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This second order effect in response cannot be \ faithfully captured unless the equilibrium equation is exactly satisfied. \ Hence engineering approximations of system stochasticity demand higher \ accuracy than their deterministic counter parts. Consequently, the \ contamination in numerical responses cannot be removed by selecting large \ Monte Carlo samples. Stochastic shape functions and stochastic Green's \ functions constitute the bases for finite and boundary elements, \ respectively. These fundamental solutions need to be modified distinctly for \ each Monte Carlo representative via symbolic manipulation of the governing \ algebraic equation. The subsequent closed form analytical integration of the \ energy density function is illustrated here for a beam problem with \ geometrical stochasticity after zero shear locking is met in a patch-test.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Introduction", "Section"], Cell[TextData[{ "Material randomness is ingrained in real-world engineering problems.\n\ During sixties Keller and his associates initiated research on stochastic \ differential equations to solve wave propagation in random media [1]. Vide \ Sobczyk [2] for details on Keller's work [3] and [4], where a number of \ innovative methods of stochastic averaging emerged. An intutively obvious \ scheme to reconstruct the partial differential equation with stochastic \ coefficients led to anamolous response since the self-adjointness was \ destroyed. In another case, where the shock front should be discontinuous, \ Monte Carlo averaging lost the discontinuity and essentially solved quite a \ different problem than the original. These situations were analyzed by ", StyleBox["Molyneux", FontFamily->"Courier New", FontSize->10], " as 'dishonest methods' [5]. Strangely enough, with the surge of \ interests in the stochastic finite element method, many authors repeated the \ same mistakes in their papers published in various engineering mechanics \ journals. Generically speaking, many such methods will violate equilibrium \ in a one-dimensional bar problem where the modulus of elasticity is a random \ function of the spatial coordinate. Their assumption of deterministic shape \ functions used in conventional finite elements was examined by the author in \ 1989, vide [6]. The conclusion led to the concept of stochastic shape \ functions. \n \n The boundary element analog of the finite element shape \ function is the Green's function [7]. Stochastic considerations of the \ associated fundamental solutions were described in [2].\n\nConstruction of \ stochastic shape functions demands algebraic manipulations including \ close-form differentiation and integration [8]. Beam vibration, wave \ propagation in beams, and beams on elastic foundations were considered there. \ For one dimensional cases workstations available in the late eighties \ sufficed. \n\nThe author completed the theoretical formulation of stochastic \ finite element stiffness matrices in 1990 and published the results in [9].\n\ \nThe deterministic shape functions for convex polygons were constructed in \ the closed algebraic form in the late eighties. A bioengineering \ application was reported in [10]. For small randomness in material properties \ and boundary geometry the perturbation technique of Nakagiri and Hisada [11] \ were implemented for convex elements with nine sides [10]. In 1997 the author \ began deveoping stochastic shape functions in the closed algebraic form for \ two-dimensional finite elements. Computers that ran at 200 MHz and contained \ 100 MB of RAM were used. The crucial results are presented in the following \ examples.\n\nFor the boundary element method the stochastic Green's functions \ were constructed by using the methods appropriate to address material \ inhomogeneity [12]. The convergence criteria of large Monte carlo ensembles \ were demonstrated in [13].\n\n" }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Organization of Examples", "Section"], Cell["\<\ System stochasticity arises out of randomness in material properties and \ nondeterminancy in boundary geometry of discrete elements. Stochastic shape \ functions were adequately explained in beam examples in an international \ conferece dedidacted to reliability analysis [16]. The author worked out in \ detail how the flux balance equation should be faithfully executed by \ constructing symbolic forms the stochastic shape functions starting from the \ random field that described the spatial variability in material properties. \ Those examples are not repeated here. In this paper only the stochastic \ aspect in geometry is focused. The stochastic strain-displacement type \ transformations can be carried out by the following code.\ \>", "Text"], Cell[CellGroupData[{ Cell["Basic codes for material randomness", "Subsubsection"], Cell[BoxData[ \(StochasticB[dDeterministic_, \ bDeterministic_, dStochastic_] := Inverse[dStochastic] . \((dDeterministic . bDeterministic)\)\)], "Input"], Cell["\<\ System stochasticity addresses secondary effects in response computation. \ Hence it is important that the associated deterministic problems be solved \ exactly. This in turn demands exact shape functions and exact integration of \ the energy density function in the element domain. The stress formulation (as \ opposed to the conventional displacement based method) was the key to address \ the material randomness criteria. The same stress elements are essential to \ satify zero locking patch tests in beam problems. The following exact \ integration routine was used for quadrilateral beam elements.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Integration in polygonal element domains", "Subsubsection"], Cell[BoxData[ StyleBox[\(BeginPackage["\", \ "\"]\n\n AreaIntegrate::usage = "\"\n\n Begin["\<`Private`\>"]\n\n\n \(Clear[IntegrateApart];\)\n\n IntegrateApart[x_, \ t_Symbol]\ := Module[{z}, \ \n\t z\ = Apart[x, \ t]; \ \n\ \ \ \ If[ Head[z]\ === Plus, \n\ \ \ \ \ \ \ Plus @@ Map[Integrate[#, \ t] &, \ \ List @@ z], \ \n\ \ \ \ \ \ \ Plus @@ Map[Integrate[#, \ t] &, Replace[ z, \ \n\ \ \ \ \ \ \ Plus[y_]\ -> List[y]]\ ]\ \ \ \ ]\n\ \ \ \ \ \ \ ]\n\ \ \ \ \ \ \ \n\ \ \ \ \ \ \ \ \n \ \ \ \ \ \ \ IntegrateApart[ x_, \ {t_Symbol, \ t1_, t2_}]\ := \(\((\n\ \ \ \ \ \ \ \((#\ /. t\ -> t2)\)\ - \((#\ /. t\ -> t1)\))\) &\)[\n\ \ \ \ \ \ \ \ \ \ IntegrateApart[ x, \ t]]\n\n \(Clear[AreaIntegrate, \ \ LineIntegrate];\)\ \n\n\n AreaIntegrate[z_, \ {x_, \ y_}, \ nodes_]\ := Module[\n{zx, \ t, \ segments}, \ \n zx\ = IntegrateApart[z, \ x]; \ \n segments\ = Partition[Append[nodes, \ nodes[\([1]\)]], \ 2, \ 1]; \ \n Plus @@ Map[ LineIntegrate[ zx, \ {x, \ y, t}, \ #] &, \ \n\ \ \ \ \ \ \ \ \ \ \ \ segments]]\n\ \ \ \ \ \ \ \ \ \ \ \ \ \n\ \ \ \ \ \ \ \ \ \ \ \ \n\ \ \ \ \ \ \ \ \ \ \ \ \n LineIntegrate[ z_, \ {x_, \ y_, t_}, \ \n\t\t{{x1_, \ y1_}, \ {x2_, y2_}}]\ := \((IntegrateApart[\n\ \((\((y2\ - y1)\)\ z)\) /. {x\ -> x1\ + t \((x2\ - x1)\), y\ -> y1\ + t \((y2\ - y1)\)}, \n\ \ {t, \ 0, 1}])\)\n\n\n End[]\n EndPackage[]\), FontFamily->"Geneva"]], "Input"], Cell["\<\ The following deterministic problem was examined to validate the above \ integration scheme and Wachspress shape functions [10].\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ A Deterministic Problem : A Benchmark Patch Test from Bathe: Page-264, Figure (b)\ \>", "Subsubsection"], Cell["\<\ From the finite element book [14] the following geometry was constructed:\ \>", "Text"], Cell[CellGroupData[{ 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0, \(-0.2999999999991366`\), 0}, {0.7999999999977696`, 0, \(-0.6999999999989109`\), 0, \(-0.8000000000005406`\), 0, 0.7000000000016782`, 0}, {\(-0.4000000000030714`\), 0, 0.5000000000004864`, 0, 0.4000000000003293`, 0, \(-0.4999999999971969`\), 0}}\)], "Output"], Cell["\<\ After a successful completion of the deterministic case the following \ stochastic problem is presented.\ \>", "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Problem -- Geometrical Randomness", "Section"], Cell[CellGroupData[{ Cell["Geometry --- stochastic", "Subsubsection", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ globalNodes = { {0,0}, {10,0},{10,2},{0,2}, \t\t\t\t{Random[Real,{2,4}],0}, \t\t\t\t{1,2},{8,0},{9,2}} \t\t\t\t \t\t\t\t connections ={{1,5,6,4},{5,7,8,6},{7,2,3,8}};\ \>", "Input", InitializationCell->True, ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ The x coordinate of the fifth node has been generated randomly.\ \>", "Text"], Cell[OutputFormData["\<\ {{0, 0}, {10, 0}, {10, 2}, {0, 2}, 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Oomoo`3oOolGOol5001JOol002eoo`03001oogoo0?mooaQoo`<005]oo`00;Goo00<007ooOol0ogoo MWoo000]Ool00`00Oomoo`3oOomfOol002eoo`<00?moogIoo`00ogooYWoo003oOonVOol00?moojIo o`00ogooYWoo003oOonVOol00?moojIoo`00ogooYWoo003oOonVOol00?moojIoo`00ogooYWoo003o OonVOol00?moojIoo`00ogooYWoo003oOonVOol00?moojIoo`00ogooYWoo003oOonVOol00?moojIo o`00ogooYWoo003oOonVOol00?moojIoo`00ogooYWoo003oOonVOol00?moojIoo`00ogooYWoo0000 \ \>"], ImageRangeCache->{{{0, 420.5}, {293.563, 0}} -> {-1.45534, -1.31468, \ 0.04766, 0.04766}}], Cell["\<\ The element strains showed zero shear (the third entry in the following \ resultss).\ \>", "Text"], Cell[CellGroupData[{ Cell["(* MatrixForm[elementStrains] *)", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[OutputFormData["\<\ MatrixForm[{{0.02000000000000263 - 0.02000000000000268*y, -0.005000000000000683 + 0.005000000000000671*y, 0}, {0.02000000000000146 - 0.0200000000000014*y, -0.005000000000000269 + 0.005000000000000351*y, 0}, {0.02000000000000029 - 0.02000000000000023*y, -0.004999999999997506 + 0.005000000000000057*y, 0}}]\ \>", "\<\ 0.02 - 0.02 y -0.005 + 0.005 y 0 0.02 - 0.02 y -0.005 + 0.005 y 0 0.02 - 0.02 y -0.005 + 0.005 y 0\ \>"], "Output", ImageRegion->{{0, 1}, {0, 1}}] }, Open ]], Cell["\<\ The axial stresses in x and y directions are linear as expected (see the \ first two columns in the above result).\ \>", "Text"], Cell["\<\ The stress results, as shown below, demonstrate that the pure bending field \ has been captured exactly. Each row represents an element. There are zero \ shear (the third column of the following table) and zero axial stress in the \ transverse direction (the second column), and the axial stress is linearly \ distributed (the first column is a linear function in y for all three \ elements)\ \>", "Text"], Cell[CellGroupData[{ Cell["(* Chop[elementStresses]//MatrixForm *)", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[OutputFormData["\<\ MatrixForm[{{0.03000000000000394 - 0.03000000000000403*y, 0, 0}, {0.03000000000000224 - 0.03000000000000211*y, 0, 0}, {0.03000000000000147 - 0.03000000000000034*y, 0, 0}}]\ \>", "\<\ 0.03 - 0.03 y 0 0 0.03 - 0.03 y 0 0 0.03 - 0.03 y 0 0\ \>"], "Output", ImageRegion->{{0, 1}, {0, 1}}] }, Open ]] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Conclusions", "Section"], Cell[TextData[{ "Finite and boundary element discretization procedures, which address \ material and geometrical randomness in engineering problems, demand higher \ accuracy than their deterministic counter parts since the response \ variability, notably the dispersion depicted by the covariance field, is of \ primary importance. This second order effect cannot be faithfully captured \ unless the equilibrium equation, which arises out of the flux balance \ consideration of continuum mechanics, is exactly satisfied at every point. \ Consequently, the displacement based finite elements yield contaminated \ results even with a large Monte Carlo sample. \n\nStochastic shape functions \ and stochastic Green's functions, which indeed comply with the requirements \ of the balance principles at each point, demand the field equation to be \ modified distinctly for each Monte Carlo representative depending upon its\n\ simulated randomness. Reliable results of system stochasticity can only be \ achieved by symbolically manipulating the algebraic form of the governing \ equation. The crucial component of the closed form analytical integration of \ the energy density function is illustrated here for a beam problem after \ zero shear locking is met in a patch-test. Since material stochasticity was \ presented in Murmansk, Russia in 1998 at the International Arctic Seminar \ \[Dash] the sister conference of the International Mathematica Symposium, \ only the geometrical stochasticity is numerically described here.\n\nKeller's \ fundamental formulations of wave propagations in random media [15] and the \ various analytical techniques described by Sobczyk in [2] cannot be achieved \ without symbolic computation. The matrix operations, differentiation and \ integration can be elegantly carried out using ", StyleBox["Mathematica", FontSlant->"Italic"], ". Some numerical results for two dimensional elasticity problems, where \ randomness in material properties and boundary geometry play a crucial role, \ are presented here. In these examples the exact results are reproduced." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["References", "Section"], Cell[TextData[{ "[1] Keller, J. B. \nWave Propagation in Random Media,\nSIAM-AMS \ Proceedings,\nNo. 13, 1962, American Mathematical Society,\nProvidence, RI, \ 1962 pp. 227-246.\n\n[2] Sobczyk, K. \nStochastic Wave Propagation, \n\ Elsevier, New York, 1985.\n\n[3] Keller, J. B.\nStochastic Equations and Wave \ Propagation in Random Media.\nSIAM-AMS Proceedings,\nNo. 16, 1964, American \ Mathematical Society,\nProvidence, RI, 1964 pp. 145-170.\n\n[4] Keller, J. B. \ and Karal, F. C.\nEffective Dielectric Constant, Permeability and \ Conductivity of a Random Medium \nand the Velocity and Attenuation of \ Coherent Waves, \nJournal of Mathematical Physics, vol. 7, 1966 pp. 661-670.\n\ \n[5] Molyneux, J. E. \nAnalysis of \"Dishonest\" Methods in the Theory of \ Wave Propagation in a Random Medium, \nJ. Optical Soc. of America, vol. 58, \ No. 7, July 1968 pp. 951- 957.\n\n[6] Dasgupta, G. and Yip, S.-C.\n\ Nondeterministic Shape Functions for Finite Elements with Stochastic Moduli, \ \nProc. ICOSSAR 89, American Soc. Civil Engrs., 1989, pp. 1065-1072.\n\n[7] \ Stakgold, I.\nGreen's functions and boundary value problems, \nJohn Wiley, \ NY, 1979.\n", StyleBox["\n", FontFamily->"Arial", FontSize->10], "[8] Dasgupta, G.\nA Computational Scheme to Analyze Nonlinear Stochastic \ Dynamic Systems by Finite Elements, \nResearch Report: April 1987, Department \ of Civil Engineering and Engineering Mechanics,\nColumbia University, New \ York, NY, 1987\n(prepared for presentation at: IUTAM Symposium on \nNonlinear \ Stochastic Dynamical Engineering Systems, \nIgls, Innsbruck, Austria, June \ 21-26, 1987).\n\n[9] Dasgupta, G. \nApproximate Dynamic Responses in Random \ Media, \nActa Mechanica, Springer Verlag, Wien, Austria, 1992, pp. 99-114.\n\n\ [10] McAlarney, M. E., and G. Dasgupta\nAnatomical Macro Element in the Study \ of Cranial Facial Rat Growth.\nJournal of Cranial Facial Growth and \ Development Biology, \nNew York, NY, Vol. 12, 1992 pp. 3- 12.\n\n[11] \ Nakagiri, S. and Hisada, T.\nAn Introduction to Stochastic Finite Element \ Method, (in Japanese), \nBai Fu Kan, Tokyo, Japan, 1985.\n\n[12] Dasgupta, \ G., Green's Functions for Inhomogeneous Media for\nBoundary Elements, \ Advances in Boundary Elements,\nComputations and Fundamentals, Brebbia, C. A. \ and J. J. Connor (eds),\nEleventh International Conference on Boundary \ Element\nMethods, Cambridge, Mass., 1989, pp. 37-46.\n\n[13] Dasgupta, G.\n\ Iterative Simulation for Stochastically Nonlinear Large Variability, \n\ J.Aero.Enggr.ASCE, vol.13, no.1, January 2000, pp.11--16.\n\n[14] Bathe, \ K.-J.\nFinite Element Procedures, Prentice Hall, 1996.\n\n[15] Keller, J. B.\n\ Stochastic equations and wave propagation in random media,\nSIAM Proc., \ American Mathematical Society, \nProvidence, RI, 1964 pp. 145-170." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Acknowledgement", "Section"], Cell["\<\ The research was supported by the National Science Foundation, grant number \ cms-9820353.\ \>", "Text"] }, Open ]] }, Open ]] }, FrontEndVersion->"4.1 for Microsoft Windows", ScreenRectangle->{{0, 1024}, {0, 695}}, AutoGeneratedPackage->None, WindowToolbars->{}, WindowSize->{1016, 668}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, PrintingCopies->1, PrintingPageRange->{1, Automatic}, Magnification->1.5 ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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