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Inversive distance circle packings introduced by Bowers-Stephenson arenatural generalizations of Thurston's circle packings on surfaces. To findpiecewise Euclidean metrics on surfaces with prescribed combinatorialcurvatures, we introduce the combinatorial Calabi flow, the fractionalcombinatorial Calabi flow and the combinatorial $p$-th Calabi flow for theEuclidean inversive distance circle packings. Due to the singularities possiblydeveloped by these combinatorial curvature flows, the longtime existence andconvergence of these combinatorial curvature flows have been a difficultproblem for a long time. To handle the potential singularities along thesecombinatorial curvature flows, we do surgery along these flows by edge flippingunder the weighted Delaunay condition. Using the discrete conformal theoryrecently established by Bobenko-Lutz for decorated piecewise Euclidean metricson surfaces, we prove the longtime existence and global convergence for thesolutions of these combinatorial curvature flows with surgery. This provideseffective algorithms for finding piecewise Euclidean metrics on surfaces withprescribed combinatorial curvatures.

Combinatorial curvature flows with surgery for inversive distance circle packings on surfaces