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An exact transient analytical solution of the temperature fields and heat fluxes for the modified transient plane source (MTPS) method is presented. The MTPS sensor is made of a thin disc with a heater/sensor and a heated guard ring on its surface, providing a one-dimensional heat flow to a semi-infinite sample in contact with its surface. The MTPS sensor measures the thermal effusivity of the sample. Two different solutions are derived, one is in the form of an infinite series with no thermal resistance, the other is in integral form and includes thermal resistance between the MTPS sensor surface and the measured sample. The theoretical model is supplemented by experimental results.

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This paper presents a method to identify the exact Pareto front for a multi-objective optimization problem. The developed technique addresses the identification of the Pareto frontier in the cost space and the Pareto set in the design space for both constrained and unconstrained optimization problems. The proposed approach identifies a n – 1 dimensional hypersurface for a multi-objective problem with n cost functions, a subset of which constitute the Pareto front. The n – 1 dimensional hypersurface is identified by enforcing a singularity constraint on the Jacobian of the cost vector with respect to the optimization parameters. Since the boundary is identified in the design space, the relation of design points to the exact Pareto front in the cost space is known. The proposed method is proven effective in the Pareto identification for a set of previously released challenge problems. Six of these examples are included in this paper; 3 unconstrained and 3 constrained.