The principle of virtual work and a principle of virtual heat as a basis for nonlinear finite element analysis

A principle of virtual heat, analogous to the principle of virtual work, is used to derive the finite element equations in stress, temperature, density, entropy and material velocities, and material accelerations. Constitutive equations for the finite dimensional generalized stresses and temperature are derived from energy equation and entropy production law, as in the case of the continuum model. The resulting equations are ordinary differential equations with respect to time. Spatial finite elements are used for flow problems and material finite elements for deformation problems. The material finite elements are adjoining material bodies; the spatial finite elements are fixed subdomains of space with matter flowing through the boundaries.