The static optimal heating of solids by a boundary heat flux has been formulated for a solid of arbitrary geometry. Through the use of a Lagrange multiplier the problem is reduced to an unconstrained optimization problem. The necessary conditions for optimally of a performance index that characterizes the physical objectives are first found by calculus of variations. An iterational numerical procedure is then proposed in which the elliptic equations for the temperature and the Lagrange multiplier are solved by the boundary element method, and better estimates for the boundary heat flux are found by the minimization of the performance index through the conjugate gradient method of optimization. Numerical results are also provided for a two-dimensional problem.

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