Abstract
The two-dimensional differential equation y’ = f(x,y) can be interpreted as a direction field. Commercial Finite Element (FE) programs can be used for this integration task without additional programming, provided that these programs have options for the calculation of orthotropic heat conduction problems. The differential equation to be integrated with arbitrary boundaries is idealized as an FE model with thermal 2D elements. Its orthotropic thermal conductivities are specified as k1 = 1 and k2 = 0. In doing so, k1 is parallel to y´, and k2 is oriented perpendicular to this. For this extreme case, it is shown that the isotherms are identical to the solution of y’ = f(x,y). The direction fields, for example, can be velocity vectors in fluid mechanics or principal stress directions in structural mechanics. In the case of the latter, possibilities for application in the construction of fiber-reinforced plastics (FRP) arise, since fiber courses, which follow the local principal stress directions, make use of the superior stiffness and strength of the fibers.
Highlights
Orthotropic materials can have extremely different thermal conductivities, for example, in printed circuit boards
Should the guided conduction paths be directed parallel to an arbitrary direction field y’ = f(x,y), the following hypothesis shall be mathematically verified: Hypothesis: The isotherms of an orthotropic steady-state 2D thermal conduction problem with the thermal conductivities k1 and k2 are tangential to an arbitrarily prescribed direction field y = f(x,y) provided that the local orientation of k1 follows the direction field y, and perfect insulation exists perpendicular to this (k2 = 0)
Should the designer change the fiber volume ratio for reasons of strength, the procedure must be repeated from the beginning. This problem is substantially simplified through the hypothesis set out above, which can be modeled by means of the orthotropic heat conduction in Finite Element (FE) programs
Summary
Orthotropic materials can have extremely different thermal conductivities, for example, in printed circuit boards. Visualization of stress trajectories should permit detection and identification of bifurcation to provide insight into the time-dependent structural evolution These issues are beyond the scope of this paper. Each start element (marked in red) in Figure 1a provides two individual fiber courses with the procedure just described, corresponding to the two PS directions. Should the designer change the fiber volume ratio for reasons of strength, the procedure must be repeated from the beginning This problem is substantially simplified through the hypothesis set out above, which can be modeled by means of the orthotropic heat conduction in FE programs. The practicability of the integration method is based on independence from thermal boundary conditions; the isotherms always follow the direction field in nonsingular areas. The influence of the orthotropy ratio k1/k2, as well as the influence of singularities on the course of the isotherms, is investigated
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