March 17, 2023 at 8:58 amFAQParticipant
Let us assume a given grid. How can one evaluate the number of unknowns? What can be said about it? What is the marginal cost of an additional element? What is the cost of a given interpolation? There is No general statement, biut there are some general considerations: 1- It is generally difficult to draw a general rule for evaluating the calculation cost of a finite element simulation. Indeed, next to the number of elements that clearly affects the cost, the topology of the grid also plays a significant role. In order to illustrate this, let us mention that, with the same number of elements, a grid where the elements are toplogically located along a preferred direction is usually cheaper than a grid where elements are equally distributed in all directions. In other words, a grid with 10x10x80 elements will be computationally less expensive than a grid with 20x20x20 elements. 2- Cost of an interpolation: Instead of evaluating the cost of a grid, which is rather difficult to estimate, one can evaluate the relative costs of various interpolations. For this, let us consider a sufficiently large 3D grid, and we try to evaluate the marginal cost of an additional element. In the figure FEM_calculation_cost_3D_grid.gif attached, we have such a grid, and we add a new element, identified in red. Depending on the simulation and on the selected interpolation, that element will bring additional unknowns. Let us focus on the velocity and on the pressure. Three interpolations are considered for the velocity: – linear interpolation (3 componenets at the additional blue node) – mini element (3 comonents at the additional blue node and 1 component at each green node) – quadratic interpolation (3 components at each additional node, blue, green, magenta, brown) Two interpolations are considered for the presure: – constant interpolation (1 additional value over the element) – linear interpolation (1 additional value at the blue node) When using linear velocity and constant pressure, each additionalelement brings 4 new unknowns When using the mini element, each additionalelement brings 7 new unknowns When unsing quadratic velocity and linear pressure, each additional element brings 25 unknowns. Similar statements can be made e.g. for non-isothermal or viscoelastic flows. Basically, one has to count the marginal cost of each new element.
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