Abstract
In order to improve the precision of data transfer in fluid-structure interaction (FSI) analysis, this paper puts forward an improved inverse isoparametric mapping (IIM) method, which can improve the load transfer accuracy by means of increasing the quantity of Gauss integral points. As displacement transfer happens, the method can perform the function of revising the displacement between the point to be interpolated and projection point, by which the example emulation is also carried out. Some practical solutions are presented, aiming to grapple with pertinent problems such as local interpolation burr and global interpolation distortion which are originated from oversimplified structure dynamic models and inconsistent structural deformation. The calculation results show that the improved IIM method has higher interpolation precision, which can consequently provide a valuable source of reference for FSI analysis by improving the precision of data transfer and satisfying the conservation of total force, total torque, and total virtual work in the meantime.
1. Introduction
At present, fluid-structure interaction (FSI) analysis is being used more widely in various fields [1–4], such as aerospace, sailing, bridge construction, and medicine. The reasonableness and effectiveness of the data transfer process are crucial to FSI analysis, where interpolation technique is one of the most essential techniques to transmit surface load and deformation information between unmatched meshes efficiently.
Since 1970’s, many different coupling interpolation methods have been proposed so as to deal with data transfer problems for FSI analysis [5, 6], generally including IPS (infinite-plate splines) [7], FPS (finite-plate splines) [8], MQ (multiquadric-biharmonics) [9–12], TPS (thin-plate splines) [13], NUBS (nonuniform B-splines) [14], RBF (radial basis function) [15–20], MEM (mortar element method) [21, 22], and IIM (inverse isoparametric mapping) [6, 23, 24]. Although, some of these methods are limited in adaptability, such as IPS, TPS, FPS, MQ, and NUBS. When using these interpolation methods to carry out FSI data transfer, satisfactory results can be only obtained when the fluid and structural surfaces coincide with each other. In the meantime, the RBF method has strong robustness, however, when applying it to complex structural models, local deformation and load distribution distortions appear frequently.
The IIM method possesses high-fidelity for FSI analysis of complex models, which is widely used in many fields, such as medicine [25], aero-engine thermal-structure coupling [26], civil and hydraulic engineering [27], and petrochemical industry [28]. Nevertheless, relevant studies concerning data transfer based on IIM for FSI analysis are still comparatively immature, and the research results are limited in number. Some earlier studies [23] suggest that this method is only suitable for settling two-dimensional problems rather than three-dimensional ones, which hinders the development of it.
Murti et al. [29, 30] proposed some practical IIM method in three-dimensional problems. In 1995, Smith et al. [24] indicated that this method had great research value and tremendous application potential for FSI analysis. Su et al. [6] made a summary of the process of interpolation method based on IIM. Shen et al. [31] described the details of the IIM method. Also, to improve the interpolation efficiency of data transfer in multi-field coupling analysis, Yue [32] designed a new IIM calculation model based on ANSYS and MATLAB, whose computation process was more efficient. Based on the inverse transformation of isoparametric elements, Cai et al. [33] specifically discussed the accuracy of the algorithm on the singular element, and it is found that the method has good adaptability to singular elements.
A number of researches have shown that total force, total moment, and virtual total work should be in conservation in data transfer of FSI as far as possible. However, at present, the mesh of structural dynamics model is relatively coarse, and the aerodynamic mesh is relatively dense so that the structural surface cannot fit well with the aerodynamic surface, which makes it difficult to ensure the conservation of the moment.
In order to discuss complex problems in multifield coupling analysis, an improved IIM method is proposed, aiming to improve the accuracy of data transfer and provide a source of reference for FSI analysis. Additionally, some practical solutions are also proposed to grapple with pertinent problems that local interpolation bur and global interpolation distortion during FSI simulation, which are originated from oversimplified structure dynamics models and inconsistent structural deformation.
2. Principle of IIM Method
2.1. The Interpolation Process of FSI
In generally FSI analysis, the nodal displacements on the structure surface should be interpolated into the aerodynamic surface nodes, and then, the aerodynamic loads on the surface nodes should be interpolated into the structure surface nodes. The whole process can be formulated as equation (1) to equation (6).
Equation (1) is used to calculate the aerodynamic force after structural deformation, equation (3) indicates that the aerodynamic force on the aerodynamic mesh nodes is transferred to the structural mesh, equation (5) indicates that the structural displacement is solved by the structural equation, and equation (6) indicates that the displacement on the structural mesh nodes is transferred to the aerodynamic mesh nodes. where is a complex function composed of aerodynamic control equations and boundary conditions, is the force vector on the aerodynamic mesh nodes, represents the node coordinates on the original aerodynamic mesh, and represents the deformation of the aerodynamic mesh on the object plane, which can be written as follows,
The subscript letter represents the aerodynamic mesh, and is the node number of the aerodynamic mesh. where is the transforming matrix of aerodynamic force of the structural nodes, and is the interpolation load vector participating in interpolation on the structural surface nodes, which has the following form,
The subscript letter represents the structural unit, and is the number of nodes participating in interpolation on the structural surface. where is the stiffness matrix, and represents the displacement of structural nodes. where is the transforming matrix of the structural displacement transferring to the aerodynamic surface mesh.
In order to ensure the accuracy of data transfer, initially, the virtual work during the process should be conserved. The following equation can be obtained from the principle of virtual work,
Then, we have
Combining (6) and (2) gives the following equation,
It can be seen from equation (9) that the data transfer matrix of structural meshes transferring to aerodynamic surface meshes and the data transfer matrix of aerodynamic surface meshes transferring to structural meshes interact as both transposed matrixes.
2.2. The Calculation Method of Natural Coordinates of Projection Points of Adjacent Units in IIM Method
In the essence of IIM method, the physical coordinates of a point on the unit can be obtained through the physical coordinates of interpolation nodes, and the isoparametric transformation is a concept which describes the fact that the interpolating function of coordinate transformation is consistent with that of physical quantity. However, different from the usual concept, the coordinate transformation here is used to describe the process transforming physical coordinates into natural coordinates, hence, the coordinate-transformation matrix and the interpolation matrix of physical quantities just interact as both inversed matrixes. Therefore, calling this method inverse isoparametric mapping (IIM) method is rather appropriate.
The basic routine of the method is as follows: first, find the target unit (e.g., aerodynamic surface mesh unit) nearest to the interpolation point (e.g., structural surface unit node); second, make the orthographical projection of the point yet to be interpolated onto the unit plane, and make the projection point be the perpendicular foot; finally, confirm whether the perpendicular foot falls inside the unit plane. Because the shape function of IIM method has a poor effect in extrapolation, it is necessary to make sure that the perpendicular foot falls inside the unit plane; if not, start to find the next nearest point until the requirement is met that the interpolation point is relatively closest and the perpendicular foot falls inside the plane.
The projection of the point to be interpolated on the target unit is shown in Figure 1, where (a), (c), and (e) represent the projection point outside the unit plane, and (b), (d), and (f) represent projection point in the unit plane.
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When the target unit is determined, the natural coordinates of the source unit can be converted into global coordinates by using formula (10). In actual operation, the global coordinates exactly need converting into natural coordinates, so the reverse transformation of formula (10) is used. where is a shape function, represents the unit influence quantity of node on node ; is the global coordinate of the node on the structural unit, and is the natural coordinate of the node ; is the number of unit nodes, as for line unit, triangle unit, and quadrilateral unit, it is referred as 2, 3, and 4, respectively.
For line unit, the shape function can be expressed as follows,
For triangle unit, the shape function can be written in the following form,
For quadrilateral unit, the shape function takes the following form, where is the global coordinate, is the length of the line unit, and are area coordinates.
The shape function has better properties, as shown in equation (9). Equation (14a) ensures that the coordinates of nodes are converted exactly to its global coordinates; equation (14b) ensures that the transformation is complete and the characteristics of rigid-body translation are still the same. Besides, the interpolation process can ensure the consistency of units’ boundary before and after the transformation.
In order to judge whether the projection point is inside the unit plane or not, a practical way is to substitute the global coordinates into the left side of equation (10); then, the natural coordinates of the projection point can be obtained. If the three obtained natural coordinates’ values all satisfy equation (15), it can be ensured that the projection point is inside the target unit [6].
When searching proximate units, it is necessary to go through all the units in the target source, so at least times of searches should be conducted, where represents the number of units. Therefore, the performance of the search algorithm directly influences the efficiency of interpolation. Chen et al. have done a lot of research about the related search algorithms, especially in UAVs [34–36] and real-time systems [37].
2.3. Transformation Process of Physical Information
After locating the proximate units by means of the basic routine as mentioned in Section 2.2, the projection of point to be interpolated is inside, and then, the physical quantities of point to be interpolated can be replaced by that of the projection points in the unit. The physical quantities of the projection point can be obtained by weighting the physical quantities of the node, which takes the following form, where is unit shape function, and and are physical quantity functions in natural coordinates.
The basic routine of the interpolation method based on IIM has been thoroughly discussed in Section 2.2, but how can total force, total moment, and total virtual work during data transfer process be in conservation?
Based on the weak formulation of the conservation of loads or displacements over the interface (shown as equation (17)), Boer et al. [38] used a weighted residual method to transfer information (shown as equation (18)). However, owing to the introduction of the shape functions of structural unit and aerodynamic unit, both of which are inconsistent in integration domains, it appears difficult to effectively deal with the mixed product global part in the equation, as shown in equation (19). To avoid calculating mixed product, some improvements will be proposed in Section 2.4. where and are deformation quantities of structural mesh nodes and aerodynamic mesh nodes, respectively; and are interpolation shape functions of mesh units on the structural surface and fluid surface.
Since equation (17) always holds, its equivalent form can be written as where is an arbitrary continuous and smooth function.
Apply Galerkin’s method to weaken equation (20) to obtain weak solutions and take , then, equation (20) can be written as
And equation (21) can be simplified as
2.4. Improvement in Conservation during Information Transfer
On account of the inconsistency of the meshes on structural surface and aerodynamic surface in densities, it seems difficult to keep the conservation during information transfer.
In general, the structural mesh is much larger than the aerodynamic mesh, a structural surface mesh unit may cover several, even more than a dozen of aerodynamic units, and the aerodynamic load also has a greater gradient in some places. As a result of the facts mentioned above, when replacing the load value of the projection point by that of the structure node, serious errors often occur.
Structural units cover a lot of aerodynamic units, but in some algorithms, only the physical information of aerodynamic units near the structural nodes is used, most of the information of other covered aerodynamic units is not used. In order to avoid the errors caused by such algorithm, the Gaussian global method is used in the structural units. The value of the Gaussian global point can be also obtained through the interpolation function mentioned above. In this way, more units’ physical information on aerodynamic surface can be used to make the calculation results more accurate; nevertheless, the calculation and searching work will be enhanced. The structural unit with Gauss integral points is shown in Figure 2.
Here, we introduce two approaches to transferring the aerodynamic load to the nodes on structural surface: (1) transfer the pressure directly to the structural nodes, then, interpolate the pressure value at any point on the surface of each structural unit using the shape function, finally, obtain the nodal forces of structure unit by integrating pressure on the structural unit; (2) integrate the pressure on the surface with the aerodynamic unit, obtain the load by integration and distribute it to the nodes of aerodynamic unit, finally, interpolate the aerodynamic force on the aerodynamic mesh node to the structural mesh node. Theoretically, both methods are feasible, yet when aerodynamics calculator is processing, the flow field always fluctuates, furthermore, the results of the pressure calculation on some structural mesh nodes are sometimes inaccurate. Hence, in order to ensure the accuracy of the result, the second method is generally adopted.
For the data transfer of displacement information, due to the smoothness of the structure deformation, multiple aerodynamic mesh nodes may fall into a structural unit, then, the displacement of the projection point of the aerodynamic node can be calculated by equation (16), whose result has pinpoint accuracy. However, since the distance between the projection point and the original point to be interpolated cannot be neglected, replacing the deformation of the point to be interpolated with that of the projection point will bring some errors. Especially in the calculation of torque, such replacement will bring considerable torque errors because of the arm variability caused by the distance. In order to prevent such calculating errors, it can be reputed that the interpolated point has the rigid connection with its projection point. If the structure and aerodynamic meshes completely overlap with each other, then, we have formula (23). Unfortunately, it is often difficult for the aerodynamic mesh surface and the structural mesh surface to overlap with each other completely, so a better result can be obtained by optimizing equation (24). The correction diagram of the displacement between the point to be interpolated and projection point is shown in Figure 3. where , , and are deformation components, is the angle vector at the projection point, and .
3. Some Methods to Deal with the Discontinuity of Modal Vibration Shape and the Serious Inconsistency between the Structural Shape and the Aerodynamic Shape
In many cases, at the junctions of different parts, the vibration shape of the partitioned organization components tends to be interrupted. Even some components are so rigid that they can be directly simplified into mass elements with characteristics of mass and inertia, concurrently, some details such as edges and corners in part were cut off directly. The operation mentioned above can make structural dynamics modeling easier, but it will bring new problems to the interpolation of FSI; in order to solve these problems, some solutions are brought up.
For the discontinuity of modal vibration shape, carrying out subdivision interpolation seems to be a more practical way, which can improve searching performance and also reduce searching steps.
Some parts of the structure are simplified with a large extent, for example, some rigid parts (such as the auxiliary tank, the hanging bomb, etc.) are directly simplified into intermediate mass elements, yet there exists a significant difference between the structural shape and the aerodynamic shape, which will cause great troubles to the interpolation. In order to obtain better interpolation results, the structural model can be replenished with virtual elements, and these elements will not affect the mass and stiffness characteristics of the original finite element model.
4. Example Analyses
In this section, we take a certain wing as an example and analyze the interpolation accuracy for load and displacement based on IIM method. The structure finite element model of the wing is shown in Figure 4, which is composed of three-dimensional solid element and shell element. Aerodynamic wall mesh is shown in Figure 5.
4.1. Validation of the Interpolation Accuracy for Load
In order to analyze the interpolation accuracy for load based on IIM method, the steady flow with 0.85 ma, 5 degrees angle of attack, and 10000 pa flow rate are interpolated to structural nodes. The pressure coefficient (Cp) distribution map of upper and lower surfaces of aerodynamic meshes is shown in Figures 6 and 7, where the structure is utilized as a rigid model. There is a shock wave at the leading edge of the upper surface. Figures 8 and 9 show the force vectors and moment vectors of aerodynamic load interpolated to the structural nodes, respectively. It can be observed from Figure 8 that the force also reversed at the shock wave of upper surface.
The comparison of total force and total torque between aerodynamic surface and structural surface after aerodynamic interpolation is shown in Tables 1 and 2, where the center of moment is (0, 0, 0). It can be seen that the total force and total torque are almost identical.
According to the statistical results of the program, the interpolation accuracy for load can reach to 10e-9. Therefore, the IIM method is effective for improving the interpolation accuracy of load transfer.
4.2. Validation of the Interpolation Accuracy for Displacement
In order to analyze the interpolation accuracy of displacement based on IIM method, the interpolation accuracy of displacement based on RBF method was carried out, and the comparison of the two techniques was done.
With mass characteristics being added to the structural finite element model, the first four modes of vibration shapes of the wing model were studied to verify the accuracy of displacement interpolation. The structure’s mode shapes were shown in Figure 10.
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The interpolation of the vibration shapes based on IIM method and RBF method is shown in Figures 11 and 12, respectively, where red represents the structure’s mode shapes, and blue and green represent the interpolation results of aerodynamic wall mesh. It can be seen from the images that the interpolated structural surface and aerodynamic surface based two methods coincide with each other very well.
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Three sets of nodes at half span, quarter span, and three quarters span of the wing model were selected to analyze the displacements in the first four modes. Maximum relative errors of the interpolation displacements based on IIM to RBF are shown in Table 3. Analysis result shows that the maximum of interpolation displacement error between the two methods is 1.75%, which proved that the improved IIM method has high interpolation accuracy in displacement.
5. Conclusions
Based on the systematic summary of the traditional IIM method, the paper proposed an improved IIM method, which can improve the load transfer accuracy by adding Gauss integral points; for displacement transfer, we add the displacement correction between the point to be interpolated and projection point to the interpolation equation to further improve the transfer accuracy of the total virtual work and the total moment. Example emulations are carried out to validate the interpolation accuracy for data transfer, and for displacement transfer, simulations are carried out by comparing the case presented using the IIM method and the RBF method already in use.
In addition, according to massive previous engineering experience, some problems, which are caused by discontinuity of structural deformation and inconsistency with aerodynamic shape in FSI analysis of complex structures, can be solved by means of subdivision interpolation and adding virtual elements to the structural model, etc.
The method is a local interpolation method, which requires a high degree of geometric matching between the difference objects. Compared with global interpolation method, the IIM method has high interpolation accuracy of data transfer; however, its robustness is poor. The improved method is mainly to improve the matching degree of the geometry of both sides by means of dividing blocks and supplementing elements, so as to improve its robustness.
Numerical results indicate that the improved IIM method features better interpolation smoothness, high robustness, high accuracy, and saving of storage. The method has great application value in solving multifield coupling problems, which can provide a valuable source of reference for FSI analysis of complex-shaped structures.
Data Availability
The underlying data supporting the results of my study can be found in my manuscript.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This investigation was supported by the Natural Science Foundation of Shaanxi Province (no. S2021-JC-YB-0590). The authors would like to thank Shaohui Li from Advanced Dynamics Incorporated for providing the data of wing and his valuable suggestions.