A linear elastic problem described by the general equations of mechanics (balance of momentum, kinematics for small displacements and isotropic linear elastic material law) is considered. The Young’s modulus
E is assumed to depend on the spatial position
\({\varvec{x}}\) and some new identification parameters
\({\varvec{\theta }}\), reflecting damage. A log-normal distribution is assumed, following the suggestions of the European design standards [
46] for material parameters. The log-normal distribution avoids negative Young’s moduli. Therefore, the Young’s modulus is given by a log-normal random field
\(\varkappa _{LN}({\varvec{x}},{\varvec{\theta }})\) as
$$\begin{aligned} \begin{aligned}&E({\varvec{x}},{\varvec{\theta }}) := E_0 \, \varkappa _{LN}({\varvec{x}}, {\varvec{\theta }})\\&\quad = E_0 \, \mathrm {exp}(\underbrace{\mu ({\varvec{x}}) + \sum \limits _{j=1}^K \sqrt{\lambda _j} \, {{\varvec{V}}}_j \, ({\varvec{x}}) \, \theta _j}_{\text {normal random field}}), \end{aligned} \end{aligned}$$
(22)
which is modeled as a transformation of an underlying normal random field [
47] represented with a Karhunen-Loéve expansion (e.g., [
48‐
50]). In Eq. (
22),
\( \mu ({\varvec{x}})\) is the mean value,
\({V}_{j}\) are the eigenvectors,
\(\lambda _{j}\) the eigenvalues of the covariance matrix (kernel) and
\({\varvec{\theta }}\) the
K variables (normally distributed with zero mean) of the underlying normal random field. Note that, due to this nonlinear transformation, the correlation of the log-normal field is not identical to the prescribed one in the normal field. The covariance matrix
\({\varvec{C}}\) simulates the spatial dependency of the variables and is given by the covariance function
\(c({\varvec{x}}_i, {\varvec{x}}_j)\), describing the correlation between two spatial points
i and
j. Among the many possible covariance functions [
48], the squared exponential kernel function (i.e. belonging to the Matérn family) is used:
$$\begin{aligned} c(r)= \sigma ^2 \, \exp \left( -\frac{r^2}{2\varrho ^2}\right) , \end{aligned}$$
(23)
where
r denotes the distance between two points
\(\vert {\varvec{x}}_i-{\varvec{x}}_j \vert \),
\(\sigma \) is the standard deviation and
\(\varrho \) the correlation length of the random field. Based on a standard finite element discretization of the random field variables with a mesh size much smaller then the correlation length, the solution of the generalized eigenvalue problem for the correlation matrix
\( {\varvec{M}}^T \, {\varvec{C}} \, {\varvec{M}} \, {\varvec{V}} = \lambda \, {\varvec{M}} \, {\varvec{V}} \) with the mass matrix
\({\varvec{M}}\) leads to the eigenvalues and eigenvectors describing the underlying normal random field in Eq. (
22). The number of random field variables
K depends on the ratio between the correlation length and the structural dimension. For a study of the influence of the random field discretization on the identification process, the reader is referred to [
47]. In this example, the number
K is chosen based on the ratio of the eigenvalues
$$\begin{aligned} \sqrt{\lambda _K}/\sqrt{\lambda _0} < tol_{RF} \end{aligned}$$
(24)
with a prescribed tolerance (given in the examples). The random field parameters (
\(\mu \),
\(\sigma \) and
\(\varrho \)) are assumed to be known. The identification parameters are the
K unknown field variables
\({\varvec{\theta }}\) defined in Eq. (
22).
The PGD approach is applied to generate a numerical abacus, giving the solution as a function of the spatial directions
\({\varvec{x}}\) and the unknown model parameters
\({\varvec{\theta }}\), here the random field variables. For this reason, the displacement field
\({\varvec{u}}\) is approximated by a sum of
n terms as
$$\begin{aligned} {\varvec{u}}^n({\varvec{x}},{\varvec{\theta }}) = \sum \limits _{i=1}^{n} {\varvec{F}}_0^i({\varvec{x}}) \, F_1^i(\theta _1) \, F_2^i(\theta _2) \dots F^i_{K}(\theta _K). \end{aligned}$$
(25)
The PGD modes are numerically determined by solving the weak form of linear elasticity with the approximation
\({\varvec{u}}^n\) from Eq. (
25) over the multi-dimensional space
\(\Omega ^D = \Omega ^{{\varvec{x}}}\bigotimes \limits _{j=1}^K \Omega ^{\theta _j}\), with its Neumann and Dirichlet boundaries
\(\Gamma _N\) and
\(\Gamma _D\),
$$\begin{aligned} \begin{aligned}&\int _{\Omega ^D} \, \delta {\varvec{\varepsilon }}^n({\varvec{x}},{\varvec{\theta }}) : {\varvec{C}}_{el}^{n_c}({\varvec{x}},{\varvec{\theta }}) : {\varvec{\varepsilon }}^n({\varvec{x}},{\varvec{\theta }}) \, d\Omega ^D \\ +&\int _{\Omega ^D} \, \delta {\varvec{u}}^n({\varvec{x}},{\varvec{\theta }}) \, {\varvec{f}}^{n_f}({\varvec{x}},{\varvec{\theta }}) \, d\Omega ^D \\ -&\int _{\Gamma _N} \, \delta {\varvec{u}}^n({\varvec{x}},{\varvec{\theta }}) \, {\varvec{t}}^{n_t}({\varvec{x}},{\varvec{\theta }}) \, d\Gamma _N= 0. \end{aligned} \end{aligned}$$
(26)
In the latter,
\(\delta \bullet \) denotes the variation of
\(\bullet \). The strains
\({\varvec{\varepsilon }}^n\) are given considering the kinematic law
\({\varvec{\varepsilon }}^n = \mathrm{grad}({\varvec{u}}^n)\) in separated form affine to Eq. (
25).
\({\varvec{f}}^{n_f}\) is a given volume load,
\({\varvec{t}}^{n_t}\) a specified surface load and
\({\varvec{C}}^{n_c}_{el}\) refers to the linear elasticity tensor, all depending on the space
\({\varvec{x}}\) and the random field parameters
\({\varvec{\theta }}\). The Dirichlet condition
\({\varvec{u}}^n = {\varvec{u}}^*\) completes the description of the problem above. For an efficient PGD computation, all
\({\varvec{\theta }}\)-depending functions in Eq. (
26) must be expressed as an affine separate representation to Eq. (
25), allowing the separation of the multi-dimensional integral into
D lower dimensional integrals [
51]. This separated form is denoted by the subscripts (
\(n_f\),
\(n_t\),
\(n_c\)) in Eq. (
26) given the number of required terms for each function. In the investigated case, the loads are not depending on the parameters
\({\varvec{\theta }}\) so that their separation is trivial. The linear elasticity tensor explicitly depends on the random field parameters
\({\varvec{\theta }}\) and the space
\({\varvec{x}}\) by the given Young’s modulus definition in Eq. (
22)
$$\begin{aligned} C_{el,ijkl}({\varvec{x}},{\varvec{\theta }}) = \varkappa _{LN}({\varvec{x}},{\varvec{\theta }})\, C^0_{el,ijkl} \end{aligned}$$
(27)
with the averaged stiffness
$$\begin{aligned} \begin{aligned} C^0_{el,ijkl}&= \frac{E_0}{(1+\nu )} \, \frac{1}{2}(\delta _{il}\delta _{jk}+\delta _{ik}\delta _{jl}) \\&+\frac{E_0\, \nu }{(1+\nu )\,(1-2\nu )} \, \delta _{ij}\delta _{kl} \end{aligned} \end{aligned}$$
(28)
for the averaged Young’s modulus
\(E_0\) and the Poisson’s ratio
\(\nu \). Since
\(\varkappa _{LN}\) is a log-normal random field defined in Eq. (
22), the separation of Eq. (
27) is more complex than e.g. prescribing the Young’s modulus as a normal random field [
33] and requires an approximation of the exponential function. Here, a Taylor series expansion of the field
\(\varkappa _{LN}\) around the mean
\({\varvec{\theta }}_0\) with a flexible number of terms is proposed to separate the dependency on
\({\varvec{x}}\) and each
\(\theta _i\). The Taylor series expansion can be reordered to obtain a separated representation (see appendix C)
$$\begin{aligned} \begin{aligned} \varkappa _{LN}&\approx {\hat{\varkappa }}_{LN}({\varvec{x}},{\varvec{\theta }})\\&= \sum \limits _{i=1}^{n_c} X_0^i({\varvec{x}}) \, X_1^i(\theta _1) \, X_2^i(\theta _2) \dots X^i_{K}(\theta _K), \end{aligned} \end{aligned}$$
(29)
affine to the assumed PGD model in Eq. (
25). Inserting this separation into Eq. (
27) leads to the required separated function of the linear elasticity tensor in Eq. (
26)
$$\begin{aligned} {\varvec{C}}_{el}^{n_c}({\varvec{x}},{\varvec{\theta }})=\sum \limits _{i=1}^{n_c} \underbrace{C^0_{el,ijkl}\,X_0^i({\varvec{x}})}_{{\varvec{X}}_0^i({\varvec{x}})} \, X_1^i(\theta _1) \, X_2^i(\theta _2) \dots . \end{aligned}$$
(30)
There exist different ways to solve Eq. (
26) in a discrete way computing the single PGD modes
\(F_j^i\) [
52]. Here, the progressive PGD strategy [
53,
54] is used, where the problem is solved iteratively for each new set of basis functions. The computation is based on a fixed-point iteration involving the finite element computation of
D small dimensional problems – one for each PGD coordinate. The detailed equations of each problem and the used convergence criteria are summarized in appendix D.
After a more elaborate computation – compared to a standard finite element model – of all PGD modes, the PGD reduced forward model
\({\mathcal {M}}_{PGD}\) depending on the identification parameters
\({\varvec{\theta }}\), required in the variational inference (Sect.
2), can be written as
$$\begin{aligned} {\mathcal {M}}^n_{PGD}: \, {\varvec{g}}^n_s({\varvec{\theta }}) = \sum \limits _{i=1}^{n} {\varvec{F}}_0^i({\varvec{x}}_{s}) \, \prod \limits _{j=1}^K F_{j}^i (\theta _j), \end{aligned}$$
(31)
where
\({\varvec{x}}_s\) denotes the coordinates of the sensors with measurement data. Note, the mode functions
\(F_j^i\) are only once computed and just need to be evaluated for each parameter set in the inference run. Furthermore, the spatial modes
\({\varvec{F}}_1^i({\varvec{x}}_{s})\) can be precomputed and cached at these positions for an even more efficient computation during the inference process. Furthermore, the Jacobian with respect to the identification parameters, required for nonlinear forward models (Eq. (
7)), can be computed by the derivative of the single one-dimensional modes also in a very efficient way, including only function evaluations
$$\begin{aligned} \begin{aligned} {\varvec{J}}_s^n({\varvec{\theta }}) =&\left[ \sum \limits _{i=1}^{n} {\varvec{F}}_0^i({\varvec{x}}_{s}) \, \frac{dF_1^i(\theta _1)}{d\theta _1} \prod \limits _{j=1}^K F_{j}^i (\theta _j), \right. \\&\left. \sum \limits _{i=1}^{n} {\varvec{F}}_0^i({\varvec{x}}_{s}) \, F_1^i(\theta _1) \, \frac{dF_2^i(\theta _2)}{d\theta _2} \prod \limits _{j=3}^K F_{j}^i (\theta _j), \dots \right] . \end{aligned} \end{aligned}$$
(32)
For the proposed iterative inference process in algorithm
2, it is important that the accuracy of the proposed PGD surrogate is mainly influenced by two aspects: the discretization error for each coordinate solving the discrete PGD modes and the truncation error (
\(=\) number of mode sets
n) [
51]. Therefore, the accuracy of the PGD surrogate of each step
n in algorithm
2 can be increased by either refining the mesh solving the PGD modes based on the discretized weak form Eq. (
26) or increasing the number of modes. For that reason, using a PGD model in the proposed iterative inference algorithm
2 leads to two nested loops: one over the mesh refinement and one over the number of modes. Therefore, the PGD model accuracy degree will, from now on, be given by two superscripts
\({\mathcal {M}}_{PGD}^{(k,n)}\), to differentiate between both aspects, where
k refers to the mesh refinement degree and
n to the number of modes of the current model. Furthermore,
\(KL^{modes}\) and
\(BF^{modes}\) denote the comparison of PGD models with the same mesh resolution
k and different mode numbers, whereas
\(KL^{mesh}\)/
\(BF^{mesh}\) refer to different mesh resolution by a fixed number of modes
n.