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The bond-propagation (BP) algorithm for the specific heat of the twodimensional Ising model is developed and that for the internal energy iscompleted. Using these algorithms, we study the critical internal energy andspecific heat of the model on the square lattice and triangular lattice withfree boundaries. Comparing with previous works [X.-T. Wu {\it et al} Phys. Rev.E {\bf 86}, 041149 (2012) and Phys. Rev. E {\bf 87}, 022124 (2013)], we reachmuch higher accuracy ($10^{-26}$) of the internal energy and specific heat,compared to the accuracy $10^{-11}$ of the internal energy and $10^{-9}$ of thespecific heat reached in the previous works. This leads to much more accurateestimations of the edge and corner terms. The exact values of some edge andcorner terms are therefore conjectured. The accurate forms of finite-sizescaling for the internal energy and specific heat are determined for therectangle-shaped square lattice with various aspect ratios and various shapedtriangular lattice. For the rectangle-shaped square and triangular lattices andthe triangle-shaped triangular lattice, there is no logarithmic correctionterms of order higher than 1/S, with S the area of the system. For thetriangular lattice in rhombus, trapezoid and hexagonal shapes, there existlogarithmic correction terms of order higher than 1/S for the internal energy,and logarithmic correction terms of all orders for the specific heat.

Accurate expansions of internal energy and specific heat of critical two-dimensional Ising model with free boundaries