Analysis of complexfluid-structureinteraction problems using a triangular boundary element for the fluid dynamics A. Hopf, G. Hailfmger, R. Krieg Forschungszentrum Karlsruhe GmbH, Institutfur D- 7^027 Abstract For fluid-structure interaction problems with complex geometries boundary element methods are quite suitable to describe the transient fluid dynamics. In this paper a triangular boundary element will be introduced to solve such problems. Both the pressure and its normal derivative are distributed linearly over the element. It was possible to carry out the integrations with regular and singular integrands analytically. To demonstrate the method, it has been applied to the blowdown accident of a nuclear reactor. By postulating a sudden break of a pressure vessel nozzle, the incipient coolant flow around the control rods inside the vessel has been investigated. 1 Introduction In safety investigations for chemical and nuclear plants, fluidstructure interaction problems play an important role. Initiating events such as impulsive loadings or postulated failures of pipes and pressure vessels, cause strong accelerations of the fluid inside these components or incipient flows through break openings. As a consequence, high transient pressures occurring at the fluid boundaries lead to dynamic deformations of the structures, and thus to changes of the fluid boundaries. In other words, there is a feed back from the structural dynamics on the fluid dynamics. For suchfluid-structureinteraction problems, boundary elements (BE) have been applied to the fluid dynamics while finite elements (FE) have been used to describe the structural dynamics. This ap-
178 Boundary Element Technology proach takes advantage of the fact that only the solution at the fluid boundary is of interest and that for many problems the BE discretization at the fluid boundary is partly identical with the FE discretization of the structure (shell structure, for example). In the Seventies the BE problem for thefluiddynamics was solved by an indirect method using rectangular boundary elements with uniform dipole distributions [1]. A major drawback was that these elements were not very suitable to describe non-rectangular surfaces and that the shapes of the elements and the distributions over the elements were different from those preferred for the structural part of the problem. Therefore, work has been started to develop triangular boundary elements for the fluid dynamics. In order to obtain useful solutions at the element corners, a direct method is applied, where both the pressure and its normal derivative are distributed linearly over the element [2]. 2 Assumptions and Basic Equations For many of the transient flow problems under discussion here, the fluid can be assumed to be incompressible, in viscid and irrotational. Sources and body forces are not present and the term %pv2, wherep is the fluid density and v the fluid velocity, is negligible in comparison to the fluid pressure. Then for the flow field, the Laplace equation applies Ap = 0, (1) where p represents thefluidpressure [3]. At thefluidboundary, F, the normal derivative, dp/dn, which is taken with respect to the outside direction of the boundary describes the acceleration of the boundary, an = 1/p dp/dn. The boundary is subdivided into FI, usually representing a free surface, and F^ formed by the structure. At Fj the pressure p is given and the derivative dp/dn yields the acceleration <%%. At F2 the acceleration a%, and thus the derivative dp/dn is given and the pressure p has to be calculated. It represents the structural loading. For classical steady state flow problems the dynamic pressure, VzpvZ, is not negligible and the flow field is governed by the Laplace equation A<p = 0, (2) where (p is the flow potential. At thefluidboundary F the normal derivative d<p/dn taken with respect to the outside direction of the boundary describes the velocity of the boundary, u% = d<p/dn. To be consistant with the transient problem above, the boundary is subdivided into FI where (p is given and d<p/dn has to be calculated, and Fa where d(p/dn is given and 0 has to be determined.
Boundary Element Technology 179 The descriptions for the transient flow problems and for steady state problems are mathematically identical. However, the physical meaning is quite different because of the different assumptions (for transient problems, often neglecting fluid viscosity is quite acceptable, while for steady state problems it usually represents a more severe restriction). In the literature, the second formulation is very common, and therefore, it will be used as the basis for the mathematical treatment. It should be taken in mind, however, that the solution will be applied for the transient flow problem described by the Laplace equation (1). 3 Introduction of Green's Formula and Boundary Elements The Laplace equation (2) can be replaced by Green's formula -)<" (3, where the potential fa at any point i in thefluidfieldcan be calculated from the potential $ and its normal derivative d$/dn = q at the boundary T. In equ. (3), r; is the distance between the point i and the particular boundary point and c, is the space angle of the fluid field around the point i. For a point inside the field c, = 4n> for a point at a smooth boundary a = 2n, for a rectangular corner as part of the boundary ci = n/2. More information about Green's formula may be found by Brebbia [4]. For numerical evaluation, the boundary T is subdivided into triangular boundary elements F<> with boundary nodes j. The potential $ and its normal derivative q at thefluidboundary are approximated by their values $/ and %, respectively, at the boundary nodes and by linear interpolations between these nodes. In addition, the point i is assumed to be identical with the boundary nodes (collocation). Green's formula then yields the following set of linear equations: The coefficients Cy represent a diagonal matrix consisting of the space angles a, while the coefficients f% and Gy consist of integrals hi and gi over the boundary elements FQ [4] (4) 4»,
180 Boundary Element Technology N<p and Nq are linear functions over the boundary elements FQ adopting the value 1 at the node j and vanishing for all the other nodes. According to the boundary condition at FI, the values qj, and at r%> the values <pj represent the unkowns. All other values are known. Using equ. (4), the unknowns can be determined. They are an approximate solution for the flow problem specified in section 2. 4 Transient Flow Problems with Flexible Structures For transient flow problems with flexible structures the boundary conditions have to be modified. At 7^, both the pressure p and the acceleration an are unknown. With respect to equ. (4), it means that at F2, both the values $>j and qj are unknown. Equation (4) can be transformed into the following form: 0 = Y M.. q. + P. Consider that now i and j denote only those nodes which are related to boundary F2. For the transient flow problem, <pi approximates the pressure and qj the acceleration of boundary f^>. Consequently, My approximates the mass effect of the flow. PI describes the given pressure at boundary FI. Finally, if the pressure at F<? is interpreted as a structural load and introduced into the structural dynamics relations, My (multiplied with a constant) has to be added to the structural mass matrix. Therefore, My is called the "added mass matrix". Usually My is fully populated, while the structural mass matrix is of band type. 5 Analytical Solution of the Integrals hi and g{ The evaluation of the integrals (5) represented a major problem. In cases, where the point i is identical with a corner point of a boundary element, TI approaches zero and thus the integrands approach infinity during the integration process. However, it turns out that the results of the integration approach regular values. Under these conditions analytical solutions of the integrals hi and gi are highly desirable. In the literature some analytical solutions have been published [5, 6], but the resulting mathematical formulae are very lengthy and very difficult to evaluate. Thus mistakes could hardly be eliminated. Revised solutions, which are more suitable for the numerical evaluation, have been obtained by Hopf [2], where details of the cumbersome analytical integration procedures and methods to check them are described. Some results will be discussed in this paper.
Boundary Element Technology 181 5.1 Point i is identical with a corner point of the boundary element The integrals hi vanish. For evaluation of the integrals g, consider the boundary element shown in Fig. 1, where the point, i, is identical with the corner. Figure 1: Notation of the corner of a boundary element Three cases occur: - Node J is identical with the corner : A a + b + c g = ' c a + b - c Node j is identical with the corner : + c* A "CJTl 2c a Node j is identical with the corner : * A b-a c c _ A b-a 7 c In these formulae, a, b, c are the side lengths of the boundary element as shown in Fig. 1, and A is its area. 5.2 Point / is located at the plane of the boundary element Again the integrals hi vanish. For evaluation of the integrals #, subproblems are superimposed as indicated in Fig. 2. The solutions of these subproblems are of the types discussed in section 5.1. The resulting formulae are rather lengthy and they will not be presented here. Figure 2: Solution of the problem by superposition of the solution of subproblems according to section 5.1
182 Boundary Element Technology 5.3 Point i is located outside of the plane of the boundary element The evaluation of both integrals hi and gi requires superposition according to Fig. 2. Again the resulting formulae are lengthy and will not be presented here. 6 Discussion of the Solutions HI and gi The solutions h[ and gi for an equilateral triangle, with side length 1, are shown in Fig. 3. However, for better demonstration the linear functions N<p and Nq have been assumed to be constant over the area of the triangle. It is assumed that the point i, where hi and gi are related, moves along four different axes I, II, III, IV, which are perpendicular to the boundary element. The solutions obtained are depicted as functions over these axes, hi at the left hand side and gi at the right hand side of the figure. As mentioned in the last section, hi vanishes in the plane of the element. However, a step occurs between the lower and the upper side of the element. At the edge of the element (axis I) the step is 2n, inside the element (axis II) it is 4n, at a corner point of the element (axis III) it is 2a, and outside of the element (axis IV) no step occurs. For the solution gi, a sudden change of the slope occurs between the lower and the upper side of the element. n I DT hi 9i Figure 3: Solutions hi and gi over different axes I,..., IV, perpendicular to the boundary element (N$ and Nq assumed to be constant)
Boundary Element Technology 183 7 Slowdown Flow around the Control Rods in a Nuclear Reactor Pressure Vessel A postulated sudden break of a pressure vessel nozzle in a nuclear reactor would lead to the so-called blowdown accident. It must be shown that the resulting incipient coolant flow around the control rods will not deform these rods significantly. For a reliable calculation of the structural loads, a sufficient three-dimensional resolution of all the spaces between the rods is necessary; but the influence of the coolant viscosity is expected to play a minor role. In the past, this problem had been analyzed with the former BEmethod [7, 81. Recently, the problem has been investigated by the method described in this paper. Fig. 4 shows the BE-dhcretization of a quarter of the blowdown flow field in the upper part of the pressure vessel. For most of the boundary elements the normal acceleration is assumed to vanish, i.e., the structures are assumed to be rigid. Only between the central part of the model and the front surface of the nozzle a pressure difference of 40 bar is applied causing an accelerating coolant flow to the nozzle. The resulting pressures at the rod surfaces have been integrated in order to obtain the total forces acting at the rods. As shown in Fig. 5, the forces are directed toward the nozzle. Fluid forces increase near the nozzle. Based on these results the deformations of the rods can be determined. References [1] R. Krieg, B. Goller, G. Hailfmger: Transient, Three-Dimensional Potential Flow Problems and Dynamic Response of the Surrounding Structures. Part I and II, J. Comput. Physics 34,139-183 (1980) [2] A. Hopf: BE/FE-Analyse von Fluid-Struktur-Problemen unter Verwendung eines dreieckigen Boundary Elements mit linearem Ansatz und vollstandiger analytischer Losung. University Karlsruhe, Ph.D Thesis (to appear) [3] O.C. Zienkiewicz: The Finite Element Method. McGraw-Hill, London (1977) [4] C.A. Brebbia, J.C.F. Telles, L.C. Wrobel: Boundary Element Techniques. Springer-Verlag, Berlin /Heidelberg/New York /Tokyo (1984) [5] K. Davey, S. Hinduja: Analytical Integration of Linear Three-Dimensional Triangular Elements in BEM. Appl. Math. Modelling 13,450-461 (1989) [6] E.E. Okon, R.F. Harrington: The Potential Integral for a Linear Distribution over a Triangular Domain. Int. J. Numer. Meth. Engng. 18,1821-1828 (1982) [7] R. Krieg, et al.: Core Support Columns in the Upper Plenum of a Pressurized Water Reactor under Blowdown Loading. Nucl. Engng. Des. 73, 23-44 (1982) [8] G. Hailfmger, et al.: Blowdown Loading of the Control Rods and the Core Support Columns in the Upper Plenum of a PWR. SMIRTll, Vol. B, Tokyo (1991)
184 Boundary Element Technology central part ( high pressure level outer surface control rod surfaces front surface of the blowdown nozzle flow pressure level) Figure 4: Boundary element discretization to describe the blowdown flow in the upper part of a nuclear reactor pressure vessel Figure 5: Calculated forces acting at the controle rods