A mixed element method is introduced to solve Darcy-Forchheimer equation, in which the velocity and pressure are approximated by mixed element such as RaviartThomas, Brezzi-Douglas-Marini element. We establish the existence and uniqueness of the problem. Error estimates are presented based on the monotonicity owned by the Forchheimer term. An iterative scheme is given for practical computation. The numerical experiments using the lowest order Raviart-Thomas (RT 0 ) mixed element show that the convergence rates of our method are in agree with the theoretical analysis.
Figure 1: Left: self-supporting surfaces with unsupported (top) and supported (bottom) boundary constraints. Unsupported boundary vertices and their corresponding power cells are colored in orange. Top right: initial self-supporting mesh. Spikes appear due to extremely small reciprocal areas. Bottom right: applying our smoothing scheme (5 iterations) improves mesh quality. The power diagrams (black) show how power cell area is distributed more evenly.
AbstractMasonry structures must be compressively self-supporting; designing such surfaces forms an important topic in architecture as well as a challenging problem in geometric modeling. Under certain conditions, a surjective mapping exists between a power diagram, defined by a set of 2D vertices and associated weights, and the reciprocal diagram that characterizes the force diagram of a discrete self-supporting network. This observation lets us define a new and convenient parameterization for the space of self-supporting networks. Based on it and the discrete geometry of this design space, we present novel geometry processing methods including surface smoothing and remeshing which significantly reduce the magnitude of force densities and homogenize their distribution.
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