TRENDS IN COMPUTATIONAL STRUCTURAL MECHANICS W.A. Wall, K.U. Bletzinger and K. Schweizerhof (Eds.) CIMNE, Barcelona, Spain 2001 FUZZY FINITE ELEMENT METHOD AND ITS APPLICATION B. Möller*, M. Beer, W. Graf and J.U. Sicert Department of Ciil Engineering, Institute of Structural Analysis Dresden Uniersity of Technology, Dresden, Germany Email: stati@rcs.urz.tudresden.de web page: http:// rcswww.urz.tudresden.de/~stati/htmlhome/homest.html Abstract: In the Fuzzy Finite Element Method (FFEM) uncertain geometrical, material and loading parameters are treated as fuzzy alues. Uncertainties of the characteristic fuzziness may be obsered, e.g. in earthquae loading, impact loads, storm loading, terrorist attacs, damage processes or shortcomings in construction wor. Spatially distributed uncertainties are described in the method of finite elements using fuzzy functions. These fuzzy functions describe a bunch of functions with uncertain bunch parameters. The introduced fuzzy functions enable fuzziness to be accounted for in the principle of irtual displacements or the particular ariational principle adopted. In order to sole the system of fuzzy differential equations and the system of fuzzy equations of the FFEM αleel optimization is applied. The FFEM approach is demonstrated by way of an application example. Key words: Fuzzy finite element method (FFEM), fuzzy function, αleel optimization 1 REMARKS ON UNCERTAINTY The uncertainty of data and information may be characterized by randomness, fuzziness or fuzzyrandomness; it is also possible to distinguish between stochastic and nonstochastic uncertainty. The property randomness is lined to specific preconditions: in particular, all elements of a sample must be generated under constant reproduction conditions. Informal and lexical uncertainty as well as uncertainty associated with human error and expert nowledge, on the other hand, possess the property fuzziness. These forms of uncertainty may be obsered, e.g. in earthquae loading, impact loads, storm loading, terrorist attacs, damage and aging processes or shortcomings in construction wor. In finite element analysis geometrical, material and loading data as well as model parameters may possess uncertainty. If sufficiently reliable stochastic data are not aailable, their uncertainty may be modeled by fuzziness. When specifying fuzziness, both objectie and subjectie information is taen into consideration.
B. Möller, M. Beer, W. Graf and J.U. Sicert In contrast to the latter, fuzzyrandomness is exhibited by data material possessing partly random properties, whose deried random parameters cannot be modeled with absolute certainty. Fuzzyrandomness combines randomness and fuzziness and is specified by means of fuzzy probability distribution functions. Fuzzyrandomness may arise, e.g. in the following context: the stochastic data material exhibits fuzziness, i.e. the sample alues hae doubtful accuracy or were obtained under unnown or nonconstant reproduction conditions. It is presupposed in the following deeloped fuzzy finite element method that the uncertainty may be specified as fuzziness. 2 MODELING OF UNCERTAINTY WITH FUZZY FUNCTIONS Physical parameters possessing fuzziness with regard to external loading or material, geometrical and model parameters may occur at all points of a structure. Depending on the dimensionality of the structure ß 1 bar structures, ß 2 plane structures, ß 3 3D structures % it is proposed to describe fuzziness using fuzzy functions. Fuzzy functions may describe uncertainty in different ways 1. Starting from a classical function f: X 6 Y, uncertainty may be present in the argument of the function, e.g. in the form f(x ), in the function f itself, e.g. in the form f(x) or simultaneously in the argument and the function itself, e.g. in the form f (x ). The symbol ~ denotes fuzziness. f(x ) is referred to as the fuzzy extension of nonfuzzy functions and f (x) is referred to as a fuzzy function with nonfuzzy arguments. The form f (x) appears to be particularly suitable for describing the fuzziness of physical parameters within the framewor of a finite element analysis. To eery crisp point x = {x 1, x 2, x 3 } it is possible to assign an uncertain physical parameter, whose distribution in ß n is described by the fuzzy function f (x). The permissibility of introducing fuzziness, also with regard to the position ector x, requires fuzzy functions of the form f (x ). Using f(x), on the other hand, it is only possible to describe uncertain geometrical parameters with restrictions. For structures in ß 1 and ß 2 fuzziness is permitted for the coordinate x 3. All geometrical parameters dependent on x 3, such as the thicness of structure elements or leel specifications for reinforcement in a crosssection, may be expressed as fuzzy parameters. Fuzzy functions with nonfuzzy arguments may be formulated in two different ways, depending on whether the image of x 0 X is a fuzzy set of y ' f(x) on Y or whether x is mapped onto Y using a fuzzy set of functions. In the first case, socalled fuzzifying functions are obtained, whereas in the second case, a fuzzy bunch of functions is obtained. The second form enables the numerical ealuation of the goerning finite element equations to be performed using αleel optimization; this forms the basis of the following considerations. A fuzzy bunch of functions f: X 6 Y is a fuzzy set F on Y X, i.e. each function f from X to Y has a membership alue µ F (f) in F and is a crisp function, which is referred to as an original. A fuzzy bunch is thus a multialued fuzzy relationship, due to the fact that if f 1 (x) and f 2 (x) are elements of the fuzzy set F, it holds that ý x * f 1 (x) = f 2 (x) = y with µ S (f 1 ) ú µ S (f 2 ). When applying αdiscretization the fuzzy bunch of functions may be represented to adantage in the form P (x) ' P (x); µ P (x) * P (x) ' [Pmin, (x); P max, (x)]; µ P (x) ' æ ' (0;1] (1) 2
B. Möller, M. Beer, W. Graf and J.U. Sicert In Figure 1 a cut through the fuzzy bunch at the point i is represented for the position ector x = x i. Figure 1: Cut through the fuzzy bunch at point i The fuzzy function P (x) is approximated by means of fuzzy alues at suitably distributed interpolation nodes in ß n. Fuzzy alues at interpolation nodes are fuzzy numbers, which describe the fuzziness of the physical parameter concerned at discrete points i. These discrete points may be (but not necessarily) identical to nodes in the finite element mesh. The functional alues at the nodes must, howeer, be determinable from the P (x). Three fuzzy alues at interpolation nodes x 1, x 2, x 3 are shown in Figure 2. P (x) is linearly approximated between these points. Figure 2: Approximation of the fuzzy function P (x) in ß 1 using fuzzy numbers at x 1, x 2, x 3 3
B. Möller, M. Beer, W. Graf and J.U. Sicert On each αleel the P α (x) of Eqn. (1) may be expressed by the bunch parameter s α. For the fuzzy function P (x) shown in Figure 2, P α (x) between x 1 and x 2 may be expressed in the form P (x) ' s % m @ x * s ' s min, ;s max, ; æ ' (0; 1] (2) Using the fuzzy bunch parameter s, the form P (x) ' s % m @ x (3) is also possible. The fuzzy bunch parameter s is shown in Figure 3. If all fuzzy bunch parameters s r are combined together in a bunch parameter ector s, P (x, s) then holds. Figure 3: Fuzzy bunch parameter s 3 FUZZY FINITE ELEMENTS The introduced fuzzy functions enable fuzziness to be accounted for in the principle of irtual displacements as well as in the underlying ariation principle adopted. Fuzzy extension is demonstrated here for structures in ß 1 and ß 2 for the principle of irtual displacements. For the position ectors x = {x 1, x 2, x 3 } the coordinates x 1 and x 2 must not exhibit fuzziness; the coordinate x 3, howeer, may possess fuzziness. A displacement field (x) with fuzziness is chosen. (x) ' M(x) @ p with x ' (x 1,x 2 ) (e) ' A @ p p ' A &1 @ (e) (x) ' M(x) @ A &1 @ (e) ' N(x) @ (e) (4) With the restriction of a linear relationship between strains and displacements, the following is obtained Moreoer, a linear material law is assumed. ḡ (x) ' B @ p ' B @ A &1 @ (e) ' H @ (e) (5) (x) ' Ē(x) @ ḡ (x) ' Ē(x) @ H @ (e) (6) 4
B. Möller, M. Beer, W. Graf and J.U. Sicert For irtual displacements and irtual strains, the following is chosen: (x) ' M(x) @ p ' N(x) @ (e) ḡ(x) ' H @ (e) The irtual internal fuzzy wor follows from (7) Ā i ' mv ḡ T (x) @ (x) dv Ā i ' & T (e) @ mv H T @ Ē(x) @ HdV @ T (e) (8) with the fuzzy element stiffness matrix K (e) ' mv H T @ Ē(x) @ HdV (9) For the irtual external fuzzy wor the following holds under consideration of timedependent fuzzy surface forces as well as mass, inertial and damping forces possessing fuzziness: A a ' T (e) @ F(e, t) T (x) @ p M (x) dv % mv irtual wor of nodal forces irtual wor of mass forces % mo T (x) @ p % (x, t) da O & mv T (x) @ (x) @ (x, t) dv irtual wor of timedependent surface forces irtual wor of inertial forces & mv T (x) @ d(x) @ 0 (x, t) dv irtual wor of damping forces (10) Ā a Time t is still introduced into as a deterministic alue. Using the familiar abbreiations for element e, the following is obtained from Ā i & Ā a ' 0 F (e, t) ' F (e, t) % K (e) @ (e, t) % M (e) @ (e, t) % D (e) @ 0 (e, t) (11) This yields a system of second order fuzzy differential equations for the structure which, for the static case, reduces to M @ % D @ 0 % K @ ' F, (12) If nonlinearities are present, incremental forms must be used. K @ ' F (13) 5
B. Möller, M. Beer, W. Graf and J.U. Sicert 4 SOLUTION TECHNIQUES The fuzziness of the n FF fuzzy functions P (x, s ), = 1, 2,..., n FF entering Eqns. (12) and (13) is expressed by the fuzzy elements of the bunch parameter ector s ' s 1,s 2,..., s j,..., s. As each fuzzy function r P(x, s) consists of a bunch ofcrisp functions f with membership alues µ P (f) the originals Eqns. (12) and (13) may be soled on an originaltooriginal wise by familiar methods. The two equations represent a mapping operator for each original of the form ' f s 1,s 2,..., s j,..., s r. (14) The goerning originals are determined for each αleel by αleel optimization 2. Figure 4: Mapping of the input alues and onto the fuzzy result alue s1 s 2 j 6
B. Möller, M. Beer, W. Graf and J.U. Sicert All fuzzy bunch parameters are discretized using the same number of αleels. The α leel set s j, is assigned to each bunch parameter s on the leel α, and all form j s j, the crisp subspace S. With the aid of the mapping operator of Eqn. (14), elements of the αleel set j, of the fuzzy result alue may be computed on the αleel α. The j mapping of all elements of S yields the crisp subspace V of the result space. In the case of conex fuzzy result alues it is sufficient to determine the largest and smallest element of the αleel set j,. Two points of the membership function of the result alue j are then nown on αleel α. This method is illustrated in Figure 4 for two bunch parameters s1 und s 2. 5 EXAMPLE The FFEM approach is demonstrated by way of an example. The reinforced concrete folded plate structure shown in Figure 5 is computed under consideration of the goerning nonlinearities of the reinforced concrete and fuzziness associated with the superficial load, the concrete compressie strength and the position of the reinforcement. Figure 5: Geometry, finite element model 7
B. Möller, M. Beer, W. Graf and J.U. Sicert The extension for eery plane of the folded plate structure in ß 2 is crisp. The system is meshed using 48 finite elements. Each element, with an oerall thicness of 10 cm, was modeled using 7 equidistant concrete layers, with two smeared mesh reinforcement layers each on the upper and lower surfaces (see Figure 5). The concrete material law according to Kupfer/Lin and a bilinear material law for reinforcement steel were applied. Tensile cracs in the concrete were accounted for in each element on a layertolayer basis according to the concept of smeared fixed cracs. The superficial load with fuzziness was increased incrementally up to the prescribed serice load. Selected result alues were computed at this stage of loading. Under consideration of the fuzziness of the uncertain input alues, the result alues are also fuzzy numbers. Inestigation 1: The superficial load and the concrete compressie strength are fuzzy input alues. The superficial load is specified as a linear fuzzy function with 3 bunch parameters, which are assigned to the fuzzy alues at interpolation nodes at the corners 3, 5, 6. The fuzzy alues at interpolation nodes are specified by the membership function µ(p) gien in Figure 6. A crisp function from the bunch of functions is shown in Figure 6 for one αleel and three alues s α =[s min, α ; s max, α ]. A linear fuzzy function is also chosen for the concrete compressie strength. Assuming that each crisp function of the respectie fuzzy bunch of functions describes a plane parallel to the folded plate planes 1 and 2 at the same distance from the folded plate planes, one bunch parameter is sufficient for the purposes of definition. This bunch parameter is assigned to the membership function µ(β R ) (Figure 6). As an example of these results the fuzzy load displacement dependency for the displacement 3 of node 63 under incremental load increase up to the serice load (ν = 1) is shown in Figure 7. Figure 6: Fuzzy function, membership functions 8
B. Möller, M. Beer, W. Graf and J.U. Sicert 0.0 0.5 1.0 ν 0.5 1.0 µ( 3 ~ ) 1 0 1.5 2.0 2.5 ~ 3 [cm] Figure 7: Fuzzy loaddisplacement dependency for 3 of node 63 Inestigation 2: The fuzzy input alues in this case are position of reinforcement layer 11 and the superficial load. It is also assumed for both assigned fuzzy functions that each crisp function of the fuzzy bunch of functions describes a plane parallel to plane 2 of the folded plate. Two bunch parameters are thus required, which must be chosen from µ(h 1 ) and µ(p) (Figure 8). In this inestigation the concrete compressie strength is taen to be a deterministic alue equal to the alue of the fuzzy bunch parameter for µ(β R ) = 1. From these results the fuzzy loaddisplacement dependency for the displacement 3 of node 63 is presented (Figure 9). Figure 8: Membership functions of the input alues 9
B. Möller, M. Beer, W. Graf and J.U. Sicert Figure 9: Fuzzy loaddisplacement dependency for 3 of node 63 6 REFERENCES [1] Dubois, D.; Prade, H. Fuzzy Sets and Systems Academic Press, New Yor, 1980 [2] Möller, B.; Graf, W.; Beer, M. Fuzzy structural analysis using αleel optimization Computational Mechanics 26 (2000) 6, 547565 [3] Bandemer, H.; Näther, W. Fuzzy Data Analysis Kluwer Academic Publishers, Dordrecht, 1992 10