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A numerically efficient, accurate, and easily implemented integration schemeover convex Voronoi polyhedra (VP) is presented for use in {\it ab-initio}electronic-structure calculations. We combine a weighted Voronoi tessellationwith isoparametric integration via Gauss-Legendre quadratures to providerapidly convergent VP integrals for a variety of integrands, including thosewith a Coulomb singularity. We showcase the capability of our approach by firstapplying to an analytic charge-density model achieving machine-precisionaccuracy with expected convergence properties in milliseconds. For contrast, wecompare our results to those using shape-functions and show our approach isgreater than $10^{5}$ faster and $10^{7}$ more accurate. A weighted Voronoitessellation also allows for a physics-based partitioning of space thatguarantees convex, space-filling VP while reflecting accurate atomic size andsite charges, as we show within KKR methods applied to Fe-Pd alloys.

Efficient Isoparametric Integration over Arbitrary, Space-Filling Voronoi Polyhedra for Electronic-Structure Calculations