Patent Publication Number: US-2019179986-A1

Title: Method for analyzing vibration damping structure

Description:
TECHNICAL FIELD 
     The present disclosure relates to a method for analyzing a vibration damping structure of a tube bundle disposed in a fluid. 
     BACKGROUND ART 
     A tube bundle disposed in a fluid, for instance, a heat-transfer tube bundle used in a heat exchanger such as a steam generator adopts a vibration damping structure in which an anti-vibration bar is disposed in a gap between each tube for suppressing vibration of the tubes in order to prevent the tubes from largely vibrating due to the fluid flowing outside the tubes. It has been recently indicated that self-excited vibration phenomena, such as fluid elastic vibration, can occur along a flow direction of the fluid in this type of tube bundle, namely, a tube bundle having a U-bent portion including a U-shaped tube. The U-shaped tube is supported by an anti-vibration member (vibration damping member) provided in a gap between the tubes. The vibration phenomenon along the flow direction is suppressed by friction between the tube and the anti-vibration member. Since such friction is closely related to the size of a gap between the tube and the anti-vibration member, the suppression of self-excited vibration requires a structural design that generates an appropriate friction force. 
     Besides, in an actual structure, members including the heat-transfer tube and the anti-vibration bar have manufacturing tolerances such as variation in size, twist, and warpage, thus having some deviation from a designed profile. For instance, Patent Document 1 discloses a tolerance analyzing/calculating system for analyzing and examining a tolerance of each component of a designed structure so that a size tolerance at the time of assembly of the structure is within a predetermined range. More specifically, this document discloses that variance or deviation with respect to the size tolerance of each component is measured, and an appropriate tolerance which satisfies required quality for a design specification value is examined based on the results. 
     CITATION LIST 
     Patent Literature 
     
         
         Patent Document 1: JP2009-146162A 
       
    
     SUMMARY 
     Problems to be Solved 
     An actual structure having an error within an allowable range is substantially ignorable. However, the above-described vibration damping structure adopted in the steam generator can cause, if an unintended gap exists therein due to an error, significant vibration of the heat-transfer tube or the anti-vibration member by an inner fluid force when the steam generator is operated. Thus, this structure requires a more severe design. In particular, such vibration can cause friction and contact between the members, which can lead to a situation such as breakage of the members. Although the above patent document discloses an analysis method for a typical structure, this method is difficult to precisely evaluate an error of the vibration damping structure disposed in a fluid and showing a complicated behavior. A novel analysis method is therefore desired. 
     At least one embodiment of the present invention was made in view of the above, and an object thereof is to provide a method for analyzing a vibration damping structure whereby it is possible to perform structural analysis with high precision, taking into consideration an error factor of the vibration damping structure disposed in a fluid. 
     Solution to the Problems 
     (1) To solve the above problems, according to at least one embodiment of the present invention, a method for analyzing a vibration damping structure in which a tube bundle disposed in a fluid is supported by a vibration damping member disposed in a gap between tubes included in the tube bundle comprises: a model making step of making a FEM model corresponding to the vibration damping structure, an error setting step of setting an error parameter for a parameter relating to an element included in the FEM model; and an analysis step of performing structural analysis by a finite-element method using the FEM model in which the error parameter is set. 
     In the above method (1), the vibration damping structure to be analyzed is represented by the FEM model to perform structural analysis by the finite-element method. At this time, the error parameter is set for the parameter relating to the element included in the FEM model, which enables structural analysis with high precision, in consideration of an error factor of the vibration damping structure disposed in a fluid. 
     (2) In some embodiments, in the above method (1), the model making step includes making the FEM model so as to include a first element corresponding to the tube bundle, a second element including a one-dimensional element corresponding to the vibration damping member, and a third element corresponding to a gap amount between the tube bundle and the vibration damping member. 
     In the above method (2), the vibration damping structure to be analyzed is represented by the first to third elements. Thus, since the FEM model corresponding to the vibration damping structure can be efficiently constructed by limited variable parameters, it is possible to perform structural analysis precisely with reduced computation load. 
     (3) In some embodiments, in the above method (2), the first element includes a plurality of first one-dimensional elements extending to respectively correspond to the tubes included in the tube bundle, the second element includes a second one-dimensional element extending to correspond to the vibration damping member, and the third element includes a gap element representing a shortest distance between each first one-dimensional element and the second one-dimensional element. 
     In the above method (3), the first and the second elements are represented by one-dimensional elements while the third element is represented by a gap element disposed therebetween. Thereby, it is possible to construct a FEM model having a considerably simple structure. Such a FEM model efficiently reduces the variable parameters and thus has low computation load and is available for high-speed arithmetic processing. 
     (4) In some embodiments, in the above method (2) or (3), the analysis step includes imposing a loading condition set so that, when the gap amount is less than a predetermined value, a contact force applied to each first one-dimensional element from the second one-dimensional element increases with an increase in interference. 
     In the above method (4), the gap amount set as an element (third element) of the FEM model corresponds to the amount of the gap between the tube and the vibration damping member in an actual vibration damping structure. Thus, the loading condition is imposed so that, when the gap amount exceeds a predetermined value, the contact force applied to the first one-dimensional element from the second one-dimensional element increases with an increase in gap amount, whereby it is possible to appropriately simulate the behavior of the contact force mutually acting on the tube and the vibration damping member when they are in contact by computing analysis. 
     (5) In some embodiments, in the above method (2), the first element includes a plurality of first one-dimensional elements extending to respectively correspond to the tubes included in the tube bundle, the second element includes a second one-dimensional element extending along a longitudinal direction of the vibration damping member and a third one-dimensional element extending along a width direction of the vibration damping member, and the third element includes a pair of gap elements representing a shortest distance between each end of the third one-dimensional element and each first one-dimensional element. 
     In the above method (5), the second element corresponding to the vibration damping member is represented by the second and third one-dimensional elements in a two-dimensional manner, and the pair of gap elements is provided as the third element between each end of the third one-dimensional element. Thereby, it is possible to reproduce a high-dimensional behavior, such as twist of the vibration damping member with respect to the tube bundle, as a behavior close to reality, and it is possible to achieve more precise structural analysis. 
     (6) In some embodiments, in the above method (1) to (5), the error parameter includes a random number. 
     With the above method (6), since the error parameter includes a random number, it is possible to randomize an error of the parameter for which the error parameter is to be set. Generally, errors of actual structural members are statistically uniform. Thus, setting the error parameter in this manner enables accurate reproduction of errors of an actual vibration damping structure and enables precise analysis. 
     (7) In some embodiments, in the above method (1) to (5), the error parameter is set based on data measured on the vibration damping structure. 
     With the above method (7), the error parameter is set based on data measured on the vibration damping structure. Thus, even if an actual structure has statistically nonuniform errors due to some factors, appropriate analysis can be performed by setting the error parameter based on the measurement data. 
     (8) In some embodiments, in the above method (1) to (7), the error parameter includes at least one of an outer diameter, a thickness, and a warpage of a tube included in the tube bundle, and a thickness and a warpage of the vibration damping member. 
     With the above method (8), since the parameters which tend to affect analysis results in this type of vibration damping structure are set as the error parameter, it is possible to simulate the behavior of an actual structure more closely, and it is possible to perform analysis with high precision. 
     (9) In some embodiments, in the above method (1) to (8), each of the tubes included in the tube bundle has a first straight tube part positioned on a fluid inlet side, a second straight tube part positioned on a fluid outlet side, and a bent part positioned between the first straight tube part and the second straight tube part, the first straight tube part and the second straight tube part are inserted into a plurality of through holes formed in a tube support plate for supporting the tube bundle, and in the analysis step, the error parameter includes an eccentricity amount of an insertion position, at which the first straight tube and the second straight tube part are inserted into the through holes, to perform the structural analysis. 
     In the above method (9), structural analysis is performed on the vibration damping structure including the tube bundle and the vibration damping member by the finite-element method using the FEM model, in consideration of a structure in which the straight tube parts of each tube are inserted to the through holes formed in the tube support plate for supporting the tube bundle. Further, in the above method (9), the eccentricity amount of the insertion position of the straight tube parts in the through holes is included in the error parameter to perform the above structural analysis. 
     Accordingly, with the above method (9), it is possible to perform the structural analysis, in consideration of increase and decrease in contact load when the contact load applied to the vibration damping member from the bent part adjoining the straight tube parts of each tube increases or decreases due to the eccentricity amount of the insertion position of the straight tube parts in the through holes. Further, with the above method (9), it is possible to perform the structural analysis, in consideration of variation due to an error of the eccentricity amount of the insertion position of the straight tube parts in the through holes. 
     Advantageous Effects 
     According to at least one embodiment of the present invention, there is provided a method for analyzing vibration damping structure whereby it is possible to perform structural analysis with high precision, in consideration of an error factor of the vibration damping structure disposed in a fluid. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         FIG. 1  is a perspective view of a U-bent portion of a heat-transfer tube bundle according to an embodiment. 
         FIG. 2  is a diagram showing an example of a support structure by anti-vibration members viewed from an in-plane direction. 
         FIG. 3  is a diagram showing an example of a support structure by anti-vibration members viewed from an out-of-plane direction. 
         FIG. 4A  is a diagram showing a computer device for performing a structural analysis method according to an embodiment. 
         FIG. 4B  is a diagram showing an internal configuration of a computing unit of the computer device depicted in  FIG. 4A . 
         FIG. 5  is a flowchart of an execution procedure of a structural analysis method according to an embodiment. 
         FIG. 6  is a diagram showing a vibration damping structure in which a heat-transfer tube is supported by friction between two anti-vibration members, using one-dimensional elements and gap elements. 
         FIG. 7  is a diagram visually showing an image when the entire heat-transfer tube bundle having a U-bent portion shown in  FIG. 1  is modeled by the finite-element method. 
         FIG. 8  is a diagram showing an example of a manufacturing tolerance distribution. 
         FIG. 9A  is a diagram showing an example of various types of manufacturing tolerances caused when a heat-transfer tube and an anti-vibration member are manufactured. 
         FIG. 9B  is a diagram showing another example of various types of manufacturing tolerances caused when a heat-transfer tube and an anti-vibration member are manufactured. 
         FIG. 9C  is a diagram showing another example of various types of manufacturing tolerances caused when a heat-transfer tube and an anti-vibration member are manufactured. 
         FIG. 9D  is a diagram showing another example of various types of manufacturing tolerances caused when a heat-transfer tube and an anti-vibration member are manufactured. 
         FIG. 10  is a graph showing that a contact force acting between one-dimensional elements increases in accordance with gap amount. 
         FIG. 11A  is a diagram schematically showing an appearance of an anti-vibration member. 
         FIG. 11B  is a diagram showing an example of twist amount in a thickness direction of the anti-vibration member of  FIG. 11A . 
         FIG. 11C  is a diagram showing another example of twist amount in a thickness direction of the anti-vibration member of  FIG. 11A . 
         FIG. 12  is a diagram showing a vibration damping structure in which a heat-transfer tube is supported by friction between two anti-vibration members, using one-dimensional elements and gap elements. 
         FIG. 13  is a diagram showing eccentricity amount of heat-transfer tube straight tube parts inserted into through holes provided in a tube support plate. 
     
    
    
     DETAILED DESCRIPTION 
     A vibration damping structure analysis method according to some embodiments of the present invention will now be described with reference to the accompanying drawings. It is intended, however that unless particularly specified, dimensions, materials, shapes, relative positions and the like of components described in the embodiments shall be interpreted as illustrative only and not intended to limit the scope of the present invention. The vibration damping structure analysis method according to some embodiments of the present invention can be applied to any tube bundle structure in which multiple tubes disposed in a fluid are supported by friction with a supporting member against a fluid force. The structure of the heat-transfer tube bundle shown in  FIGS. 1 to 3  will be described as an example of the tube bundle structure to which the vibration damping structure analysis method according to some embodiments of the present invention can be applied. Next, process details of the vibration damping structure analysis method will be described with reference to  FIGS. 4 to 12 . 
       FIG. 1  is a perspective view of a U-bent portion  10   a  of a heat-transfer tube bundle  10  according to an embodiment.  FIG. 2  is a side view of the heat-transfer tube bundle  10  viewed from an in-plane direction D 2  in  FIG. 1  (array direction d 2  in  FIG. 1 ).  FIG. 3  is a side view of the heat-transfer tube bundle  10  viewed from an out-of-plane direction D 1  in  FIG. 1  (array direction d 1  in  FIG. 1 ). In  FIG. 1 , some components are omitted for clarity. The components omitted in  FIG. 1  are illustrated in  FIGS. 2 and 3 , which are side views of the heat-transfer tube bundle  10  in  FIG. 1 . 
     In some embodiments, the heat-transfer tube bundle  10  includes a plurality of heat-transfer tubes  3  and a tube support plate  7  into which the plurality of heat-transfer tubes  3  are inserted. The heat-transfer tube bundle  10  is configured to generate steam by heat exchange with a fluid flowing through the plurality of heat-transfer tubes  3 . Each of the heat-transfer tubes  3  has a first straight tube part  4  positioned on a fluid inlet side, a second straight tube part  5  positioned on a fluid outlet side, and a bent part  6  positioned between the first straight tube part  4  and the second straight tube part  5 . The tube support plate  7  is provided with a plurality of through holes into which the first straight tube part  4  and the second straight tube part  5  are inserted. The through holes disposed in the tube support plate  7  for inserting the first straight tube part  4  and the second straight tube part  5  will be described in detail with reference to  FIG. 13 . 
     The heat-transfer tube bundle  10  is composed of a plurality of heat-transfer tubes  3  each having a U-shaped bent part  6 . The bent parts  6  of the plurality of heat-transfer tubes  3  form a U-bent portion  10   a . In the structure shown in  FIG. 1 , the heat-transfer tubes  3  are concentrically arranged around a common curvature center in the same plane (along the in-plane direction D 2 ) so that a heat-transfer tube  3  having a bent part  6  with a larger curvature radius is positioned at an outer portion (upper portion in  FIG. 1 ) in a radial direction of the bent part  6  (tube array  8  in  FIG. 1 ).  FIG. 3  shows that a plurality of tube arrays  8 , each including the heat-transfer tubes  3  arranged along the in-plane direction D 2 , are located in row. These tube arrays  8  are arranged in a direction (out-of-plane direction D 1  in  FIG. 1 ) perpendicular to the plane containing the bent parts  6 . 
     As shown in  FIGS. 1 and 3 , the curvature radius of the bent part  6  of the heat-transfer tube  3  positioned on the outermost peripheral side in each of the tube arrays  8  varies with the position of the corresponding tube array  8  in the out-of-plane direction D 1 . In this way, a semi-spherical U-bent portion  10   a  is formed at an upper end of the heat-transfer tube bundle  10  by changing the curvature radius of the bent part  6  and stacking the plurality of tube arrays  8  in the out-of-plane direction D 1 . As a result, as shown in  FIG. 1 , a plurality of bent parts  6   a   1 ,  6   a   2 ,  6   a   3  . . . with different curvature radii are arranged along the in-plane direction D 2 , and a plurality of bent parts  6   a   1 ,  6   b   1 ,  6   c   1  . . . with the same curvature radius are arranged along the out-of-plane direction D 1 . 
     While the present embodiment shows a case where the heat-transfer tubes  3  constituting the heat-transfer tube bundle  10  are arranged in square, the technical ideas of the present application can also be applied to a case where the bent parts are arranged in another way, for instance, by triangle arrangement in which the bent parts with the same curvature radius are arranged in every other row. 
     In the heat-transfer tube bundle  10 , an anti-vibration member  12  is inserted between bent parts  6  of adjacent heat-transfer tubes  3  in the out-of-plane direction D 1  perpendicular to the plane containing the bent parts  6  to restrict movement of the plurality of heat-transfer tubes  3  (bent parts  6 ) in the out-of-plane direction D 1 . For instance, in  FIG. 1 , a plurality of anti-vibration members  12  are inserted along the in-plane direction D 2  on both sides of each of the tube arrays  8  arranged in the out-of-plane direction D 1  so as to restrict movement of the bent parts  6  of the plurality of heat-transfer tubes  3  belonging to the corresponding tube array  8  in the out-of-plane direction D 1 . 
     As shown in  FIG. 1 , a first retention member  11  is an arc-shaped rod member mounted along the outer circumference of the U-bent portion  10   a , i.e., the outer circumference of the semi-spherical U-bent portion  10   a . The anti-vibration members  12  extend inward in a radial direction of the semi-spherical U-bent portion  10   a  from the first retention member  11 . To end parts  12   a  of the anti-vibration members  12 , as shown in  FIG. 1 , the first retention member  11  is welded to connect the end parts  12   a  of the plurality of anti-vibration members  12 . The first retention member  11  is perpendicular to the tube arrays  8 , in which the heat-transfer tubes  3  are stacked along the in-plane direction D 2 , and extends along the semi-spherical surface of the U-bent portion  10   a.    
     As shown in  FIGS. 2 and 3 , a plurality of first retention members  11  may be connected by a second retention member (bridge)  14 . The second retention member  14  is an arc-shaped plate member disposed along the outer circumference of the U-bent portion  10   a , i.e., the outer circumference of the semi-spherical U-bent portion  10   a . The second retention member  14  extends along a direction of extending the bent parts  6  of the heat-transfer tubes  3  in the U-bent portion  10   a . A plurality of second retention members  14  may be arranged in the out-of-plane direction D 1 . 
     In the heat-transfer tube bundle  10 , the anti-vibration members  12  inserted between the bent parts  6  of the adjacent heat-transfer tubes  3  in the out-of-plane direction restrict movement of the plurality of heat-transfer tubes  3  (bent parts  6 ) in the out-of-plane direction D 1 . Thus, when an exciting force is applied in the out-of-plane direction D 1 , only a section of the bent part  6  of each heat-transfer tube  3  between two adjacent anti-vibration members  12  vibrates. However, a series of the heat-transfer tubes  3  (tube array  8  in  FIG. 1 ) arranged in the in-plane direction D 2  along the plane containing the bent parts  6  is not connected to the anti-vibration members  12  on both sides. The restriction is based on only fiction with the anti-vibration members  12  on both sides. As a result, the direction in which each heat-transfer tube  3  vibrates is substantially coincident with the in-plane direction D 2 , and a contact load caused by collision between the heat-transfer tube  3  and the adjacent anti-vibration member  12  is mostly derived from friction in the in-plane direction D 2 . 
     In an illustrative embodiment, the heat-transfer tube bundle  10  described with reference to  FIGS. 1 to 3  may be a heat-transfer tube bundle of a steam generator for heat exchange between primary cooling water and secondary cooling water in a pressurized water reactor (PWR) type nuclear power generating plant. In this case, the secondary cooling water flows from just below the U-bent portion  10   a  to just above the U-bent portion  10   a  along a flow direction G perpendicular to the out-of-plane direction D 1  and the in-plane direction D 2  shown in  FIG. 1 , thereby performing heat exchange with the primary cooling water flowing through the heat-transfer tubes  3 . Thus, the flow of the secondary cooling water is cross-flow perpendicular to the bent parts  6  of the heat-transfer tubes  3  at the uppermost part of the U-bent portion  10   a . From the above, the vibration damping structure analysis method according to some embodiments of the present invention may be performed to previously evaluate whether a sufficient damping force for suppressing self-excited vibration of the heat-transfer tube bundle  10  is achieved in the heat-transfer tube bundle  10  when the flow of the secondary cooling water along the flow direction G in  FIG. 1  acts as an excitation force. 
     As described above, in a case where the heat-transfer tube bundle  10  constitutes a steam generator provided in a pressurized water reactor, the heat-transfer tubes  3 , through which primary cooling water supplied from the reactor flows, are arranged in parallel to form the heat-transfer tube bundle  10 , and secondary cooling water flows over an outer surface of a heat transfer portion of the beat-transfer tube bundle  10  to perform heat exchange. In this steam generator, the heat-transfer tubes need to be arranged densely to improve the efficiency of heat exchange. In this case, a gap between the heat-transfer tubes for the secondary cooling water becomes small, and consequently, the flow rate of the secondary cooling water can increase. If the flow rate exceeds a certain limit flow rate, self-excited vibration can occur in the heat-transfer tube bundle  10 . This self-excited vibration is an unstable structural behavior in which movement of the heat-transfer tube bundle  10  and fluid flow are mutually affected. Its vibration amplitude increases over time and leads to damage to the heat-transfer tube bundle  10 . Thus, self-excited vibration is a large problem. 
     In order to prevent self-excited vibration of the heat-transfer tube bundle in the steam generator, the plurality of heat-transfer tubes  3 , whose lower end is supported by the tube support plate  7 , are supported by inserting the plurality of anti-vibration members  12  at the U-bent portion  10   a  at the upper portion. That is, at the U-bent portion  10   a  of the steam generator, the tube arrays  8 , composed of the plurality of heat-transfer tubes  3  arranged along the same plane, are supported by the anti-vibration members  12  inserted therebetween. In this case, a contact load which acts between the anti-vibration member  12  and the bent part  6  of the heat-transfer tube  3  serves as a damping force to reduce energy of self-excited vibration caused by a fluid force of the secondary cooling water, and it is beneficial to previously evaluate whether a sufficient damping force for suppressing self-excited vibration is achieved in the heat-transfer tube bundle  10 . For instance, if interference exists between the anti-vibration member  12  and the bent part  6  of the heat-transfer tube  3  in a state where the heat-transfer tube bundle  10  shown in  FIG. 1  is assembled, a bending load or a compression load occurs in the anti-vibration member  12  and the bent part  6  in accordance with the interference. By this load, the bent part  6  receives a contact force from the anti-vibration member  12 . Accordingly, it is beneficial to previously evaluate whether the contact force applied between the anti-vibration member  12  and the bent part  6  is necessary and sufficient for suppressing self-excited vibration of the heat-transfer tube bundle  10  in a state where the heat-transfer tube bundle  10  is assembled. 
     In some embodiments described below, evaluation of the damping force is mainly performed with respect to the bent part  6  of each beat-transfer tube  3  constituting the U-bent portion  10   a  of the heat-transfer tube bundle  10 . Accordingly, in some embodiments below, the U-bent portion  10   a  of the heat-transfer tube bundle  10  is simply referred to as the heat-transfer tube bundle  10 , and the bent part  6  of each heat-transfer tube  3  is simply referred to as the heat-transfer tube  6  or the tube  6 . 
     Next, the vibration damping structure analysis method according to some embodiments of the present invention and a computer device for performing the vibration damping structure analysis method will be described with reference to  FIGS. 4 to 6 .  FIG. 4A  is a diagram showing an overall configuration of a computer device  20  for implementing the vibration damping structure analysis method according to some embodiments. The computer device  20  includes a computing unit  21 , a storage unit  22 , an output unit  23 , and an input unit  24 . In an illustrative embodiment, the computing unit  21  may be configured as an arithmetic circuit which reads and executes a program  22   a  stored in the storage unit  22  to perform the vibration damping structure analysis method for evaluating self-excited vibration of the heat-transfer tube bundle  10  disposed in a fluid and supported by the anti-vibration member  12 . In this embodiment, data which needs to be read and written by the computing unit  21  for performing the vibration damping structure analysis method may be stored as data  22   b  in the storage unit  22 . 
     The output unit  23  is an output device for presenting a part of computing results of the computing unit  21  or the data  22   b  stored in the storage unit  22  to a user. In an illustrative embodiment, the output unit  23  may include display means such as a display device as output means. The input unit  24  is an input device for inputting external data representative of various information and parameters into the computing unit  21  by operation of the user. In an illustrative embodiment, the input unit  24  may include input means such as a keyboard or a mouse. 
       FIG. 4B  is a diagram showing an internal configuration of the computing unit  21  included in the computer device  20 . With reference to  FIG. 4B , the computing unit  21  includes a FEM model making part  211 , an error parameter setting part  212 , and an analysis part  213 . In an example, the computing unit  21  may be implemented by a general purpose processor. In this case, the FEM model making part  211 , the error parameter setting part  212 , and the analysis part  213  may be implemented as program modules generated in the computing unit  21  when the computing unit  21  reads the program  22   a  from the storage unit  22 . 
     The FEM model making part  211  makes a FEM model corresponding to the vibration damping structure of the heat-transfer tube bundle  10 . The error parameter setting part  212  sets an error parameter for a parameter relating to an element included in the FEM model made by the FEM model making part  211 . The analysis part  213  performs structural analysis by the finite-element method using the FEM model with the error parameter set by the error parameter setting part  212 . Detail operation performed by the FEM model making part  211 , the error parameter setting part  212 , and the analysis part  213  will now be described in accordance with the flowchart shown in  FIG. 5 . 
     The flowchart shown in  FIG. 5  starts with step S 1 . In step S 1  of  FIG. 5 , the FEM model making part  211  makes a FEM model corresponding to the vibration damping structure of the heat-transfer tube bundle  10 . More specifically, contact surfaces between each of the heat-transfer tubes  6  and each of the anti-vibration members  12  are modeled by using contact elements of partial contact including the amount of a gap (gap amount) between each of the heat-transfer tubes  6  and each of the anti-vibration members  12 . For example, in an illustrative embodiment shown in  FIG. 6 , in step S 11 , the FEM model making part  211  may make the FEM model so as to include a first element  53  corresponding to each of the heat-transfer tubes  6  constituting the heat-transfer tube bundle  10 . Further, in step S 12 , the FEM model making part  211  may make the FEM model so as to include a second element  50  including a one-dimensional element corresponding to the anti-vibration member  12 . Further, in step S 13 , the FEM model making part  211  may make the FEM model so as to include a third element  62  corresponding to a gap amount (the amount of a gap) between the heat-transfer tube  6  and the anti-vibration member  12 . 
     As shown in  FIG. 6 , the first element  53  may include a plurality of first one-dimensional elements  53 A to  53 C extending to respectively correspond to the heat-transfer tubes  6 ( 1 ) to  6 ( 3 ) included in the heat-transfer tube bundle  10 . More specifically, the first one-dimensional elements  53 A to  53 C may be modeled as one-dimensional elements each passing through the center of a circular cross-section of a corresponding one of the heat-transfer tubes  6 ( 1 ) to  6 ( 3 ) and extending along a length direction of the corresponding one of the heat-transfer tubes  6 ( 1 ) to  6 ( 3 ). Further, as shown in  FIG. 6 , the second element  50  may include second one-dimensional elements  50 A and  50 B extending to correspond to the anti-vibration members  12 ( 1 ) and  12 ( 2 ). More specifically, the second one-dimensional elements  50 A and  50 B may be modeled as one-dimensional elements each passing through a central portion with respect to a width direction of a corresponding one of the anti-vibration members  12 ( 1 ) and  12 ( 2 ) and extending along a length direction of the corresponding one of the anti-vibration members  12 ( 1 ) and  12 ( 2 ). 
     Further, as shown in  FIG. 6 , the third element  62  may include gap elements  62 A to  62 D each representing the shortest distance between the second one-dimensional element  50 A,  50 B and the first one-dimensional element  53 B,  53 C. In this regard.  FIG. 6  shows four gap elements  62 A to  62 D corresponding to four gaps between the first one-dimensional elements  53 B,  53 C and the second one-dimensional elements  50 A,  50 B. More specifically, the gap elements  62 A to  62 D are defined as follows. 
     In the example shown in  FIG. 6 , the second one-dimensional element  50  ( 50 A and  50 B) and the first one-dimensional element  53  ( 53 B and  53 C) extend in perpendicular directions. Here, consider a case where the second one-dimensional element  50  ( 50 A and  50 B) and the first one-dimensional element  53  ( 53 B and  53 C) are viewed from a direction perpendicular to the contact surface between the heat-transfer tube  6  ( 6 ( 1 ) to  6 ( 3 )) and the anti-vibration member  12  ( 12 ( 1 ),  12 ( 2 )). In this case, a distance between the second one-dimensional element  50  ( 50 A and  50 B) and the first one-dimensional element  53  ( 53 B and  53 C) is shortest at an intersection between the second one-dimensional element  50  ( 50 A and  50 B) and the first one-dimensional element  53  ( 53 B and  53 C). Then, as shown in  FIG. 6 , this shortest distance between the second one-dimensional element ( 50 A and  50 B) and the first one-dimensional element ( 53 B and  53 C) is modeled as the gap element  62 A to  62 D. In brief, the gap element  62 A to  62 D is a model of the gap amount between each of the heat-transfer tubes  6  and each of the anti-vibration members  12 . 
     As described above, all of the heat-transfer tubes  6  and the anti-vibration members  12  constituting the heat-transfer tube bundle  10  are modeled as the one-dimensional elements  53  and  50 , and further, all of the gap amounts separating the heat-transfer tubes  6  and the anti-vibration members  12  in the heat-transfer tube bundle  10  are modeled as the gap elements  62 . As a result, an image which visually represents the whole structure of the heat-transfer tube bundle  10  by the FEM model is obtained as shown in  FIG. 7 . 
     Then, the flowchart shown in  FIG. 5  proceeds to step S 2 , and the error parameter setting part  212  sets an error parameter ε for a parameter relating to an element included in the FEM model made by the FEM model making part  211 . More specifically, first, in step S 21 , the error parameter setting part  212  sets a probability distribution of the error parameter ε as shown in  FIG. 8 , based on a manufacturing tolerance distribution of each of the heat-transfer tubes  6  and each of the anti-vibration members  12 . In an example, the error parameter ε may include a random number distributed in accordance with the probability distribution determined based on the manufacturing tolerance distribution. In an alternative embodiment, the error parameter may be set based on data measured on the vibration damping structure of the heat-transfer tube bundle  10 . 
     Then, in step S 22 , a tolerance is provided to the FEM model, in which the amount of a gap between each of the heat-transfer tubes  6  and each of the anti-vibration members  12  is modeled, so that the error parameter ε distributed in accordance with the probability distribution as shown in  FIG. 8  is reflected. For instance, in the example shown in  FIG. 6 , a tolerance is provided to the gap elements  62 A to  62 D so as to reflect the error parameter ε distributed in accordance with the manufacturing tolerance distribution. In the example shown in  FIG. 6 , the gap elements  62 A to  62 D are FEM models corresponding to the shortest distances between the first one-dimensional elements  53 A to  53 C corresponding to the heat-transfer tubes  6 ( 1 ) to  6 ( 3 ) and the second one-dimensional elements  50 A and  50 B corresponding to the anti-vibration members  12 ( 1 ) and  12 ( 2 ). 
     In an illustrative embodiment, as shown in  FIGS. 9A to 9D , the error parameter s may be based on a manufacturing tolerance caused by at least one of the outer diameter, the plate thickness, and warpage of the heat-transfer tubes  6  included in the heat-transfer tube bundle  10 , and the plate thickness and warpage of the anti-vibration members  12 . That is, the error parameter corresponds to one or more of variation in thickness of the anti-vibration member  12  perpendicular to the contact surface with the heat-transfer tube  6 , deviation from flatness of the anti-vibration member  12  caused by strain of the contact surface, variation in outer diameter in the cross-sectional shape of the heat-transfer tube  6 , and deviation from the designed profile along the length direction of the heat-transfer tube  6  caused by waviness of the heat-transfer tube  6  in the length direction, due to manufacturing tolerances of the heat-transfer tubes  6  and the anti-vibration members  12 .  FIGS. 9A to 9D  depict manufacturing tolerances regarding the outer diameter, the plate thickness, and warpage of the heat-transfer tubes  6  included in the heat-transfer tube bundle  10 , and the plate thickness and warpage of the anti-vibration members  12 . 
     The manufacturing tolerances described herein are for illustrative purpose only, and the error parameter ε may be set for other types of manufacturing tolerances, based on the same spirit. 
     For instance, when the thickness is defined as the dimension of the anti-vibration member  12  along a direction perpendicular to the contact surface between the anti-vibration member  12  and the heat-transfer tube  6 ,  FIG. 9A  shows an error added to the normal thickness, which is a designed dimension relating to the thickness of the anti-vibration member  12  due to the manufacturing tolerances. Accordingly, if the actual thickness of the anti-vibration member  12  is larger than the designed dimension due to the manufacturing tolerances, the amount of the gap between the anti-vibration member  12  and the heat-transfer tube  6  is smaller than the designed dimension. Conversely, if the actual thickness of the anti-vibration member  12  is smaller than the designed dimension due to the manufacturing tolerances, the amount of the gap between the anti-vibration member  12  and the heat-transfer tube  6  is larger than the designed dimension. 
       FIG. 9B  shows an error corresponding to the amount of deviation (warpage amount) from flatness of the anti-vibration member  12 , due to the manufacturing tolerances, caused by warpage in the thickness direction of the anti-vibration member  12  designed to be flat in the thickness direction. Accordingly, when the anti-vibration member  12  is not flat in the thickness direction due to the manufacturing tolerances, the amount of the gap between the anti-vibration member  12  and the heat-transfer tube  6  is larger or smaller than the designed dimension in accordance with the warpage amount in the thickness direction of the anti-vibration member  12 . 
       FIG. 9C  shows an error caused by an irregular outer diameter of the heat-transfer tube  6  designed to have a circular cross-section and disposed between two anti-vibration members  12 , due to the manufacturing tolerances. That is, the outer diameter of the heat-transfer tube  6  designed to have a circular cross-section should be uniform over the entire circumference of the heat-transfer tube  6 . However, in practice, the heat-transfer tube  6  includes a phase portion having a relatively large outer diameter and a phase portion having a relatively small outer diameter due to the manufacturing tolerances; thus, the heat-transfer tube  6  has an elliptical cross-section. For instance, as shown in  FIG. 9C , in the heat-transfer tube  6 , the outer diameter in a direction perpendicular to the contact surface with the anti-vibration member  12  can be larger than the inner diameter in a direction parallel to the contact surface. As a result, the amount of the gap between the anti-vibration member  12  and the heat-transfer tube  6  is smaller than the designed dimension. 
       FIG. 9D  shows a state where the shape of the heat-transfer tube  6  in the length direction is deviated from the designed profile along the length direction of the heat-transfer tube  6  due to waviness of the heat-transfer tube  6  in the length direction. When the deviation of the shape in the length direction of the heat-transfer tube  6  acts as an error for the amount of the gap between the heat-transfer tube  6  and the anti-vibration member  12 , the amount of the gap between the heat-transfer tube  6  and the anti-vibration member  12  becomes larger or smaller than the designed dimension. As described above, the error parameter setting part  212  provides a tolerance to the FEM model which represents the amount of the gap between each of the heat-transfer tubes  6  and each of the anti-vibration members  12  so that the error parameter ε as illustrated in  FIGS. 9A to 9D  is reflected. 
     Then, the flowchart shown in  FIG. 5  proceeds to step S 3 , and the analysis part  213  performs structural analysis by the finite-element method using the FEM model (first one-dimensional element  53 , second one-dimensional element  50 , and gap element  62 ) with the error parameter set by the error parameter setting part  212 . In step S 31  shown in  FIG. 5 , the analysis part  213  sets a later-described loading condition for the gap amount represented by the gap element  62 A to  62 D shown in  FIG. 6 . For instance, in the example shown in  FIG. 6 , the loading condition is set so that, when the gap amount represented by the gap element  62 A to  62 D is less than a predetermined value, a contact force applied to the first one-dimensional element  53  from the second one-dimensional element  50  increases with an increase in interference. The gap amount represented by the gap element  62 A to  62 D is an amount whose initial value between the heat-transfer tube  6  and the anti-vibration member  12  at the time of designing the dimension is represented, for instance, by zero and which decreases as a side edge of the heat-transfer tube  6  facing the contact surface with the anti-vibration member  12  approximates to the contact surface of the anti-vibration member  12 . 
     The initial gap amount may be not zero. For instance, the initial gap amount may be set to have a positive value. 
     That is, the loading condition imposed on the gap element  62 A to  62 D causes the contact force applied to the first one-dimensional element  53  ( 53 A to  53 C) from the second one-dimensional element  50  ( 50 A,  50 B) to change with an increase in interference represented by the gap elements  62 A to  62 D, as shown by a fold line in  FIG. 10 . With reference to  FIG. 10 , under this loading condition, as long as the gap amount exists between the first one-dimensional element  53  and the second one-dimensional element  50 , i.e., a physical gap amount exists between the heat-transfer tube  6  and the anti-vibration member  12 , the contact force applied to the first one-dimensional element  53  from the second one-dimensional element  50  is left zero. However, when the gap amount between the first one-dimensional element  53  and the second one-dimensional element  50  negatively increases, and the heat-transfer tube  6  physically interferes with the anti-vibration member  12 , the contact force increases with the increase in gap amount in proportion to the physical interference. 
     When the contact force increases with the increase in gap amount in proportion to the physical interference, the slope of the straight line of  FIG. 10  corresponding to the proportionality constant is modeled by a spring constant K. That is, assuming that the gap amount is displacement amount applied to a spring by an external force in proportion to the physical interference, the contact force is equivalent to a spring load obtained by multiplying this displacement amount by the spring constant K. In this way, the above-described loading condition is set for the four gap elements  62 A to  62 D corresponding to the four gaps between the first one-dimensional elements  53 B,  53 C and the second one-dimensional elements  50 A,  50 B. 
     Then, significance of providing the loading condition described with reference to  FIG. 10  to the FEM model of the heat-transfer tube bundle  10  will be described. The loading condition allows the FEM model to reflect the following physical interaction between the heat-transfer tube  6  represented by the first one-dimensional element  53  and the anti-vibration member  12  represented by the second one-dimensional element  50 . When the interference corresponding to the gap element  62  exceeds the physical gap amount between the heat-transfer tube  6  and the anti-vibration member  12 , the exceedance corresponds to the interference between the heat-transfer tube  6  and the anti-vibration member  12 . As a result, a compression load or a bending load occurs in the heat-transfer tube  6  and the anti-vibration member  12 ( 2 ) in accordance with the interference between the heat-transfer tube  6  and the anti-vibration member  12 ( 2 ). The compression load or the bending load which occurs in the heat-transfer tube  6  and the anti-vibration member  12 ( 2 ) causes a contact load between the heat-transfer tube  6  and the anti-vibration member  12  and suppresses vibration caused in the heat-transfer tube bundle  10  due to the flow of the fluid. As described above, once the analysis part  213  sets the loading condition for the gap amount corresponding to the gap element  62 , the flowchart of  FIG. 5  proceeds to step S 32 . 
     In step S 32 , a finite-element analysis is performed on the whole of the FEM model of the heat-transfer tube bundle  10  in which the error parameter c and the loading condition are set as described above. The significance of performing the finite-element analysis on the gap amount represented by the gap element  62  as the amount of the gap between the first one-dimensional element  53  and the second one-dimensional element  50  immediately after step S 31  of  FIG. 5  will be described. That is, the gap amount immediately after step S 31  of  FIG. 5  does not consider an interaction of the compression load or the bending load which acts between the plurality of anti-vibration members  12  and the plurality of heat-transfer tubes  6  constituting the heat-transfer tube bundle  10  over the whole heat-transfer tube bundle  10 . Accordingly, the finite-element analysis is repeatedly performed until the compression load or the bending load which acts between the plurality of anti-vibration members  12  and the plurality of heat-transfer tubes  6  constituting the heat-transfer tube bundle  10  is converged to an equilibrium state over the whole heat-transfer tube bundle  10 . 
     As described above, as a result of the finite-element analysis, it is possible to achieve a state where the compression load or the bending load which acts between the plurality of anti-vibration members  12  and the plurality of heat-transfer tubes  6  constituting the heat-transfer tube bundle  10  is in equilibrium over the whole heat-transfer tube bundle  10 . Then, the analysis part  213  can estimate the contact force applied to the heat-transfer tube  6  from the anti-vibration member  12  in the equilibrium state in a numerical analysis manner. In this way, once the analysis part  213  completes the finite-element analysis, the flowchart of  FIG. 5  proceeds to step S 33 , and the analysis part  213  calculates a distribution of the contact force applied to the heat-transfer tube  6  from the anti-vibration member  12 , based on the gap amount between the first one-dimensional element  53  and the second one-dimensional element  50 . As a result, the contact force acting between the plurality of anti-vibration members  12  and the plurality of heat-transfer tubes  6  constituting the heat-transfer tube bundle  10  is determined as a contact force distribution over the whole heat-transfer tube bundle  10 . Thus, it is possible to evaluate the vibration damping property of the whole heat-transfer tube bundle  10  in a numerical analysis manner. 
     As described above, in the embodiment described with reference to  FIGS. 4 to 10 , the vibration damping structure to be analyzed is represented by the FEM model to perform structural analysis by the finite-element method. At this time, the error parameter ε is set for the parameter relating to the element included in the FEM model, which enables structural analysis with high precision, in consideration of an error factor of the vibration damping structure disposed in a fluid. Further, in this embodiment, the vibration damping structure to be analyzed is represented by the first to third elements. Since the FEM model corresponding to the vibration damping structure can be efficiently constructed by limited variable parameters, it is possible to perform structural analysis precisely with reduced computation load. 
     Further, in this embodiment, the first and the second elements are represented by one-dimensional elements while the third element is represented by a gap element disposed therebetween. Thereby, it is possible to construct a FEM model having a considerably simple structure. Such a FEM model efficiently reduces the variable parameters and thus has low computation load and is available for high-speed arithmetic processing. Further, in this embodiment, the gap amount set as an element (third element) of the FEM model corresponds to the amount of the gap between the heat-transfer tube  3  and the anti-vibration member  12  in an actual vibration damping structure. Thus, the loading condition is imposed so that, when the gap amount is less than a predetermined value, the contact force applied to the first one-dimensional element from the second one-dimensional element increases with an increase in interference, whereby it is possible to appropriately simulate the behavior of the contact force mutually acting on the heat-transfer tube  3  and the anti-vibration member  12  when they are in contact by computing analysis. 
     Further, in this embodiment, since the error parameter ε includes a random number, it is possible to randomize an error of the parameter for which the error parameter ε is to be set. Generally, errors of actual structural members are statistically uniform. Thus, setting the error parameter in this manner enables accurate reproduction of errors of an actual vibration damping structure and enables precise analysis. Further in this embodiment, the error parameter ε is set based on data measured on the vibration damping structure. Thus, even if an actual structure has statistically nonuniform errors due to some factors, appropriate analysis can be performed by setting the error parameter based on the measurement data. 
     In some embodiments, the error parameter ε may include at least one of the outer diameter, the plate thickness, and warpage of the heat-transfer tubes  3  included in the heat-transfer tube bundle  10 , and the plate thickness and warpage of the anti-vibration members  12 . In these embodiments, since the parameters which tend to affect analysis results in this type of vibration damping structure are set as the error parameter, it is possible to simulate the behavior of an actual structure more closely, and it is possible to perform analysis with high precision. 
     In an illustrative embodiment, as described later, focusing on the fact that, when the anti-vibration member  12  is viewed from a cross-sectional direction perpendicular to the longitudinal direction of the anti-vibration member  12 , the anti-vibration member  12  has a twist amount (see  FIG. 11B  and  FIG. 11C ) with respect to an extension direction of the heat-transfer tube  6 , the FEM model may be made in consideration of the twist amount. The twist amount ( FIG. 11B  and  FIG. 11C ) of the anti-vibration member  12  with respect to the extension direction of the heat-transfer tube  6  is caused by the manufacturing tolerance of the anti-vibration member  12 , and the twist is one of manufacturing errors randomly distributed in accordance with the manufacturing tolerance distribution. 
     That is, in the structure of an actual heat-transfer tube bundle  10 , the contact surface, at which the anti-vibration member  12  is in contact with the heat-transfer tube  3 , has a width along the length direction of each heat-transfer tube  6  and has a twist amount as illustrated in  FIGS. 11B and 11C , with respect to the length direction of the heat-transfer tube as a reference. Accordingly, the width direction of the contact surface, at which the anti-vibration member  12  is in contact with the heat-transfer tube  3 , has an angular offset with respect to the length direction of each heat-transfer tube  6 . As a result, the contact surface of the anti-vibration member  12  having the twist amount abuts on the heat-transfer tube  6  only at one end in the width direction of the contact surface. 
     In accordance with this embodiment, the FEM model for modeling an interaction between the heat-transfer tube  6  and the anti-vibration member  12  will now be described with reference to  FIG. 12 . The embodiment shown in  FIG. 12  is same as the embodiment described with  FIG. 6  in that the FEM model includes the plurality of first one-dimensional elements  53  extending to represent the respective heat-transfer tubes  6  included in the heat-transfer tube bundle  10 . Also, the embodiment shown in  FIG. 12  is same as the embodiment described with  FIG. 6  in that the FEM model includes the second one-dimensional element  50  extending to along the longitudinal direction of the anti-vibration member  12 . However, the embodiment shown in  FIG. 12  differs from the embodiment described with  FIG. 6  in that the FEM model further includes a third one-dimensional element  51  extending along the width direction of the anti-vibration member  12 . The third one-dimensional element  51  includes a pair of gap elements  72  which represent the shortest distance between each end of the third one-dimensional element  72  and the first one-dimensional element  53 . 
     In the embodiment shown in  FIG. 12 , the FEM model element which describes the anti-vibration member  12  further includes, in addition to the second one-dimensional element  50  extending along the longitudinal direction, the third one-dimensional element  51  extending along the width direction of the anti-vibration member  12 . In addition, in the embodiment shown in  FIG. 12 , the gap element modeling the amount of gap between each heat-transfer tube  6  and each anti-vibration member  12  is defined as the pair of gap elements  72  which represent the shortest distance between each end of the third one-dimensional element  51  and the first one-dimensional element  53  extending along the longitudinal direction of each heat-transfer tube  6 . 
     For instance, in  FIG. 12 , a third one-dimensional element  51 D extending along the width direction of the anti-vibration member  12 ( 2 ) is provided to represent the twist amount in accordance with the angular offset in the width direction of the anti-vibration member  12 ( 2 ) with respect to the extension direction of the heat-transfer tube  6 ( 2 ). Further, a pair of gap elements  72 E and  72 D are set between the first one-dimensional element  53 B corresponding to the heat-transfer tube  6 ( 2 ) and the third one-dimensional element  51 D corresponding to the anti-vibration member  12 ( 2 ) to represent an interaction between the heat-transfer tube  6 ( 2 ) and the anti-vibration member  12 ( 2 ). Similarly, a third one-dimensional element  51 E extending along the width direction of the anti-vibration member  12 ( 2 ) is provided to represent the twist amount in accordance with the angular offset in the width direction of the anti-vibration member  12 ( 2 ) with respect to the extension direction of the heat-transfer tube  6 ( 3 ). Further, a pair of gap elements  72 A and  72 B are set between the first one-dimensional element  53 C corresponding to the heat-transfer tube  6 ( 3 ) and the third one-dimensional element  51 E corresponding to the anti-vibration member  12 ( 2 ) to represent an interaction between the heat-transfer tube  6 ( 3 ) and the anti-vibration member  12 ( 2 ). 
     Similarly, a third one-dimensional element  51 A extending along the width direction of the anti-vibration member  12 ( 1 ) is provided to represent the twist amount in accordance with the angular offset in the width direction of the anti-vibration member  12 ( 1 ) with respect to the extension direction of the heat-transfer tube  6 ( 2 ). Further, a pair of gap elements  72 G and  72 H are set between the first one-dimensional element  53 B corresponding to the heat-transfer tube  6 ( 2 ) and the third one-dimensional element  51 A corresponding to the anti-vibration member  12 ( 1 ) to represent an interaction between the heat-transfer tube  6 ( 2 ) and the anti-vibration member  12 ( 1 ). Similarly, a third one-dimensional element  51 C extending along the width direction of the anti-vibration member  12 ( 1 ) is provided to represent the twist amount in accordance with the angular offset in the width direction of the anti-vibration member  12 ( 1 ) with respect to the extension direction of the heat-transfer tube  6 ( 3 ). Further, a pair of gap elements  72 C and  72 D are set between the first one-dimensional element  53 C corresponding to the heat-transfer tube  6 ( 3 ) and the third one-dimensional element  51 C corresponding to the anti-vibration member  12 ( 1 ) to represent an interaction between the heat-transfer tube  6 ( 3 ) and the anti-vibration member  12 ( 1 ). 
     Here, consider a case where, in  FIG. 12 , the interference represented by the gap element  72 A between the heat-transfer tube  6 ( 3 ) and the anti-vibration member  12 ( 2 ) exceeds the physical gap amount between the heat-transfer tube  6 ( 3 ) and the anti-vibration member  12 ( 2 ). In this case, the contact force corresponding to the twist load of the anti-vibration member  12 ( 2 ) acts between one end in the width direction of the anti-vibration member  12 ( 2 ) and the contact surface of the heat-transfer tube  6 ( 3 ). Similarly, consider a case where the interference represented by the gap element  72 B between the heat-transfer tube  6 ( 3 ) and the anti-vibration member  12 ( 2 ) exceeds the physical gap amount between the heat-transfer tube  6 ( 3 ) and the anti-vibration member  12 ( 2 ). In this case, the contact force corresponding to the twist load of the anti-vibration member  12 ( 2 ) acts between the other end in the width direction of the anti-vibration member  12 ( 2 ) and the contact surface of the heat-transfer tube  6 ( 3 ). It is needless to say that the same interaction acts between the heat-transfer tube  6 ( 2 ) and the anti-vibration member  12 ( 2 ), and the same interaction acts between the heat-transfer tube  6 ( 2 ) and  6 ( 3 ) and the anti-vibration member  12 ( 1 ). 
     Accordingly, the following loading conditions are imposed on the gap elements  72 A and  72 B representing the interaction between the heat-transfer tube  6 ( 3 ) and the anti-vibration member  12 ( 2 ). That is, when the interference represented by the gap elements  72 A or  72 B is less than the physical gap amount between the heat-transfer tube  6 ( 3 ) and the anti-vibration member  12 ( 2 ), the contact force between the anti-vibration member  12 ( 2 ) and the heat-transfer tube  6 ( 3 ) is left zero. On the other hand, when the interference exceeds the physical gap amount between the heat-transfer tube  6 ( 3 ) and the anti-vibration member  12 ( 2 ), the contact force corresponding to the twist load acts on either end in the width direction of the anti-vibration member  12 ( 2 ). Accordingly, in this case, a spring constant model generating a spring load in accordance with exceedance by which the gap amount exceeds the physical gap amount is set for the gap element  72 A or  72 B. The same loading condition is imposed on the gap elements  72 C to  72 H as the loading condition imposed on the gap elements  72 A and  72 B. 
     As described above, in the FEM model shown in  FIG. 12 , the contact surface of the anti-vibration member  12  has a width along the length direction of each of the heat-transfer tubes  6  so as to establish a surface contact with each of the heat-transfer tubes  6 . Further, in the FEM model shown in  FIG. 12 , the width direction on the contact surface of the anti-vibration member  12  with the above width has an angular offset with respect to the length direction of the heat-transfer tube  6 , so that the contact surface of the anti-vibration member  12  has twist with respect to the length direction of the heat-transfer tube  6 . Accordingly, in the FEM model shown in  FIG. 12 , the contact surface of the anti-vibration member  12  abuts on the heat-transfer tube  6  at least partially on the width in accordance with the twist. As a result, in the embodiment shown in  FIG. 12 , it is possible to appropriately simulate, based on the FEM model, a structural dynamic characteristic in which only the twist load of the anti-vibration member  12  acts on the heat-transfer tube  6  in a state where the anti-vibration member  12  abuts on the tube only at one end in the width direction of the contact surface. 
     Further, in this embodiment, the error parameter c reflected in the FEM model by the error parameter setting part  212  further includes an error parameter ε reflected in the gaps  72 A to  72 H by the error parameter setting part  212 . More specifically, the error parameter ε reflected in the gaps  72 A to  72 H corresponds to a tolerance which represents variation in twist amount of the anti-vibration member  12  caused by an angular offset when the contact surface of the anti-vibration member  12  has the angular offset with respect to the extension direction of the heat-transfer tube  3 . 
     As described above, in this embodiment, the second element corresponding to the anti-vibration member  12  is represented by the second one-dimensional element  50  and the third one-dimensional element  51  in a two-dimensional manner, and the pair of gap elements  72  are provided as the third element between each end of the third one-dimensional element  51 . Thereby, it is possible to reproduce a high-dimensional behavior, such as twist of the anti-vibration member  12  with respect to each heat-transfer tube  6  constituting the heat-transfer tube bundle  10 , as a behavior close to reality, and it is possible to achieve more precise structural analysis. 
     In another illustrative embodiment, the FEM model is generated for the heat-transfer tube bundle  10  including the heat-transfer tubes  3  each of which has a first straight tube part  4  positioned on a fluid inlet side, a second straight tube part  5  positioned on a fluid outlet side, and a bent part  6  positioned between the first straight tube part  4  and the second straight tube part  5 . In this embodiment, as shown in  FIG. 13 , the first straight tube part  4  and the second straight tube part  5  are inserted into through holes  9  ( 9 ( 1 ) to  9 ( 6 )) formed in the tube support plate  7  for supporting the heat-transfer tube bundle  10 . 
     As described with reference to  FIG. 1 , the anti-vibration member  12  in the heat-transfer tube bundle  10  is configured to restrict movement of the plurality of heat-transfer tubes  3  in the out-of-plane direction D 1  perpendicular to the plane including the bent parts  6 . Additionally, in the heat-transfer tube bundle  10  shown in  FIG. 1 , the through holes  9  ( 9 ( 1 ) to  9 ( 6 )), for receiving the first straight tube parts  4  and the second straight tube parts  5  of the heat-transfer tubes  3 , are arranged in accordance with a pattern in a top view of the tube support plate  7  so that the bent parts  6  with different curvature radii are arranged along the in-plane direction D 2  parallel to the above plane, and the bent parts  6  with the same curvature radius are arranged along the out-of-plane direction D 1 . 
     This embodiment will now be described in detail with reference to  FIG. 13 . In this embodiment, the bending load which acts between the heat-transfer tube  3  and the anti-vibration member  12  is evaluated further considering interference caused when a position of inserting the first straight tube part  4  and the second straight tube part  5  into the through holes  9  in the tube support plate  7  is deviated from the center of the through hole  9  among the interference between the heat-transfer tube  3  and the anti-vibration member  12 . Further, in this embodiment, the contact force applied to the bent part  6  of the heat-transfer tube  3  from the anti-vibration member  12  is evaluated based on the bending load evaluated, further considering the interference caused when the insertion position of the heat-transfer tube  3  is deviated from the center of the through hole  9 . 
     For instance, as shown in  FIG. 13 , when the heat-transfer tube  3 (X) is inserted into the through hole  9 ( 2 ) provided in the tube support plate  7 , the center of a circular cross-section of the heat-transfer tube  3 (X) is deviated from the center of the through hole  9 ( 2 ). In this case, the eccentricity amount of the insertion position of the straight tube part in the through hole  9 ( 2 ) shown in  FIG. 13  is represented by distance ΔC between the center of the circular cross-section of the heat-transfer tube  3 (X) and the center of the through hole  9 ( 2 ). Moreover, in this embodiment, the error parameter ε includes a tolerance which represents variation in magnitude of deviation of the insertion position of the first straight tube part  4  and the second straight tube part  5  in the through hole  9  in the tube support plate  7  from the center of the through hole  9 . In brief, in this embodiment, in step S 2  of  FIG. 5 , the eccentricity amount of the insertion position of the first straight tube part  4  and the second straight tube part  5  in the through holes  9  ( 9 ( 1 ) to  9 ( 6 )) shown in  FIG. 13  is included in the error parameter e. Under this condition, in this embodiment, in step S 3  of  FIG. 5 , structural analysis is performed to evaluate the damping force of the heat-transfer tube bundle  10 . 
     As described above, in this embodiment, structural analysis is performed on the vibration damping structure including the plurality of anti-vibration members  12  and the plurality of heat-transfer tubes  3  constituting the heat-transfer tube bundle  10  by the finite-element method using the FEM model, in consideration of a structure in which the straight tube parts of each heat-transfer tube  3  are inserted to the through holes  9  formed in the tube support plate  7  for supporting the heat-transfer tube bundle  10 . Further, in this embodiment, the eccentricity amount of the insertion position of the straight tube parts in the through holes  9  is included in the error parameter ε to perform the structural analysis. Accordingly, in this embodiment, it is possible to perform the structural analysis, in consideration of increase and decrease in contact load when the contact load applied to the anti-vibration member  12  from the bent part  6  adjoining the straight tube parts of each heat-transfer tube  3  increases or decreases due to the eccentricity amount ΔC of the insertion position of the straight tube parts in the through holes  9 . Further, in this embodiment, it is possible to perform the structural analysis, in consideration of variation due to an error of the eccentricity amount ΔC of the insertion position of the straight tube parts in the through holes  9 . 
     REFERENCE SIGNS LIST 
     
         
           4  First straight tube part 
           5  Second straight tube part 
           6  ( 6   a   1 ,  6   a   2 ,  6   a   3 ,  6   b   1 ,  6   c   1 ) Bent part 
           7  Tube support plate 
           8  Tube array 
           10  Heat-transfer tube bundle 
           10   a  Bent portion 
           11  First retention member 
           12  Anti-vibration member 
           12   a  End part 
           14  Second retention member 
           20  Computer device 
           21  Computing unit 
           22  Storage unit 
           22   a  Program 
           22   b  Data 
           23  Output unit 
           24  Input unit 
           50  ( 50 A,  50 B) Second one-dimensional element 
           51  ( 51 A to  51 E) Third one-dimensional element 
           53  ( 53 A,  53 B,  53 C) First one-dimensional elements 
           62  ( 62 A to  62 D) Gap element 
           72  ( 72 A to  72 H) Gap element 
           211  FEM model making part 
           212  Error parameter setting part 
           213  Analysis part 
         D 1  Out-of-plane direction 
         D 2  In-plane direction 
         G Flow direction 
         K Spring constant