Fulvio Tonon

Three-dimensional Green's functions in anisotropic piezoelectric solids

Abstract Explicit expressions for three-dimensional extended Green’s displacements in general anisotropic piezoelectric solids are derived. A very efficient procedure for the numerical evaluation of the derivatives of the extended Green’s displacements is also proposed. Numerical comparisons are carried out for a transversely isotropic piezoelectric solid for which exact closed-form solutions are available. It is found that the extended Green’s displacements and their derivatives obtained with the present explicit formulation are in perfect agreement with the exact closed-form solutions. These Green’s functions can be used in the boundary integral equations for piezoelectric solids of general anisotropy and for subsequent numerical solutions of these equations by means of the boundary element method.

Determination of parameters range in rock engineering by means of Random Set Theory

Abstract Uncertainty arising from our relationship with the real world can often lead to imprecise probabilities, whose bounds are captured by Random Set Theory. Reasons leading to this kind of uncertainty in rock engineering are investigated, as well as the limitations of the probabilistic approach. Procedures are given to handle the usual information gained either in the field or from experts opinions, taking into account not only dissonance, but also non-specificity. Examples of applications are presented with special regard to event tree analysis, rock mass classifications, and reliability-based design of tunnels by means of the empirical recommendations of rock mass classifications.

Using random set theory to propagate epistemic uncertainty through a mechanical system

Abstract The Epistemic Uncertainty Project of Sandia National Laboratories (NM, USA) proposed two challenge problems intended to assess the applicability and the relevant merits of modern mathematical theories of uncertainty in reliability engineering and risk analysis. This paper proposes a solution to Problem B: the response of a mechanical system with uncertain parameters. Random Set Theory is used to cope with both imprecision and dissonance affecting the available information. Imprecision results in an envelope of CDFs of the system response bounded by an upper CDF and a lower CDF. Different types of parameter discretizations are introduced. It is shown that: (i) when the system response presents extrema in the range of parameters considered, it is better to increase the fineness of the discretization than to invoke a global optimization tool; (ii) the response expectation differed by less than 0.5% when the number of function calls was increased 15.7 times; (iii) larger differences (4–5%) were obtained for the lower tails of the CDFs of the response. Further research is necessary to investigate (i) parameter discretizations aimed at increasing the accuracy of the CDFs (lower) tails; (ii) the role of correlation in combining information.

Numerical Models in Discontinuous Media: Review of Advances for Rock Mechanics Applications

The paper presents a description of the methods used to model rock as discontinuous media. The objective of the work is to bring to the geomechanics community recent advances in numerical modeling in the field of rock mechanics. The following methods are included: (1) the distinct element method; (2) the discontinuous deformation analysis method; and (3) the bonded particle method. A brief description of the fundamental algorithms that apply to each method is included, as well as a simple case to illustrate their use.

Green’s functions and boundary element method formulation for 3D anisotropic media

Abstract The implementation of Wang’s theoretical solution is presented for elastostatic displacement Green’s function for three-dimensional solids of general anisotropy. Excerpts from the authors’ fortran code are included. A numerical algorithm for the calculation of the derivatives of the Green’s displacements and stresses is also introduced. These implementations have been incorporated into a boundary element method (BEM) code developed by the authors. The numerical results of Green’s displacements, stresses and stress derivatives are in perfect agreement with closed-form solutions for transversely isotropic solids. The BEM code results are also in very close agreement with both exact solutions and other BEM formulations, even if coarse mesh discretizations are used.

Reliability analysis of rock mass response by means of Random Set Theory

Abstract When the parameters required to model a rock mass are known, the successive step is the calculation of the rock mass response based on these values of the parameters. If the latter are not deterministic, their uncertainty must be extended to the predicted behavior of the rock mass. In this paper, Random Set Theory is used to address two basic questions: (a) is it possible to conduct a reliable reliability analysis of a complex system such as a rock mass when a complex numerical model must be used? (b) is it possible to conduct a reliable reliability analysis that takes into account the whole amount of uncertainty experienced in data collection (i.e. both randomness and imprecision)? It is shown that, if data are only affected by randomness, the proposed procedures allow the results of a Monte Carlo simulation to be efficiently bracketed, drastically reducing the number of calculations required. This allows reliability analyses to be performed even when complex, non-linear numerical methods are adopted. If not only randomness but also imprecision affects input data, upper and lower bounds on the probability of predicted rock mass response are calculated with ease. The importance of imprecision (usually disregarded) turns out to be decisive in the prediction of the behavior of the rock mass. Applications are presented with reference to slope stability, the convergence-confinement method and the Distinct Element Method.

A RANDOM SET APPROACH TO THE OPTIMIZATION OF UNCERTAIN STRUCTURES

Abstract The single-objective optimization of structures, whose parameters are assigned as fuzzy numbers or fuzzy relations, is presented in this paper as a particular case of the random set theory and evidence theory approach to uncertainty. Some basic concepts concerning these theories are reviewed and the relationships among interval analysis, convex modeling, possibility theory and probability theory are pointed out. In this context a frequentistic view of fuzzy sets makes sense and it is possible to calculate bounds on the probability that the solution satisfies the constraints. Some special but useful cases illustrate in detail the meaning of the approach proposed and its links with a recent formulation conceived within the context of convex modeling. Some theorems allow a very efficient computational procedure to be set up in many real design situations. Two numerical examples illustrate the model presented.

Multiobjective optimization of uncertain structures through fuzzy set and random set theory

An interactive procedure to solve multicriteria optimization problems is proposed and discussed. A fuzzy set is used to model the engineers judgment on each objective function. The properties of the compromise solution obtained are investigated along with the links between the present method and those based on fuzzy logic. Uncertainty affecting the parameters is modeled by means of fuzzy relations or fuzzy numbers, whose probabilistic meaning is clarified by random set and possibility theory. Bounds to the probability that a solution satisfies a constraint can be calculated and procedures that consider the lower bound as a constraint or as an objective to be maximized are presented. Some theorems make the computational effort particularly limited in a vast class of practical problems. The relations with a recent formulation in the context of convex modeling are also stressed. Two examples show the effectiveness of the proposed approach.

Using random set theory to calculate reliability bounds for a wing structure

This paper shows how uncertainty can be propagated through a computer model when the input data consists of uncertainty affected by imprecision and randomness. The proper orthogonal decomposition (POD) is used to determine the dominant air pressure distributions on a wing, and the uncertainty in the POD modal amplitude variables is propagated through a finite element (FEM) structural model to calculate the reliability bounds for the wing. It is shown that: (i) when working with imprecise input, the traditional probabilistic approach is inadequate because it would force the designer to add unavailable information; (ii) the proposed mathematical model of uncertainty is capable of capturing the incomplete knowledge upon which important conceptual and preliminary decisions must be made; (iii) this approach indicates if and where resources should be invested to acquire more information; (iv) the proposed techniques can be easily implemented using existing numerical codes; (v) a limited number of model runs is necessary to calculate bounds on the output distributions.

Multiobjective optimization under uncertainty in tunneling: application to the design of tunnel support/reinforcement with case histories

In this paper, two questions common in tunnel design are addressed: (i) how to choose an optimum solution when more than one conflicting objective must be achieved; (ii) how to deal with data affected both by imprecision and randomness. Fuzzy Set Theory and Random Set Theory are used to develop a general interactive multiobjective procedure, which is then applied to the design of tunnel support/reinforcement. A case history illustrates how the procedure was successfully used in the preliminary design of a total of 40 km of tunnels in Central Italy. It is shown that the procedure allows the designer to become a knowledgeable decision maker because his interaction is required at the key points of the process, and because the trade-offs among the objective functions can be easily assessed. The designers personal input is valued and clearly defined in its impact on the solution. The case history demonstrates that, without an optimization procedure, it is extremely likely (probability of 99%) that a solution is chosen, which either increases the costs without increasing safety, or decreases the safety without decreasing the costs. Finally, it is shown that both imprecision and randomness can be easily taken into account in tunnel design.