The modal amplitude equations also involve co-ordinates of unknown "lift-off points" determined as a part of the solution for structural response through the constraints imposed on the solution derived by the Galerkin method using linear modal bases.
The atom map is then determined as the solution of a combinatorial optimization problem resulting in a bijective mapping of all vertices of the educt molecule graph to corresponding vertices in the product molecule graphs.
Note that static and dynamic-FAA are determined as the solution of an optimization problem where a lower bound on the effective capacity is used as the objective function.
The solution is expressed as an infinite trigonometric series, whose coefficients are determined as the solution of an infinite system of ordinary differential equations.
We note that the hydrostatic pressure in the constitutive relation of the material is to be determined as a part of the solution of the problem since it cannot be determined from the strain field.
The mixed boundary-value problem is challenging since the a priori unknown deformed indented surface of the layer contacting the rigid cylinder is to be determined as a part of the solution of the problem.
The size distribution, average size, number density and volume fraction of the Al3(Sc,Zr) particles were determined as a function of the solution treatment temperature and time.
A heat transfer process involving heating of a solid, melting, and partial vaporisation of liquid can be considered as a three-phase Stefan or two MB problem, where the time of appearance and disappearance of phases are to be determined as a part of the solution.
This value is determined as the solution of a non-linear equation.
