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We give a construction of Lagrangian torus fibrations with controlleddiscriminant locus on certain affine varieties. In particular, we apply ourconstruction in the following ways. We find a Lagrangian torus fibration on the3-fold negative vertex whose discriminant locus has codimension 2; thisprovides a local model for finding torus fibrations on compact Calabi-Yau3-folds with codimension 2 discriminant locus. Then, we find a Lagrangian torusfibration on a neighbourhood of the one-dimensional stratum of a simple normalcrossing divisor (satisfying certain conditions) such that the base of thefibration is an open subset of the cone over the dual complex of the divisor.This can be used to construct an analogue of the non-archimedean SYZ fibrationconstructed by Nicaise, Xu and Yu.

Constructing local models for Lagrangian torus fibrations