We present a novel method, called Simplex Assembly, to computeinversion-free mappings with low or bounded distortion on simpli-cial meshes. Our method involves two steps: simplex disassemblyand simplex assembly. Given a simplicial mesh and its initial piece-wise affine mapping, we project the affine transformation associatedwith each simplex into the inversion-free and distortion-boundedspace. The projection disassembles the input mesh into disjointsimplices. The disjoint simplices are then assembled to recover theoriginal connectivity by minimizing the mapping distortion and thedifference of the disjoint vertices with respect to the piecewise affinetransformations, while the piecewise affine mapping is restrictedinside the feasible space. Due to the use of affine transformations asvariables, our method explicitly guarantees that no inverted simplexoccurs, and that the mapping distortion is below the bound duringthe optimization. Compared with existing methods, our method isrobust to an initialization with many inverted elements and positionalconstraints. We demonstrate the efficiency and robustness of ourmethod through a variety of geometric processing tasks