis functions must be spent more time to calculate. Summary, there is a basic of IGA application. We hope some problems mentioned above was enough to demonstrate the effect results of this analysis.