PGD-Based Computational Vademecum for Efficient Design, Optimization and Control

Abstract

In this paper we are addressing a new paradigm in the field of simulation-based engineering sciences (SBES) to face the challenges posed by current ICT technologies. Despite the impressive progress attained by simulation capabilities and techniques, some challenging problems remain today intractable. These problems, that are common to many branches of science and engineering, are of different nature. Among them, we can cite those related to high-dimensional problems, which do not admit mesh-based approaches due to the exponential increase of degrees of freedom. We developed in recent years a novel technique, called Proper Generalized Decomposition (PGD). It is based on the assumption of a separated form of the unknown field and it has demonstrated its capabilities in dealing with high-dimensional problems overcoming the strong limitations of classical approaches. But the main opportunity given by this technique is that it allows for a completely new approach for classic problems, not necessarily high dimensional. Many challenging problems can be efficiently cast into a multidimensional framework and this opens new possibilities to solve old and new problems with strategies not envisioned until now. For instance, parameters in a model can be set as additional extra-coordinates of the model. In a PGD framework, the resulting model is solved once for life, in order to obtain a general solution that includes all the solutions for every possible value of the parameters, that is, a sort of computational vademecum. Under this rationale, optimization of complex problems, uncertainty quantification, simulation-based control and real-time simulation are now at hand, even in highly complex scenarios, by combining an off-line stage in which the general PGD solution, the vademecum, is computed, and an on-line phase in which, even on deployed, handheld, platforms such as smartphones or tablets, real-time response is obtained as a result of our queries.

Appendix: Alternating Directions Separated Representation Constructor

1.1 A.1 Computing R(x) from S(t) and W(k)

We consider the extended weighted residual form of Eq. (8):

$$ \int_{\varOmega\times\mathcal{I}_t \times \mathcal{I}_k} {u^{\ast}} \biggl( { \frac{\partial u}{\partial t}-k \cdot \varDelta u-f} \biggr)\,d\mathbf{x} \cdot dt \cdot dk=0 $$

(106)

where the trial and test functions write respectively:

$$ u^n ( {\mathbf{x},t,k} )=\sum _{i=1}^{n-1} {X_i ( \mathbf{x} )} \cdot T_i ( t )\cdot K_i ( k )+R ( \mathbf{x} )\cdot S ( t )\cdot W ( k ) $$

(107)

and, assuming S and W known from the previous iteration,

$$ u^{\ast} ( {\mathbf{x},t,k} )=R^{\ast} ( \mathbf{x} )\cdot S ( t )\cdot W ( k ) $$

(108)

Introducing (107) and (108) into (106) it results:

(109)

where ${\mathcal{R}}^{n-1}$ defines the residual related to u ⁿ⁻¹(x,t,k):

$$ {\mathcal{R}}^{n-1} = {\sum_{i=1}^{n-1} {X_i \cdot} \frac{\partial T_i }{\partial t}\cdot K_i -\sum _{i=1}^{n-1} {k\cdot \varDelta X_i \cdot T_i \cdot K_i } -f} $$

(110)

Once all functions involving time and conductivity have been determined, we can integrate Eq. (109) along its respective domains $\mathcal{I}_{t} \times\mathcal{I}_{k}$, and by taking into account the following notations:

$$ \mbox{\footnotesize $\left[ \begin{array}{c@{\quad}c@{\quad}c} w_{1} =\int_{\mathcal{I}_{k}} W^{2}\,dk & s_{1} =\int_{\mathcal{I}_{t}} S^{2}dt & r_{1} =\int_{\varOmega}R^{2}\,d\mathbf{x} \\[3pt] w_{2} =\int_{\mathcal{I}_{k}} kW^{2}\,dk & s_{2} =\int_{\mathcal{I}_{t}} S\cdot \frac{dS}{dt}\,dt & r_{2} =\int_{\varOmega}R\cdot \varDelta R\,d\mathbf{x} \\[3pt] w_{3} =\int_{\mathcal{I}_{k}} W\,dk & s_{3} =\int_{\mathcal{I}_{t}} S\,dt & r_{3} =\int_{\varOmega}R\,d\mathbf{x} \\[3pt] w_{4}^{i} =\int_{\mathcal{I}_{k}} W\cdot K_{i} \,dk & s_{4}^{i} =\int_{\mathcal{I}_{t}} S\cdot\frac{dT_{i} }{dt}\,dt & r_{4}^{i} =\int_{\varOmega}R\cdot \varDelta X_{i} \,d\mathbf{x}\\[3pt] w_{5}^{i} =\int_{\mathcal{I}_{k}} kW\cdot K_{i} \,dk & s_{5}^{i} =\int_{\mathcal{I}_{t}} S\cdot T_{i} \,dt & r_{5}^{i} =\int_{\varOmega} R\cdot X_{i} \,d\mathbf{x} \end{array} \right]$} $$

(111)

Equation (109) is reduced to:

(112)

Equation (112) defines an elliptic steady-state boundary value problem that can be solved by using any discretization technique operating on the weak form of the problem (finite elements, finite volumes, …). Another possibility consists in coming back to the strong form of Eq. (112):

(113)

that could be solved by using any classical collocation technique (finite differences, SPH, …).

1.2 A.2 Computing S(t) from R(x) and W(k)

In the present case the test function is written as:

$$ u^{\ast} ( {\mathbf{x},t,k} )=S^{\ast} ( t )\cdot R ( \mathbf{x} )\cdot W ( k ) $$

(114)

Now, the weighted residual form becomes:

(115)

that integrating in the space $\varOmega\times\mathcal{I}_{k}$ and by taking into account the notation (111) results:

(116)

Equation (116) represents the weak form of the ODE defining the time evolution of the field S that can be solved by using any stabilized discretization technique (SU, Discontinuous Galerkin, …). The strong form of Eq. (116) reads:

(117)

Equation (117) can be solved by using backward finite differences, or higher order Runge-Kutta schemes, among many other possibilities.

1.3 A.3 Computing W(k) from R(x) and S(t)

In this part of the algorithm, the test function is written as:

$$ u^{\ast} ( {\mathbf{x},t,k} )=W^{\ast} ( k )\cdot R ( \mathbf{x} )\cdot S ( t ) $$

(118)

Now, the weighted residual form becomes:

(119)

Integrating Eq. (119) in $\varOmega\times\mathcal{I}_{t}$ and considering the notations given by Eq. (111) leads to:

(120)

Equation (120) does not involve any differential operator. The strong form of Eq. (120) is:

(121)

Equation (121) represents an algebraic equation because the original model does not involve derivatives with respect to the conductivity. Thus, despite the introduction of parameters as additional model coordinates, the computational complexity remains essentially the same, however, the introduction of extra-coordinates implies in general the increase of the number of modes involved by the separated representation, and consequently the computing time.