The evolution of the law of random processes in the analysis of dynamic systems

The analysis of structures subjected to dynamic loads is now generally conducted using refined mechanical models and adequate numerical techniques. However, even when the problem is well-posed so that both the existence and the uniqueness of the solution are guaranteed, some parameters used to describe the geometric and mechanical characteristics of the structure and the external actions are generally affected by uncertainties that must be taken into account in the analysis. These input parameters are thus represented by random variables, which are defined on an appropriate probability space and whose law is supposed to be known. A stochastic process is thus obtained which is parameterized over time and has the Euclidean space as state space.

In many applications, it is necessary to determine the probability distribution of some output quantities. It is a question of determining the evolution over time of the law of some random process which is defined in terms of the solution of the dynamic system. This objective is generally achieved using the Monte Carlo method which, at least in principle, gives the possibility to consider complicated models and geometries without having to resort to unrealistic simplifying hypotheses. On the other hand, this method can require long computation times which may not be compatible with the required precision [1].

A different way to deal with the problem is to use the generalized density evolution equation, which is a consequence of the "principle of preservation of probability" [2]. This method leads to writing a linear PDE whose solution gives the probability density function for the quantities of interest. The coefficients of this equation at each instant depend on the value of the state variables and then can be obtained from the (deterministic) solution of the dynamic system. The method of the generalized density evolution equation has been implemented into the MADY code, which has already the routines for dynamic analysis of plane, three-dimensional, or beam and shell-based structures [3], and has been applied to the study of some masonry constructions [4, 5]. For these structures, the uncertainties have particular relevance, due to the constitutive characteristics of the material and the geometry.

In this paper, we deduce an equation similar to the one proposed in [2] but which allows calculating the cumulative distribution function directly, instead of the density function, of the chosen random variable. Indeed, while the former is always a locally integrable function, it may happen that the latter is defined only in a distributional sense. The deduction is made using classical theorems of the measure theory; although most notions discussed below can be presented in their intuitive sense, we prefer to give explicit definitions and proofs to avoid misunderstandings. Thus, both the approach proposed in this paper and the one proposed in [2] are rigorously justified. Moreover, with the help of the differential forms and the coarea formula, relations are deduced for the explicit computation of both the probability density function and the cumulative distribution function; these are especially useful if the number of random input parameters is greater than one.

Finally, two examples are presented in which the evolution of the probability distribution of a chosen output parameter is explicitly calculated. The solution is then compared with the numerical one obtained with the MADY code, which has meanwhile been updated with the new numerical procedures presented in this paper.

Let \((\Omega ,{\mathcal {A}},{\mathbb {P}})\) be a probability space. Here \(\Omega\) is the set of the outcomes, \({\mathcal {A}}\) is the \(\sigma\)-algebra on \(\Omega\) made of all the events and \({\mathbb {P}}: {\mathcal {A}}\rightarrow [0,1]\) is a probability measure, i.e. a positive measure on \((\Omega ,{\mathcal {A}})\) such that \({\mathbb {P}}(\Omega )=1.\) Moreover, let \({\mathbb {R}}^n\) and \({\mathcal {B}}({\mathbb {R}}^n)\) be the n-dimensional Euclidean space and its corresponding Borel \(\sigma\)-algebra, respectively.

A map \({\mathbf{X}}:\Omega \rightarrow {\mathbb {R}}^n\) is said to be a (vector) random variable if it is \({\mathcal {B}}({\mathbb {R}}^n)\)-measurable, i.e. ifwhere, as usual, \(\{{\mathbf{X}}\in {B}\}\) denotes \({\mathbf{X}}^{-1}(B)\). The measure \(\mu _X\) denotes the law of \({\mathbf{X}}\) or the image of measure \({\mathbb {P}}\) under \({\mathbf{X}}\) [6], i.e. the measure defined on \(({\mathbb {R}}^n,{\mathcal {B}}({\mathbb {R}}^n))\) byand the function \(F_X: {\mathbb {R}}^n\rightarrow [0,1]\), defined byis called cumulative distribution function of X.

Let \({\mathcal {L}}^n\) be the Lebesgue measure on \(({\mathbb {R}}^n,{\mathcal {B}}({\mathbb {R}}^n))\). Measure \(\mu _X\) is said to be absolutely continuous (with respect to \({\mathcal {L}}^n\)) if \(\mu _X(B)=0\) for every set \(B\in {\mathcal {B}}({\mathbb {R}}^n))\) with \({\mathcal {L}}^n(B)=0\). In this case, the Radon-Nikodym theorem [6] guarantees the existence of a positive integrable function \(p_X\) such thatfor every \(B\in {\mathcal {B}}({\mathbb {R}}^n))\). Function \(p_{X}\) is called the (joint) probability density function of the vector random variable X and it holdsorIf \(\mu _X\) is not absolutely continuous, then its probability density function does not exist as an integrable function. However, it can be defined as a generalized function by interpreting (2) as a distributional derivative [7].

Let \({\mathbf{X}}\) be a vector random variable and \(f:{{\mathbb {R}}^{n}}\rightarrow {{\mathbb {R}}^{m}}\) a Borel function; then \({\mathbf{Y}}=f{\circ }{\mathbf{X}}\) is again a vector random variable and, for each \({B}\in {\mathcal {B}}({{\mathbb {R}}^{n}})\),If \({\mathbf{X}}\) has a probability density function \(p_{X}\), thenby (1) and (3). Moreover, if f is such that [8]for every Borel subset B of \({\mathbb {R}}^n\), then, in view of (4), \(\mu _Y\) is (finite and) absolutely continuous with respect to \({\mathcal {L}}^m\) and thus it admits a probability density function \(p_Y\). In the particular case when \(n=m\) and f is a diffeomorphism, i.e. a bijective application such that both f and \(f^{-1}\) are continuously differentiable, thenwhere \(Jf^{-1}\) denotes the Jacobian of \(f^{-1}\).

More generally, suppose that there exists a countable family \((B_i)_{i\in {I}}\) of pairwise disjoint open sets of \({\mathbb {R}}^n\), such that (i) the eventhas probability \({\mathbb {P}}(B)\) equal to 1; (ii) for each \(i\in {I}\), the restriction \(f_i\) of f to \(B_i\) is a diffeomorphism of \(B_i\) onto an open set \(C_i\subset {\mathbb {R}}^n\). Under this assumption, definewhereis the indicator function of \(C_i\). ThenA stochastic process \(\{{\mathbf{X}}_t, t\in {D}\}\), with \(D=[0,{\bar{t}}]\) a real interval, is a family of vector random variables indexed by a parameter t, defined on a common probability space \((\Omega ,{\mathcal {A}},{\mathbb {P}})\) and all with their values in \(({\mathbb {R}}^n,{\mathcal {B}}({\mathbb {R}}^n))\), which is called the state space. By definition, for each \(t\in {D}\), \({\mathbf{X}}_t\) is an \({\mathcal {A}}\)-misurable function and, for each \(\omega \in \Omega\), \(\{{\mathbf{X}}_t(\omega ), t\in {D}\}\) is a function defined in D that is called \(\textit{sample function, realizzation}\) or \(\textit{trajectory}\) of the process.

Let be \(z\in {{\mathbb {R}}}\) and \(f:{\mathbb {R}}^m\rightarrow {\mathbb {R}}\) a smooth function. Then, the setif it is not empty, is a regular hypersurface of \({\mathbb {R}}^m\) whose orientation is determined by the unit normal vector \({\mathbf{n}}=\nabla {f}/|\nabla {f}|\). Let us putand denote by \(\mu _S\) the volume form on \(\{z=f\}\) [9], i.e. the differential form(where \(d{\hat{x}}_j\) means “omit the factor \(dx_j\)”). Recalling that on the hypersurface \(\{z=f\}\) we haveit is easy to verify that, for each \(j=1,\ldots ,n\) it turns out that [7]everywhere \(f_{x_j}\not =0\) holds.

For each \(\alpha \le {n}\), let \({\mathcal {H}}^{\alpha }\) be the Hausdorff measure in \({\mathbb {R}}^n\) [10], defined in such a way that \({\mathcal {H}}^{n}={\mathcal {L}}^{n}\). Then for every function g which is integrable on \(\{z=f\}\) we haveThe following useful result is a consequence of the coarea formula. Let \(f:{\mathbb {R}}^m\rightarrow {\mathbb {R}}\) be a smooth function such that \(|\nabla {f({{\mathbf{x}}})}|>0\), and \(g:{\mathbb {R}}^m\rightarrow {\mathbb {R}}\) be an integrable function. Moreover, let us putThenand, in particular,

The equation of motion of a body, discretized by the finite element method with respect to the space variable is reduced to a system of ODEswhere \(t\in {D}\) is the time and \({\mathbf{y}}, \dot{{\mathbf{y}}}\) and \(\ddot{{\mathbf{y}}}\) are the displacement, velocity and acceleration vector, respectively, which are defined on D and take their values in \({\mathbb {R}}^n\). \({\textbf {M}}\) is the mass matrix, \({\mathbf {f}}\) is the internal force vector (including damping and restoring force), \({\textbf {B}}\) is the input force influence matrix, \({\varvec{\xi }}\) is the external excitation vector and \({\mathbf{y}}_0\) and \(\dot{{\mathbf{y}}}_0\) are the initial displacement and velocity vector, respectively. By introducing the state vectorEquation (9) can be rewritten aswhereIf randomness is present, coming from the initial conditions, the excitations, or the properties of the system, a random state equation can be written as [2]where \({\varvec{\theta }}=(\theta _1,\theta _2,\ldots \theta _m)\in \Lambda \subset {{\mathbb {R}}^m}\) is the vector of all random parameters, i.e. the values assumed by the vector random variable \({\varvec{\Theta }}:\Omega \rightarrow {\mathbb {R}}^m\), that is supposed to be time-independent and to have a (joint) probability density function \(p _{{\varvec{\Theta }}}\).

If the (deterministic) problem (10) is well posed, then for each choice of \({\varvec{\theta }}\) and \({\mathbf{x}}_0\), Eq. (11) has one and only one solutionwith \({\mathbf {H}}:\Lambda \times {\mathbb {R}}^{2n}\times {D}\rightarrow {\mathbb {R}}^{2n}\) a suitable smooth function.

Below, for sake of simplicity, we omit to explicitly indicate the dependence on \({\mathbf{x}}_0\) and writeto denote the solution of (11).

In applications, we are interested in considering stochastic processes of the typewithand in determining the evolution of the law of \(\textit{Z}_t\). Here \(\psi :{\mathbb {R}}^{2n}\rightarrow {\mathbb {R}}\) is a deterministic smooth function. To this aim, for each \(t\in {D}\), let us consider the family of measures \(\{\mu _t^{\theta }:{\varvec{\theta }}\in {\Lambda }\}\) which are defined on \({\mathbb {R}}\) byfor every Borel subset B of \({\mathbb {R}}\), with \(\chi _B\) the indicator function of B. Measure \(\mu _t^{\theta }\) is the law of the conditional probability of \(\textit{Z}_t\), given \({\varvec{\Theta }}\), and sometimes it is denoted by \(p_{\textit{Z}\varvec{\Theta }}(\textit{z}|\varvec{\Theta }={\varvec{\theta }},t)\). Indeed, once \({\varvec{\theta }}\) and t are fixed, \(\mu _t^{\theta }\) is the Dirac measure on \({\mathbb {R}}\), concentrated in the point \(\textit{Z}_t({\varvec{\theta }})\). The cumulative distribution function of \(\mu _t^{\theta }\) is the real functionLet \({\mathcal {C}}_0\) be the space of the continuous real functions that are compactly supported in \({\mathbb {R}}\), with the maximum norm \(|\cdot | _{{\mathcal {C}}_0}\). For every function \(f\in {{\mathcal {C}}_0}\) we have(this is true for the indicator functions of measurable sets by (13) and then for simple functions so that the general case follows from an approximation argument). Therefore, the map \({\varvec{\theta }}\rightarrow \int _ {{\mathbb {R}}} f(\textit{z})\mu _t^{\theta } (d\textit{z})\) is measurable on \(\Lambda\).

If we denote by \(F_\textit{Z}(\textit{z},t)\) the cumulative distribution function of \(\mu _t\), with the help of (16) and (14) we can writewhereMoreover, under the hypothesis that, for each \(t\in {D}\), \(\textit{Z}_t\) is a smooth function withfrom (18) to (7) it followswhereIn many applications of interest, for each \(t\in {D}\) and \(B\in {{\mathcal {B}}({\mathbb {R}})}\), \(Z_t\) is such thatso that in view of (17), it also results \(\mu _t(B)=0\) and therefore \(\mu _t\) is absolutely continuous with respect to \({\mathcal {L}}\) [and \(F_Z(\cdot ,t)\) is an absolutely continuous real function] [6]. Then, by the Radon-Nikodym theorem there exists a probability density function \(p_Z(z,t)\) of \(\mu _t\), i.e. for every integrable function f it holdsThus, in this case from (21) to (8), we deduceFor all z and t, \({\{Z_t=z\}}\) is a regular hypersurface of \(\Lambda\) by (20). Moreover, in view of (5) and (6), we haveif on the hypersurface \({\{Z_t=z\}}\) the inequality \(\frac{\partial {Z_t}}{\partial {\theta _j}}\ne {0}\) holds.

As will be shown later in the examples, Eqs. (18), (21) and (22), (23) can be used for the explicit calculation of the cumulative distribution function and the probability density function, respectively, even if the dimension of \(\Lambda\) is greater than one.

In [2] the authors obtain a linear PDE in the unknownwhich once solved allows to determine the probability density function \(p_Z(z,t)\) of \(Z_t\), by integrating \(p_{Z\Theta }\) over \(\Lambda\), with respect to \({\varvec{\theta }}\). Below we propose a similar method to directly determine the cumulative distribution function \(F_Z(z,t)\) of \(Z_t\). Let’s start with some notation.

Let be \(N={\mathbb {R}}\times {\Lambda }\times {D}\subset {{\mathbb {R}}^{m+2}}\), \(G(z,{\varvec{\theta }},t)=z-Z({\varvec{\theta }},t)\) and \(\chi _K\) be the indicator function of the regionthat is(note that, for each pair z and t, region \(\{Z_t\le {z}\}\) that has been defined in (19) is a subset of K). Let us denote by \({\mathcal {D}}\) the space of infinitely differentiable real functions with compact support in N.

Consider the two distributions [7] which, for every \(\phi \in {{\mathcal {D}}}\), are defined in N bywith \({\mathbf{x}}\)=\((z,{\varvec{\theta }},t)\), andwhereis a regular hypersurface, because in N we have \(|\nabla {G}|=\sqrt{1+|\nabla {Z}|^2}\ge {1}\). Note that while the first distribution is regular, the second is concentrated on a set that is negligible with respect to the Lebesgue measure.

It can be proved [7] that \(\delta (G)\) is the distributional derivative of \(\chi _K\), i.e. the distribution such thator alsoFrom (25)\(_1\) to (25)\(_3\) we deduce the PDEwith the initial conditionswhich follows from (24) to (12).

Once \(\chi _K\) has been calculated, \(F_Z\) is obtained from the relationIn a similar way, [7] it is defined the distributional derivative \(\delta ^{\prime }\) of \(\delta\) and it is shown that it resultsfrom which the PDE that has been proposed in [2] can be obtained, i.e.The corresponding initial condition isdepending on whether \(z_0\) is deterministic or not. Once \(p_{Z\varvec{\Theta }}\) has been calculated, \(p_Z\) is obtained from the relationThus \(\chi _K\) and \(p_{Z\varvec{\Theta }}\) satisfy the same distributional PDE but with different initial conditions.

In applications, system (11) and Eqs. (26) [or (28)] are both solved numerically, after which \(F_Z(z,t)\) is determined by the relation (27) [or \(p_Z(z,t)\) is determined by (29)]. Two examples are presented below where the whole procedure can be followed explicitly. Let’s briefly mention the solution technique for Eq. (26). The analogue for Eq. (28) is dealt with in detail in [2].

Using the method of characteristics [12] for this purpose, we can write the following system of ODEs, with the corresponding initial conditionsFrom (30)\(_2\) we obtain \(t=s\), which in view of (30)\(_1\) provides the following two implicationsFrom (30)\(_3\) to (31), taking into account that \(Z({\varvec{\theta }},0)=z_0\) by (12), we deduceFinally, from (27) we obtainwhich coincides with (18).

From the results obtained in the cases examined it appears that the accuracy of the cumulative distribution function, calculated with the proposed method, is good. The relative numerical procedure can be implemented in any code that allows the solution of the dynamic system and can be easily extended to vector-valued random processes.