In this algorithm, a damped nonlinear Newton with preconditioned iterative linear solver was used to solve Dual-Primal equations.
By solving both primal and adjoint equation system, the desired (volume-normalised) topological and (surface-normalised) shape sensitivities can be computed as ∂ J ∂ α = u ⋅ v and ∂ J ∂ β = − ν ∂ n u ⋅ ∂ n v, (9).
The weights w 1, …, w T are forced to be similar by changing the SVR primal (1) to Equation 5. min w 1, …, w T, ξ 1 2 ∑ t = 1 T | | w t | | 2 + J ( w 1, …, w T ) + C ∑ i = 1 l l ∊ ( ξ i, y i ) ξ i = w t i T x i (5).
We show a general procedure for the construction of the boundary conditions such that both the primal and dual equations are well-posed.
This is especially relevant in inverse problems, when one needs to solve the partial differential equation (the primal problem) many times in an optimization algorithm.
In this paper, we explore the variational imposition of essential boundary conditions that arise from the thermodynamic derivation of the Cahn Hilliard equation in primal variables.
We moreover require that the boundary conditions for the primal and dual Navier Stokes equations converge to well-posed boundary conditions for the primal and dual Euler equations.
Moreover, the computational cost is reduced by solving the complete set of governing equations on the primal grid while only solving the magnetic induction equation on the polygonal dual mesh.
The motivation of this work is to apply the hp-version of the mortar finite-element method to the nearly incompressible elasticity model formulated as a mixed displacement-pressure problem as well as to Stokes equations in primal velocity-pressure variables.
We construct an accurate, robust and reliable hybrid method that combines a mixed finite element discretization of the momentum equations with a primal discontinuous finite volume-element discretization of the mass (or transport) equations.
