Advanced Reduced-Order Models for Moisture Diffusion in Porous Media

Abstract

It is of great concern to produce numerically efficient methods for moisture diffusion through porous media, capable of accurately calculate moisture distribution with a reduced computational effort. In this way, model reduction methods are promising approaches to bring a solution to this issue since they do not degrade the physical model and provide a significant reduction of computational cost. Therefore, this article explores in details the capabilities of two model reduction techniques—the Spectral reduced-order model and the proper generalized decomposition—to numerically solve moisture diffusive transfer through porous materials. Both approaches are applied to three different problems to provide clear examples of the construction and use of these reduced-order models. The methodology of both approaches is explained extensively so that the article can be used as a numerical benchmark by anyone interested in building a reduced-order model for diffusion problems in porous materials. Linear and nonlinear unsteady behaviors of unidimensional moisture diffusion are investigated. The last case focuses on solving a parametric problem in which the solution depends on space, time and the diffusivity properties. Results have highlighted that both methods provide accurate solutions and enable to reduce significantly the order of the model around 10 times lower than the large original model. It also allows an efficient computation of the physical phenomena with an error lower than $10^{-2}$ when compared to a reference solution.

Appendices

Appendix A: Dimensionless Values

1.1 A.1 Linear Case

Problem (5) is taken into account with $g_{\mathrm{l}, \mathrm {L}}^{\star }=g_{\mathrm{l}, \mathrm {R}}^{\star }=0$ and a Dirichlet condition on the left side:

$$\begin{aligned} c_\mathrm{m}^{\star }\frac{\partial u}{\partial t^{\star }}&=\frac{\partial }{\partial x^{\star }} \left( d_\mathrm{m}^{\star }\frac{\partial u}{\partial x^{\star }} \right) , \quad t^{\star }> 0, \quad x^{\star }\in \ \big [ 0, 1 \big ] , \end{aligned}$$

(23a)

$$\begin{aligned} u&=u_{\mathrm {L}}, \quad t^{\star }> 0, \quad x^{\star }=0 , \end{aligned}$$

(23b)

$$\begin{aligned} -d_\mathrm{m}^{\star }\frac{\partial u}{\partial x^{\star }}&=\mathrm {Bi}_{\mathrm{v},\mathrm {R}}\cdot \Bigl ( u -u_{\mathrm {R}}(t^{\star }) \Bigr ) , \quad t^{\star }> 0, \quad x^{\star }=1 , \end{aligned}$$

(23c)

$$\begin{aligned} u&=1 , \quad t^{\star }=0, \quad x^{\star }\in \big [ 0, 1 \big ] . \end{aligned}$$

(23d)

The dimensionless properties of the material are $d_\mathrm{m}^{\star }=1$ and $c_\mathrm{m}^{\star }=430$. The reference time is $t^{0}=1$ $\mathrm{h}$, thus the final simulation time is fixed to $t^{\star } =120$. The Biot number is $\mathrm {Bi}_{\mathrm{v},\mathrm {R}}=333$. The boundary conditions are expressed as:

$$\begin{aligned} u_{\mathrm {L}}&=1 , \\ u_{\mathrm {R}}(t^{\star })&=1 +1.6 \sin ^{2} \left( \frac{2\pi t^{\star }}{48}\right) . \end{aligned}$$

1.2 A.2 Parametric Case

Problem (5) is taken into account with $g_{\mathrm{l}, \mathrm {L}}^{\star }=g_{\mathrm{l}, \mathrm {R}}^{\star }=0$ and a Dirichlet condition on the left side, the same as in the previous case. The reference time is $t^{0}=1$ $\mathrm{h}$, thus the final simulation time is fixed to $t^{\star } =120$. The Biot number is $\mathrm {Bi}_{\mathrm{v},\mathrm {R}}=100$. The boundary conditions are expressed as:

$$\begin{aligned} u_{\mathrm {L}}&=1 , \\ u_{\mathrm {R}}(t^{\star })&=1 +1.6 \sin ^{2} \left( \frac{2\pi t^{\star }}{48}\right) . \end{aligned}$$

The dimensionless properties of the materials are $d_\mathrm{m}^{\star }=1$ and $c_\mathrm{m}^{\star }$ assume the following values:

i	1	2	3	4	5	6	7	8	9	10
$c_{m,i}^{\star }$	833	576	441	357	300	258	227	203	183	166

1.3 A.3 Nonlinear Case

Problem (5) is taken into account with $g_{\mathrm{l}, \mathrm {L}}^{\star }=g_{\mathrm{l}, \mathrm {R}}^{\star }=0$ and Robin condition on both boundaries. The Biot number are $\mathrm {Bi}_{\mathrm{v},\mathrm {L}}=10$ and $\mathrm {Bi}_{\mathrm{v},\mathrm {R}}=15$. The boundary conditions are expressed as:

$$\begin{aligned}&u_{\mathrm {L}}(t^{\star }) =1 +0.3 \biggl [ 1 -\cos \left( \frac{2\pi t^{\star }}{24}\right) \biggr ] , \\&u_{\mathrm {R}}(t^{\star }) =1 +0.6 \sin ^{2} \left( \frac{2\pi t^{\star }}{60}\right) . \end{aligned}$$

The reference time is $t^{0}=1$ $\mathrm{h}$, thus the final simulation time is fixed to $t^{\star } =120$. The dimensionless properties of the materials are:

$$\begin{aligned} d_\mathrm{m}^{\star }(u)&=\bigl ( 0.86 +0.25 u \bigr ) \cdot 5 \times 10^{-3} , \\ c_\mathrm{m}^{\star }(u)&=3.36 -6.11 u +3.37 u^{2} . \end{aligned}$$

Appendix B: Solving the Spectral ODE Reduced System

Consider a more general situation of the reduced-order system (14):

$$\begin{aligned} \left\{ \begin{array}{rcl} \dot{a}(t) &{} =&{} \mathcal {A}(\nu ) a +{b}(t , a(t) ;\nu ) , \\ a(t_{0})&{} =&{} a_{0} , \end{array}\right. \end{aligned}$$

(24)

where ${b}(t , a(t) ;\nu )$ depends on the solution a(t) via nonlinear boundary conditions, or it contains problem’s nonlinearities, if there are some. The dependence on parameters is the most accurate within the chosen Spectral framework. The general analytical solution to problem (24) can be written as:

$$\begin{aligned} a(t;\nu ) =\mathrm {e}^{ ( t -t_{0} ) \mathcal {A}(\nu ) } a_{0} +\int _{t_{0}}^{t} \mathrm {e}^{ (t -\tau ) \mathcal {A}(\nu )} {b}\bigl ( \tau , a(\tau ); \nu \bigr ) \mathrm {d}\tau . \end{aligned}$$

(25)

The exponential matrix is defined as the limit:

$$\begin{aligned} \mathrm {e}^{t\mathcal {A}} =\lim _{n \rightarrow \infty } \biggl (\mathrm {Id} +\dfrac{1}{n} t \mathcal {A}\biggr )^{n} , \end{aligned}$$

in which $\mathrm {Id} \in \mathrm {Mat}_{(n-2)\times (n-2)}(\mathbb {R})$ is the identity matrix. However, this method is not the best way to compute the exponential matrix. In some particular cases, the solution of Eq. (24) can be simplified and thus better exploited (Moler and Loan 2003).

Case I: If we have homogeneous boundary conditions, problem (24) becomes:

$$\begin{aligned} \left\{ \begin{array}{l} \dot{a} =\mathcal {A}(\nu ) a ,\\ a( t_{0} ) =a_{0} , \end{array}\right. \end{aligned}$$

and it can be analytically solved as:

$$\begin{aligned} a(t ; \nu ) =\mathrm {e}^{( t -t_{0} ) \mathcal {A}(\nu )} a_{0}. \end{aligned}$$

Using modern methods, the exponential matrix can be computed using $\sim 48n^{3}$ floating point operations per second (FLOPS) (Al-Mohy and Higham 2009). As an information to the reader, the previous result was $\sim 538n^{3}$ FLOPS (Kenney and Laub 1998).

However, one can notice that we do not really need to build the exponential matrix, but we want to compute its action on the initial state vector $a_{0}$. Nowadays, it can be directly done, without forming $\mathrm {e}^{( t -t_{0})\mathcal {A}( \nu )}$, explicitly to a prescribed accuracy that can be set significantly lower than the standard machine precision $\sim 10^{-16}$ (Al-Mohy and Higham 2011). If computing Eq. (24) by a MATLAB solver, for example ODE45, the standard tolerance is of order of $\sim 10^{-6}$.

Case II: If we have inhomogeneous boundary conditions constant in time, the problem from Eq. (24) becomes:

$$\begin{aligned} \left\{ \begin{array}{l} \dot{a} =\mathcal {A}( \nu ) a +{b}( \nu ) , \\ a ( t_{0} ) =a_{0} , \end{array}\right. \end{aligned}$$

which can also be analytically solved:

$$\begin{aligned} a ( t; \nu ) =\mathrm {e}^{( t -t_{0} ) \mathcal {A}(\nu )} a_{0} +( t -t_{0} ) \frac{\mathrm {e}^{( t -t_{0} ) \mathcal {A}( \nu )} -\mathrm {Id} }{( t -t_{0} ) \mathcal {A}(\nu )} {b}( \nu ) . \end{aligned}$$

Case III: If we have inhomogeneous boundary conditions are linear in time, problem Eq. (24) becomes:

$$\begin{aligned} \left\{ \begin{array}{l} \dot{a} =\mathcal {A}( \nu ) a +{b}( \nu )\cdot t , \\ a ( t_{0} ) =a_{0} , \end{array}\right. \end{aligned}$$

the solution is given by:

$$\begin{aligned} a ( t; \nu ) =\mathrm {e}^{( t -t_{0} ) \mathcal {A}( \nu )} a_{0} +( t -t_{0} )^{2} \ \frac{\mathrm {e}^{( t -t_{0} ) \mathcal {A}( \nu )} -\mathrm {Id} }{( t -t_{0} ) \mathcal {A}( \nu )} \ {b}( \nu ) . \end{aligned}$$

Case IV: Boundary condition is polynomial in time:

$$\begin{aligned} \left\{ \begin{array}{l} \dot{a} =\mathcal {A}( \nu ) a +{b}( \nu ) \cdot t^{ m-1},\quad m \ \geqslant \ 3 , \\ a ( t_{0} ) =a_{0} . \end{array}\right. \end{aligned}$$

Then, the solution is given by the following analytical formula:

$$\begin{aligned} a ( t; \nu ) =\mathrm {e}^{( t -t_{0} ) \mathcal {A}( \nu )} a_{0} +( t -t_{0} )^{m} \ \varphi _{ m} \Bigl ( ( t -t_{0} ) \mathcal {A}( \nu ) \Bigr ) {b}( \nu ) . \end{aligned}$$

Above we introduced the so-called matrix $\varphi$-functions:

$$\begin{aligned} \varphi _\mathrm{m} ( z ) =\dfrac{\varphi _\mathrm{m} ( z) -\dfrac{1}{m!}}{z}, \quad m \ \geqslant \ 0 \quad \text {with} \quad \varphi _{0} (z)\ \equiv \ \mathrm {e}^{z} . \end{aligned}$$

A few first functions are given below explicitly:

$$\begin{aligned} \varphi _{ 0} ( z )&=\mathrm {e}^{z} =1 +z +\dfrac{1}{2} z^{2} +\dfrac{1}{3!} z^{3} +\cdots \\ \varphi _{ 1} ( z )&=\dfrac{\mathrm {e}^{z}-1}{z} =1 +\dfrac{1}{2} z +\dfrac{1}{3!} z^{2} +\dfrac{1}{4!} z^{3} +\cdots \\ \varphi _{ 2} ( z )&=\dfrac{\mathrm {e}^{z}-1-z}{z^{2}} =\dfrac{1}{2} +\dfrac{1}{3!} z +\dfrac{1}{4!} z^{2} +\dfrac{1}{5!} z^{3} +\cdots \\ \varphi _{ 3} ( z )&=\dfrac{\mathrm {e}^{z}-1-z-\frac{1}{2}}{z^{3}} =\dfrac{1}{3!} +\dfrac{1}{4!} z +\dfrac{1}{5!} z^{2} +\dfrac{1}{6!} z^{3} +\cdots \end{aligned}$$

The general power series representation of $\varphi$-functions is

$$\begin{aligned}&\varphi _{ m} ( z )\ \equiv \ \sum _{k = 0}^{\infty } \dfrac{z^{ k}}{( m +k )!}. \end{aligned}$$

The exponential definitions of $\varphi _{ m} ( z )$ should not be used for practical simulations, because of severe cancelation errors for $z \ll 1$. Efficient methods for computation of $\varphi$-functions have been developed based on PadÉ-type expansions, to give an example, MATLAB’s function expm() is based on such approximations (Cox and Matthews 2002).

Case V: For a general case of linear boundary conditions, the solution of problem Eq. (24) is:

$$\begin{aligned} a ( t; \nu ) =\mathrm {e}^{ ( t -t_{0} ) \mathcal {A}(\nu )} a_{0} +&\underbrace{\int _{t_{0}}^{t} \mathrm {e}^{ (t -\tau ) \mathcal {A}(\nu )} {b}( \tau ; \nu ) \mathrm {d}\tau }_{(I)}. \end{aligned}$$

To exploit the last formula, one might employ a quadrature formula to discretize the integral (I):

$$\begin{aligned} a ( t; \nu ) =\mathrm {e}^{ ( t -t_{0} ) \mathcal {A}(\nu )} a_{0} +\Delta t \sum ^{m}_{j =1} \mathrm {e}^{ ( t -t_{0} ) \mathcal {A}(\nu )} \cdot {b}( \tau _{ j} ; \nu ), \quad \Delta t\ \mathop {\ {\mathop {:=}\limits ^{\mathrm {def}}}\ }\ \frac{t -t_{0}}{m}, \end{aligned}$$

where we employed rectangle formula for simplicity. We note that the sequence $\{\mathrm {e}^{ \Delta t \mathcal {A}(\nu )} \}_{j=1}^{m}$ can be entirely computed in an efficient manner (Al-Mohy and Higham 2011).

Case VI: Considering a general nonlinear case of boundary conditions from problem Eq. (24) and the general solution Eq. (25). To exploit a better solution, we can develop the function $\tau \mapsto {b}(\tau , a(\tau ); \nu )$ in Taylor expansion series and integrate it exactly:

$$\begin{aligned} a ( t; \nu ) =\mathrm {e}^{ ( t -t_{0} ) \mathcal {A}(\nu )} a_{0} +\sum ^{\infty }_{k =1} ( t -t_{0} )^{k} \varphi _{ k} \Bigl ( ( t-t_{0} ) \mathcal {A}(\nu ) \Bigr ) a_{k} , \end{aligned}$$

where

$$\begin{aligned} a_{k}\ \mathop {\ {\mathop {:=}\limits ^{\mathrm {def}}}\ }\ \left. \dfrac{\mathrm {d}^{ k - 1}}{\mathrm {d}t^{ k - 1}}\ {b}\Bigl ( t, a(t);\nu \Bigr ) \right| _{ t =t_{0}} . \end{aligned}$$

Finally, the series solution can be exploited by truncating it at some finite order:

$$\begin{aligned} a ( t; \nu ) =\mathrm {e}^{ ( t -t_{0} ) \mathcal {A}(\nu )} a_{0} +\sum _{k =1}^{K} ( t -t_{0} )^{k} \varphi _{k} \Bigl ( ( t -t_{0} ) \mathcal {A}(\nu ) \Bigr ) a_{k} . \end{aligned}$$