An uneven and incomplete seal ring is the main factor causing leakage, which is composed of three main scale factors. The absence of any element on any scale will result in invalid predictions of seal joint leakage. Thus, in this section, an effective geometric-based multiscale FE model was established to evaluate the impact of factors at each scale on the contact status at the seal ring.

## 2.1 The relative position of two parts during assembly

The main factors causing the change in relative position are the mating angle *α* followed by the circular runout and the flare angle *γ*. In this section, the initial relative position is investigated geometrically, and the schematic diagram is shown in Fig. 1.

To define the relative position of the two contact parts, the geometric model was built taking both the macroscopic error and mesoscopic deviation into consideration. There are three commonly occurring macroscopic errors in the assembly: the mating angle *α* between the two axes 1 of cylinders, the gap *h* in the z direction, and the gap between the two axes of cylinders. The last two errors will be reduced to the first one when two parts approach each other under the preload. Thus, only the error of angle *α* will be considered. As shown in Fig. 1a), the mating angle *α* during assembly was obtained by fixing the joint part and rotating the tube part around the axis of *ϕ *= 0. In Fig. 1c), the position of the tube was defined, and point A(*r*A,*θ*A,*ϕ*Α), used as the positioning point for the FE model, could be described through spherical coordinates:

$${\theta _{\text{A}}}=\frac{\pi }{2}+\delta \left( {\alpha ,\varphi } \right),\varphi \in \left. {\left[ { - \pi } \right.,\pi } \right)$$

2

where *θ*A is the angle between the straight-line OA and the z-axis, *a* is the length of line OA, and *ϕ* is the circumferential angle ranging from − 90° to 90°. The point B(*r*B,*θ*B,*ϕ*Β) is the top contact point of the tube, whose position can be defined by the following equation:

$${r_{\text{B}}}=\sqrt {{{(a+b\cos \beta )}^2}+{{(b\sin \beta )}^2}}$$

3

$${\theta _{\text{B}}}=\frac{\pi }{2} - \omega +\delta \left( {\alpha ,\varphi } \right),\varphi \in \left. {\left[ { - \pi } \right.,\pi } \right)$$

4

where *b* is the length of line AB, angle *ω* is the angle between straight lines OA and OB, and * β* is the angle related to the manufacturing deviation.

In Fig. 1c), the angle *θ*B is defined by two parts: the constant angle *ω* related to the structure of the tube and angle *δ*, which changes nonlinearly with the growth of mating angle *α*. As shown in Fig. 1d), a new model was built to investigate the relationship between the mating angle *α* and the corresponding angle *δ*.

$$\left\{ \begin{gathered} l= - {r_B}' \cdot \sin \delta =n \cdot \tan \alpha \hfill \\ n=m \cdot \sin \varphi \hfill \\ m={r_B}' \cdot \cos \delta \hfill \\ \end{gathered} \right.$$

5

$$\delta =\arctan (\sin \varphi \cdot \tan \alpha )$$

6

$$OO'=b \cdot \sin \beta {\prime }$$

7

The mesoscopic error is the so-called circular runout of the joint ranging from − 50 µm to + 50 µm, and it is presented by the position of point E on generating line CD. As shown in Fig. 1b) (the profile at *ϕ *= 90°), *e* is the distance of E from endpoint C, and *γ* is the angle between the line CE and the vertical coordinate **z**, which is normally equal to the flare angle of the joints.

To define the relative position of the two parts, the contact area of the joints should also be determined formulaically. Point E can be defined by the following equations:

$${r_E}=\sqrt {{{(e \cdot \sin \gamma +c)}^2}+{{(e\cos \gamma +h)}^2}}$$

8

\(\left\{ \begin{gathered} {\theta _E}=\arctan \frac{{c+e\sin {\gamma _j}}}{{h+e\cos {\gamma _j}}} \hfill \\ e=\frac{{{r_B}\cos {\theta _B} - h}}{{\cos {\gamma _j}}} \hfill \\ \end{gathered} \right.\) , \({\varphi _B} \in ( - \pi ,\pi ]\) (9)

where *h* equals the height of point C and length *c* represents the circular run-out of the joint, which is the distance of point C from the z-axis.

To date, the contact points of the joint and tube have been normalized to a unified spherical coordinate system that includes both macroscopic and mesoscopic features. In this paper, for the tube with a diameter of 16 mm, the angle *β'*= arctan (3.6/1.7), angle *ω *= arctan (3.6/9.7), and for the lengths *a* = 8 mm, *b* = 4.04 mm, *c* = 7.6 mm, and *h* = 10 mm.

## 2.2 The structural parameters

Based on the initial relative position obtained by the geometric model, the contact status of the seal ring could be simulated by reconstructing the FE model with the measured surface point cloud data.

## 2.2.1 Model generation

Through the procedure shown in Fig. 2, the two contact parts with rough surfaces were reconstructed. First, the surface characteristics were passed in the form of point clouds gained by the measured topography. The point clouds were filtered before fitting by NURBS, and then the fitted surfaces were imported into 3D design software [40]. After some stretching and combination with the fitted surfaces, solid CAE models were built and imported into the commercial finite element geometric software.

The accuracy of the reconstructed model relies on the density of the surface point cloud. The in-plane precision produced by 3D super-deep field microscope sweeping could reach as high as 0.57 µm, and the vertical accuracy can approach 0.05 µm. The in-plane accuracy of the reconstructed model used in this paper is 7.09 µm, and the vertical accuracy is 0.05 µm, counting the time consumption of subsequent computation. This balances the need to properly characterize roughness features with the need to speed up calculation. The lateral length of the tube and the joint was long enough to cover the whole conical rough seal surface, enabling the multiscale model to include the seal ring during assembly under mesoscopic geometric deviations. In addition, the width of the tube is larger than the joints to guarantee that the typical part will contact the tube during the simulation.

The multiscale solid CAE models containing both mesoscopic structural characteristics and microscopic roughness were then imported into the commercial finite element analysis software. First, the two contact parts were assembled with the prescribed relative position gain by the geometric model, and the relative positions at several typical circumferential angles were picked up.

The initial relative position shown above consists of not only the mating angle and flare angle but also the circular runout. Both the tube and the joint have circular runouts, but because the joint is made by turning rather than stamping, its manufacturing variation is substantially larger than the pipe's circular runout. Therefore, while utilizing the model developed in this work to determine the initial position of the two contact parts, only the circular runout of the joint is explored. A circular runout means that the radius of curvature varies with the circumferential angle, as illustrated in Fig. 3.

As shown in Fig. 4a) and b), to reduce the time consumption of the simulation, the parts were meshed unevenly, and unnecessary structures of the joints were ignored, such as the threaded surface. Meshes near the contact interfaces were much smaller than those far away from them. The element type of the joint and tube parts was hexahedral C3D8R elements with eight nodes. The joint part contains 143533 elements and 129600 nodes, while the tube part contains 344006 elements and 325000 nodes. The joint's mesh in-plane accuracy could reach 9.91*10.58 µm, while the tube's could reach 6.36*3.54 µm, both of which are high enough to accurately depict the degree of roughness on the contact surfaces.

## 2.2.2 FE material properties and boundary conditions

To investigate the contact behavior of the seal interface during the assembly process, a contact pair was established. The contact surface of the tube is the master surface, while the joint surface is the slave surface. The penalty function was adopted to describe the friction at the seal interface, and the friction coefficient was set to 0.15. The Newton-Raphson method was applied to the contact analysis. The joint and tube were made of stainless steel, whose mechanical properties are listed in Table 1.

Table 1

Mechanical properties of the tube and joint parts

Material | Young’s modulus (MPa) | Poisson ratio | Yield strength (MPa) |

0Cr18Ni9 | 204000 | 0.285 | 205 |

In addition, Fig. 5 shows the boundary conditions of the model. To simulate the contact behavior during the assembly process, the bottom and side surfaces in the x direction of the joint were fixed. As the typical part was selected from the whole joint, the side surfaces were restricted with symmetry constraints from moving along the circumference. To simulate the preload provided by the sleeve nut, an axial displacement was applied on the tube part through a reference point coupled to it.

The procedure for constructing the model is illustrated in detail in Fig. 6. The characteristics of the seal joints were divided into three scales: the macroscopic assembly error (mating angle *α*), the mesoscopic manufacturing deviations (flare angle *γ *and the circular runout) and the microscopic roughness. The assembly errors are on the order of 103 µm, the manufacturing deviations are on the scale of 102 µm, and the roughness is limited to the range of 10 µm. By establishing the geometric model including the mating angle *α*, the flare angle *γ * and the circular runout, the initial relative position of the two contact parts can be described. Based on the analyzed relative position, the FE model was modified, including the surface characteristics, such as the flare angle, surface material and roughness. In this way, the contact status of the seal ring could be accurately simulated.