# Integrated Tolerance and Fixture Layout Design for Compliant Sheet Metal Assemblies
### Compliant Variation Simulations  
The parts of a compliant assembly may deform during the assembly process and spring-back can occur. Therefore, finite element analysis is utilized in order to determine the deviations of these assemblies from their nominal geometry (i.e., the geometry after spring-back). The assembly process of spot-welded sheet metal assemblies can be divided into four steps [39]. The first step is to position the non-ideal parts in the assembly fixture.  
Figure 1: A fixture layout for a single sheet metal part and the locking direction of each locator [22].  
The second step is to clamp these parts into their nominal shapes. Subsequently, the parts are welded in the third step, and the clamps are released in the fourth step.  
Deviations of the parts from their nominal geometries are represented by \([D_{1}]\). Therefore, the clamping forces (in the second step) can be determined by Equation (1). In this equation, \([K_{1}]\) and \([F_{c1}]\) are the stiffness matrix of the parts before assembly and the clamping forces, respectively.  
\[[F_{c1}]=[K_{1}][D_{1}] \tag{1}\]  
The stiffness matrix of parts after welding can be determined by merging the nodes of parts in the welding areas. This matrix is indicated by \([K_{2}]\). Supposing that the welding does not imply additional deformations, the clamping forces before and after welding are equal (\([F_{c1}]=[F_{c2}]\)). Thereby, the second FE simulations can determine the deviations of the assembly after releasing the clamps, \([D_{2}]\), through Equation (2).  
\[[D_{2}]=[K_{2}]^{-1}[F_{c2}]=[K_{2}]^{-1}[F_{c1}]=[K_{2}]^{-1}[K_{1}][D_{1}] \tag{2}\]  
The distribution of deviation of the assembly can be determined by simulating thousands of random input deviations of the parts. FE simulations may have a relatively high calculation cost. Consequently, determining the variation of thousands of assemblies will be computationally expensive. [PERSON] and [PERSON] [39] have addressed this issue by assuming a linear relation between the deviations of the assemblies and parts. Thereby, a sensitivity matrix (\([S]\)) is determined by conducting two FE simulations. Hence, Equation (3) can determine the assembly deviations for every part deviation. This method is known as the method of influence coefficients (MIC).  
\[[D_{2}]=[S][D_{1}] \tag{3}\]  
[PERSON] et al. [40] demonstrated that this method may lead to severe errors in the results, because the contacts between the parts are not considered. They proposed a new MIC, in which contact modeling is considered, but it is not linear. The reason for non-linearity is that the contacts are dependent on the deviation of the parts, and these deviations vary for each part. Consequently, the stiffness matrix should be updated in every iteration of the simulations. [PERSON] et al. [41] improved this method by developing a method for automatically detecting these contacts in each iteration. [PERSON] et al. [42] further improved the efficiency of the variations by presenting a quadric optimization to solve the contact equations.  
The variation simulation techniques have become more sophisticated by considering the effects of heat [43], non-linearity of materials [44], and welding sequences [45,46]. Several commercial programs have been developed to implement these techniques in CAD models for practical uses.