Re: Numerical integration over an arbitrary 2D domain

*To*: mathgroup at smc.vnet.net*Subject*: [mg127167] Re: Numerical integration over an arbitrary 2D domain*From*: Fred Bartoli <""@news.free.fr>*Date*: Mon, 2 Jul 2012 22:22:35 -0400 (EDT)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*References*: <jsopu3$12g$1@smc.vnet.net>*Reply-to*: myname_with_a_dot_inbetween at free.fr

Alfonso Pagani a écrit : > Hi all, > I am a new user of Wolfram Mathematica. I need to integrate > (numerically of course) generic functions above complex 2D domains. > For example, i would like to find the moments of inertia of an airfoil > cross-section. Is it possible? > My idea was to import a triangular mesh (list of nodes and element > connectivity) of the integration domain. Then integrate my functions > using a trapezoidal integration. How could i do this? > Thank you very much, > > Alfonso > Alfonso, No need to mess with that. Just use a weighting function and then integrate the weighted function over a simple rectangular region that contains all your shape: (* First define the integrating region *) weightFunc[x_, y_] = Piecewise[{{1, x^2 + y^2 <= 2 && x >= 0}, {1, y >= 0 && y <= Sqrt[2] (x + 1)^2 && x >= -1 && x <= 0}, {1, y <= 0 && y >= -Sqrt[2] (x + 1) && x >= -1 && x <= 0}}, 0]; (* Then plot it and check its shape *) toLogic[v_] = Switch[v, 1, True, 0, False]; RegionPlot[toLogic@weightFunc[x, y], {x, -2, 2}, {y, -2, 2}] (* Finally integrate the moment inertia about the (1/2,0) point *) NIntegrate[((x - 1/2)^2 + y^2) weightFunc[x, y], {x, -2, 2}, {y, -2, 2}] Integrate[((x - 1/2)^2 + y^2) weightFunc[x, y], {x, -2, 2}, {y, -2, 2}] N[% - %%] -- Thanks, Fred.