### The single scale problem

Consider a fully resolved porous domain Ω=Ω^{F}⋃Ω^{S}, such as the one depicted in Figure 1(a). The domain consists of a topologically periodic substructure where Ω^{F} is the part of Ω occupied by the fluid phase and Ω^{S} the part occupied by the solid phase^{a}. The interface between the solid and fluid phases is denoted Ґ ^{int} and the part of Ґ where fluid can enter and exit the domain is denoted Ґ^{F}=*∂* Ω^{F}\ Ґ^{int}=Ґ ⋂*∂* Ω^{F}. The fluid part of the boundary Ґ^{F} is further divided into {\mathrm{\u0490}}_{\mathrm{P}}^{\mathrm{F}} where the pressure *p* is prescribed and {\mathrm{\u0490}}_{\mathrm{V}}^{\mathrm{F}} where the velocity *v* is prescribed. We hereby restrict ourselves to flows with low Reynolds numbers and purely viscous, incompressible fluids, whereby the fluid velocity field *v* can be found by minimizing the energy potential pertaining to a local viscous potential Ф(*v*⨂*∇*), defined such that \frac{\mathrm{\partial \u0424}(\mathit{v}\u2a02\nabla )}{\partial \mathit{v}\u2a02\nabla}={\mathit{\sigma}}^{\mathrm{v}} where *σ*^{v} is the deviatoric part of the Cauchy stress ^{b}. Thus, we seek \mathit{v}\u03f5\mathcal{V} that satisfies the constrained problem

\begin{array}{l}\text{minimize}\phantom{\rule{2.77626pt}{0ex}}\underset{{\Omega}^{\mathrm{F}}}{\int}\u0424(\mathit{v}\u2a02\nabla )\phantom{\rule{2.77626pt}{0ex}}\mathrm{d}V-\underset{{\mathrm{\u0490}}_{\mathrm{P}}^{\mathrm{F}}}{\int}\widehat{\mathit{t}}\bullet \mathit{v}\phantom{\rule{2.77626pt}{0ex}}\mathrm{d}S\phantom{\rule{2em}{0ex}}\end{array}

(1)

\begin{array}{l}\text{subject to:}\phantom{\rule{2.77626pt}{0ex}}\nabla \bullet \mathit{v}=0\phantom{\rule{2.77626pt}{0ex}}\text{on}\phantom{\rule{2.77626pt}{0ex}}{\Omega}^{\mathrm{F}}\phantom{\rule{2em}{0ex}}\end{array}

(2)

where \widehat{\mathit{t}}=-\widehat{p}\mathit{n} is the prescribed pressure on the boundary {\mathrm{\u0490}}_{\mathrm{P}}^{\mathrm{F}}\subset {\mathrm{\u0490}}^{\mathrm{F}} and is defined below. This can equivalently be written as the inf-sup problem

\begin{array}{l}\underset{\mathit{v}\u03f5\mathcal{V}}{inf}\underset{p\u03f5\mathcal{P}}{sup}\left\{\underset{{\Omega}^{\mathrm{F}}}{\int}\u0424\left(\mathit{v}\u2a02\nabla \right)\mathrm{d}V-\underset{{\Omega}^{\mathrm{F}}}{\int}p(\nabla \bullet \mathit{v})\mathrm{d}V-\underset{\underset{\mathrm{P}}{\overset{\mathrm{F}}{\mathrm{\u0490}}}}{\int}\widehat{\mathit{t}}\bullet \mathit{v}\mathrm{d}S\right\}\end{array}

(3)

where

\begin{array}{l}\mathcal{V}=\left\{\mathit{v}\u03f5{\left[{H}^{1}\left({\Omega}^{\mathrm{F}}\right)\right]}^{3}:\mathit{v}=\mathbf{0}\phantom{\rule{2.77626pt}{0ex}}\text{on}\phantom{\rule{2.77626pt}{0ex}}{\mathrm{\u0490}}^{\text{int}},\phantom{\rule{2.77626pt}{0ex}}\mathit{v}=\phantom{\rule{2.77626pt}{0ex}}{\widehat{v}}_{n}\mathit{n}\phantom{\rule{2.77626pt}{0ex}}\text{on}\phantom{\rule{2.77626pt}{0ex}}{\mathrm{\u0490}}_{\mathrm{V}}^{\mathrm{F}}\right\}\phantom{\rule{2em}{0ex}}\end{array}

(4)

\begin{array}{l}\mathcal{P}=\left\{p\u03f5{L}_{2}\left({\Omega}^{\mathrm{F}}\right)\right\}\phantom{\rule{2em}{0ex}}\end{array}

(5)

and *p* is a Lagrange multiplier resulting from the continuity condition. Note that due to the fact that *σ*^{v} is the deviatoric part of the Cauchy stress *σ*, *p* is interpreted as the pressure.

We proceed by splitting the domain into a finite number of *n* domains Ω_{□,i} such that \Omega ={\bigcup}_{i=1}^{n}{\Omega}_{\square ,i} and such that each subdomain retains geometric periodicity (cf. Figure 1(a) and the periodic cutout in Figure 1(d)). By the choice of function spaces and , all functions (\mathit{v},p)\u03f5\mathcal{V}\times \mathcal{P} is continuous on the whole Ω ^{F}. Rewriting Equation 2 as the sum of the energy contribution from all subdomains gives

\begin{array}{l}\underset{\mathit{v}\u03f5\mathcal{V}}{inf}\underset{p\u03f5\mathcal{P}}{sup}\left\{\underset{i=1}{\overset{n}{\Sigma}}\left(\underset{\underset{\square ,i}{\overset{\mathrm{F}}{\Omega}}}{\int}\u0424(\mathit{v}\u2a02\nabla )\mathrm{d}V-\underset{\underset{\nabla ,i}{\overset{\mathrm{F}}{\Omega}}}{\int}p(\nabla \bullet \mathit{v})\mathrm{d}V\right)-\underset{\underset{\mathrm{P}}{\overset{\mathrm{F}}{\mathrm{\u0490}}}}{\int}\widehat{\mathit{t}}\bullet \mathit{v}\mathrm{d}S\right\}\end{array}

(6)

In order to separate the macro and subscales features, we split the pressure term *p* into a smooth part {p}^{\mathrm{M}}\u03f5{\mathcal{P}}^{\mathrm{M}} and a fluctuating part {p}^{\mathrm{S}}\u03f5{\mathcal{P}}^{\mathrm{S}} such that *p*=*p*^{M}+*p*^{S} and \mathcal{P}={\mathcal{P}}^{\mathrm{M}}+{\mathcal{P}}^{\mathrm{S}}, {\mathcal{P}}^{\mathrm{S}} being the hierachial complement to {\mathcal{P}}^{\mathrm{M}}. Integration by parts on *p*^{M} in the continuity constraint in Equation 4 yields

\begin{array}{l}\underset{i=1}{\overset{n}{\Sigma}}\underset{{\Omega}_{\square ,i}^{\mathrm{F}}}{\int}{p}^{\mathrm{M}}(\nabla \bullet \mathit{v})\mathrm{d}V=\underset{i=1}{\overset{n}{\Sigma}}\left(-\underset{{\Omega}_{\square ,i}^{\mathrm{F}}}{\int}\nabla {p}^{\mathrm{M}}\bullet \mathit{v}\mathrm{d}V+\underset{{\mathrm{\u0490}}_{\square ,i}^{\mathrm{F}}}{\int}\mathit{n}\bullet \mathit{v}{p}^{\mathrm{M}}\mathrm{d}S\right)\end{array}

(7)

where *n* is the outward pointing normal. The boundary integral on the right hand side in Equation 5 vanish on all internal boundaries as *v* and *p*^{M} are continuous. Thus, after introducing the split in *p*, Equation 4 can be restated as

\begin{array}{l}\underset{\mathit{v}\u03f5\mathcal{V}}{inf}\underset{\begin{array}{c}{p}^{\mathrm{M}}\u03f5{\mathcal{P}}^{\mathrm{M}}\\ {p}^{\mathrm{S}}\u03f5{\mathcal{P}}^{\mathrm{S}}\end{array}}{sup}\left\{\right.\phantom{\rule{2em}{0ex}}\\ \underset{i=1}{\overset{n}{\Sigma}}\left(\underset{{\Omega}_{\square ,i}^{\mathrm{F}}}{\int}\u0424(\mathit{v}\u2a02\nabla )\mathrm{d}V-\underset{{\Omega}_{\square ,i}^{\mathrm{F}}}{\int}{p}^{\mathrm{S}}(\nabla \bullet \mathit{v})\mathrm{d}V+\underset{{\Omega}_{\square ,i}^{\mathrm{F}}}{\int}\nabla {p}^{\mathrm{M}}\bullet \mathit{v}\mathrm{d}V\right)\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}-\underset{{\mathrm{\u0490}}_{\mathrm{P}}^{\mathrm{F}}}{\int}\widehat{\mathit{t}}\bullet \mathit{v}\mathrm{d}S-\underset{{\mathrm{\u0490}}^{\mathrm{F}}}{\int}\mathit{n}\bullet \mathit{v}{p}^{\mathrm{M}}\mathrm{d}S\left(\right)close="\}">& \phantom{\rule{2em}{0ex}}\end{array}\n

(8)

Furthermore, the last two terms can be rewritten as

\begin{array}{l}-\underset{{\mathrm{\u0490}}_{\mathrm{P}}^{\mathrm{F}}}{\int}\widehat{\mathit{t}}\bullet \mathit{v}\mathrm{d}S-\underset{{\mathrm{\u0490}}^{\mathrm{F}}}{\int}\mathit{n}\bullet \mathit{v}{p}^{\mathrm{M}}\mathrm{d}S=-\underset{{\mathrm{\u0490}}_{\mathrm{P}}^{\mathrm{F}}}{\int}\left[\widehat{\mathit{t}}\bullet \mathit{v}+\mathit{n}\bullet \mathit{v}{p}^{\mathrm{M}}\right]\mathrm{d}S-\underset{{\mathrm{\u0490}}_{\mathrm{V}}^{\mathrm{F}}}{\int}\mathit{n}\bullet \mathit{v}{p}^{\mathrm{M}}\mathrm{d}S\end{array}

(9)

where it is noted that the last term contains the prescribed velocity, {\widehat{v}}_{n}=\mathit{v}\bullet \mathit{n}. Under the assumption that \mathit{t}=-\widehat{p}\mathit{n}=-{p}^{\mathrm{M}}\mathit{n} the integral over {\mathrm{\u0490}}_{\mathrm{P}}^{\mathrm{F}} vanishes and Equation 6 is equivalent to

\begin{array}{l}\underset{\mathit{v}\u03f5\mathcal{V}}{inf}\underset{\begin{array}{c}{p}^{\mathrm{M}}\u03f5{\mathcal{P}}^{\mathrm{M}}\\ {p}^{\mathrm{S}}\u03f5{\mathcal{P}}^{\mathrm{S}}\end{array}}{sup}\left\{\right.\phantom{\rule{2em}{0ex}}\\ \underset{i=1}{\overset{n}{\Sigma}}\left(\underset{{\Omega}_{\square ,i}^{\mathrm{F}}}{\int}\u0424(\mathit{v}\u2a02\nabla )\mathrm{d}V-\underset{{\Omega}_{\square ,i}^{\mathrm{F}}}{\int}{p}^{\mathrm{S}}(\nabla \u2022\mathit{v})\mathrm{d}V+\underset{{\Omega}_{\square ,i}^{\mathrm{F}}}{\int}\nabla {p}^{\mathrm{M}}\bullet \mathit{v}\mathrm{d}V\right)\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}-\underset{{\mathrm{\u0490}}_{\mathrm{V}}^{\mathrm{F}}}{\int}{\widehat{v}}_{n}{p}^{\mathrm{M}}\mathrm{d}S\left(\right)close="\}">& \phantom{\rule{2em}{0ex}}\end{array}\n

(10)

### Computational homogenization

Up to this point, nothing has changed since the original problem except the formulation. To proceed, we now assume separation of scales, i.e. that the subscale feature has a length scale much smaller than that of the macroscale. Furthermore, we also make the assumption that *v* and *p*^{S} are periodic over, and continuous inside, each {\Omega}_{\square}^{\mathrm{F}}, thus replacing the condition on continuity over the boundaries {\mathrm{\u0490}}_{\square}^{\mathrm{F}}. As an intermediate step, we note that by removing continuity over {\mathrm{\u0490}}_{\square}^{\mathrm{F}}, reaction forces arise, which eventually will contribute to the subsequent macrohomogeneity condition. In [3] it is shown that periodic boundary conditions satisfy the aforementioned condition. In order to impose periodicity (either in weak or strong form), we start out by following along the lines of [3] and split the subscale boundary Ґ _{□} into two parts; {\mathrm{\u0490}}_{\square}={\mathrm{\u0490}}_{\square}^{+}\bigcup {\mathrm{\u0490}}_{-}^{\square} where the +/- sign is the sign of the normal to that part of the boundary^{c}. Furthermore, we introduce the jump operator

where *x* is a point on {\mathrm{\u0490}}_{\square}^{+} and *x*^{-}(*x*) is the corresponding point on the opposite side of the RVE. The conditions for periodicity are given as

where *t*^{S+} and *t*^{S-} are the subscale tractions along the edges {\mathrm{\u0490}}_{\square}^{+} and {\mathrm{\u0490}}_{\square}^{-} respectively. By imposing the periodicity constraints in a weak sense, i.e. introducing the Lagrange multiplier *β* for the constraint *[* *v* *]*=0 and *γ* for the constraint *[* *p*^{S}*]*=0 and allow the constraints to be fulfilled in average, rather than confining the respective solution spaces, we get

where

\begin{array}{ll}{\mathcal{V}}_{\square}& =\left\{\mathit{v}\u03f5{\left[{H}^{1}\left({\Omega}^{\mathrm{F}}\right)\right]}^{3}:\mathit{v}=0\phantom{\rule{2.77626pt}{0ex}}\text{on}\phantom{\rule{2.77626pt}{0ex}}{\mathrm{\u0490}}^{\text{int}},\phantom{\rule{2.77626pt}{0ex}}\mathit{v}=\phantom{\rule{2.77626pt}{0ex}}{\widehat{v}}_{n}\mathit{n}\phantom{\rule{2.77626pt}{0ex}}\text{on}\phantom{\rule{2.77626pt}{0ex}}{\mathrm{\u0490}}_{\mathrm{V}}^{\mathrm{F}}\right\}\phantom{\rule{2em}{0ex}}\end{array}

(11)

\begin{array}{ll}{\mathcal{P}}_{\square}^{\mathrm{S}}& =\left\{p\u03f5{H}^{1}\left({\Omega}_{\square}^{\mathrm{F}}\right)\right\}\phantom{\rule{2em}{0ex}}\end{array}

(12)

\begin{array}{ll}{\mathcal{P}}^{\mathrm{M}}& =\left\{p\u03f5{H}^{1}\left({\Omega}^{\mathrm{F}}\right):p=\widehat{p}\phantom{\rule{2.77626pt}{0ex}}\text{on}\phantom{\rule{2.77626pt}{0ex}}{\mathrm{\u0490}}_{\mathrm{P}}^{\mathrm{F}}\right\}\phantom{\rule{2em}{0ex}}\end{array}

(13)

\begin{array}{ll}{\mathcal{B}}_{\square}& =\left\{\mathit{\beta}\u03f5{\left[{L}_{2}\left({\mathrm{\u0490}}_{\square}^{\mathrm{F}}\right)\right]}^{3}\right\}\phantom{\rule{2em}{0ex}}\end{array}

(14)

\begin{array}{ll}{\mathcal{G}}_{\square}& =\left\{\gamma \u03f5{L}_{2}\left({\mathrm{\u0490}}_{\square}^{\mathrm{F}}\right)\right\}\phantom{\rule{2em}{0ex}}\end{array}

(15)

The Lagrange multiplier *β* can be interpreted as the traction needed to maintain periodicity on *v* and *γ* as the flux needed to maintain periodicity on *p*^{S}. The infimum on *γ* is further discussed in Remark 1. In order to allow for strong (essential) boundary conditions on {\mathrm{\u0490}}_{\mathrm{P}}^{\mathrm{F}} in the subsequent macroscale problem, the function space {\mathcal{P}}^{\mathrm{M}} is confined, replacing the former integral formulation of the condition.

#### Remark1.

In order to motivate the infimum on *γ*, consider the supremum of the term containing *p*^{S} (which already is a Lagrange multiplier) in Equation 8.

\begin{array}{l}\underset{{p}^{\mathrm{S}}\u03f5\underset{\square}{\overset{\mathrm{S}}{\mathcal{P}}}}{sup}\underset{i=1}{\overset{n}{\Sigma}}-\underset{\underset{\square ,i}{\overset{\mathrm{F}}{\Omega}}}{\int}{p}^{\mathrm{S}}\left(\nabla \bullet \mathit{v}\right)\mathrm{d}V\end{array}

(16)

which, when adding the constraint [*p*^{S}]=0 becomes, locally,

or in weak form

We now introduce the total energy potential П which is split into *n* RVE potentials {\u041f}_{i}^{\text{int}} which, in turn, can be expressed using the RVE mean potential {\u041f}_{\square ,i}=\frac{{\mathrm{\u041f}}_{i}^{\text{int}}}{\u2502{{\Omega}_{\square}}_{,i}\u2502} as

\begin{array}{ll}\u041f\left({p}^{\mathrm{M}},\mathit{v},{p}^{\mathrm{S}},\mathit{\beta},\gamma \right)& =\underset{i=1}{\overset{n}{\Sigma}}{\u041f}_{i}^{\text{int}}\left(\nabla {p}^{\mathrm{M}},\mathit{v},{p}^{\mathrm{S}},\mathit{\beta},\gamma \right)-{\u041f}^{\text{ext}}\left({p}^{\mathrm{M}}\right)\phantom{\rule{2em}{0ex}}\\ =\underset{i=1}{\overset{n}{\Sigma}}{\pi}_{\square ,i}\left(\nabla {p}^{\mathrm{M}},\mathit{v},{p}^{\mathrm{S}},\mathit{\beta},\gamma \right)\u2502{{\Omega}_{\square}}_{,i}\u2502-{\u041f}^{\text{ext}}\left({p}^{\mathrm{M}}\right)\phantom{\rule{2em}{0ex}}\end{array}

(17)

Here, the RVE potential is given as

and the external load as

\begin{array}{l}{\u041f}^{\text{ext}}\left(\phantom{\rule{0.3em}{0ex}}{p}^{\mathrm{M}}\right)=\underset{{\mathrm{\u0490}}_{\mathrm{V}}^{\mathrm{F}}}{\int}{\widehat{v}}_{n}{p}^{\mathrm{M}}\mathrm{d}S\end{array}

(18)

Furthermore, we note that by introducing separation of scales, i.e. for each coordinate \stackrel{-}{\mathit{x}}\u03f5\Omega there exist one RVE, thus, the RVE mean potential functions can be written as

\begin{array}{l}{\pi}_{\square}\left(\nabla {p}^{\mathrm{M}},\mathit{v},{p}^{\mathrm{S}},\mathit{\beta},\gamma ,\stackrel{-}{\mathit{x}}\right)={\pi}_{\square ,i}\left(\nabla {p}^{\mathrm{M}},\mathit{v},{p}^{\mathrm{S}},\mathit{\beta},\gamma \right)\end{array}

(19)

where *i* is the number of the RVE occupyed by coordinate \stackrel{-}{\mathit{x}}. Here, we define the RVE such that \stackrel{-}{\mathit{x}} is the centroid of Ω_{□}. By the assumption that the RVE is small compared to the macroscale, we identify │Ω_{□}_{,i}│ as a volume element on the macroscale and rewrite the sum in Equation 16 as an integral. It should be noted that the term │Ω_{□}_{,i}│ in the definition of the RVE mean potential is left unchanged during the transition from sum to integral as we are interested in the mean potential in the vicinity of \stackrel{-}{\mathit{x}}. We give the RVE mean potential *π*_{□} on explicit integral form as

We proceed by writing the total potential on compact form as

\begin{array}{l}\u041f\left({p}^{\mathrm{M}},\mathit{v},{p}^{\mathrm{S}},\mathit{\beta},\gamma \right)=\underset{\Omega}{\int}{\pi}_{\square}\left(\nabla {p}^{\mathrm{M}},\mathit{v},{p}^{\mathrm{S}},\mathit{\beta},\gamma ,\stackrel{-}{\mathit{x}}\right)\mathrm{d}V-{\u041f}^{\text{ext}}\left(\phantom{\rule{0.3em}{0ex}}{p}^{\mathrm{M}}\right)\end{array}

(20)

### Nested saddle-point formulation

We proceed by introducing the macroscale potential function ψ(*p*^{M}) as

\begin{array}{ll}\underset{\mathit{v}\u03f5{\mathcal{V}}_{\square}}{inf}\underset{\begin{array}{c}{p}^{\mathrm{M}}\u03f5{\mathcal{P}}^{\mathrm{M}}\\ {p}^{\mathrm{S}}\u03f5\underset{\square}{\overset{\mathrm{S}}{\mathcal{P}}}\\ \mathit{\beta}\u03f5{\mathcal{B}}_{\square}\end{array}}{sup}\underset{\gamma \u03f5{\mathcal{G}}_{\square}}{inf}& \u041f\left({p}^{\mathrm{M}},\mathit{v},{p}^{\mathrm{S}},\mathit{\beta},\gamma \right)\phantom{\rule{2em}{0ex}}\\ =\underset{{p}^{\mathrm{M}}\u03f5{\mathcal{P}}^{\mathrm{M}}}{sup}\underset{\mathit{v}\u03f5{\mathcal{V}}_{\square}}{inf}\underset{\begin{array}{c}{p}^{\mathrm{S}}\u03f5\underset{\square}{\overset{\mathrm{S}}{\mathcal{P}}}\\ \mathit{\beta}\u03f5{\mathcal{B}}_{\square}\end{array}}{sup}\underset{\gamma \u03f5{\mathcal{G}}_{\square}}{inf}\u041f\left({p}^{\mathrm{M}},\mathit{v},{p}^{\mathrm{S}},\mathit{\beta},\gamma \right)=\underset{{p}^{\mathrm{M}}\u03f5{\mathcal{P}}^{\mathrm{M}}}{sup}\psi \left\{{p}^{\mathrm{M}}\right\}\phantom{\rule{2em}{0ex}}\end{array}

(21)

where we refer to Appendix "Commutativity of inf and sup" for a proof on the commutativity of the inf and sup operators. We now introduce the macroscale pressure \stackrel{-}{p} and the macroscale pressure gradient \stackrel{-}{\mathit{g}} as

\begin{array}{ll}\stackrel{-}{p}\stackrel{\text{def}}{=}{\u3008p\u3009}_{\square}& \stackrel{-}{\mathit{g}}\stackrel{\text{def}}{=}\nabla \stackrel{-}{p}\end{array}

(22)

and assume 1st order homogenization, i.e. the macroscale pressure *p*^{M} varies linearily inside the RVE. Thus, we have

\begin{array}{l}{p}^{\mathrm{M}}=\stackrel{-}{p}\left(\stackrel{-}{\mathit{x}}\right)+\stackrel{-}{\mathit{g}}\left(\stackrel{-}{\mathit{x}}\right)\bullet \left[\mathit{x}-{\stackrel{-}{\mathit{x}}}^{\mathrm{F}}\right]\end{array}

(23)

where {\stackrel{-}{\mathit{x}}}^{\mathrm{F}} is the center of mass of {\Omega}_{\square}^{\mathrm{F}}. We can now express *ψ*{*p*^{M}} in terms of the macroscale pressure \stackrel{-}{p} as

\begin{array}{l}\psi \left\{\stackrel{-}{p}\right\}=\underset{\Omega}{\int}{\mathit{\psi}}_{\square}\left\{\stackrel{-}{\mathit{g}}\right\}\mathrm{d}\Omega -{\u041f}^{\text{ext}}\left(\phantom{\rule{0.3em}{0ex}}\stackrel{-}{p}\right)\end{array}

(24)

where we have introduced the local macroscale potential

\begin{array}{l}{\mathit{\psi}}_{\square}\left\{\stackrel{-}{\mathit{g}}\right\}:=\underset{\mathit{v}\u03f5{\mathcal{V}}_{\square}}{inf}\underset{\begin{array}{c}{p}^{\mathrm{S}}\u03f5\underset{\square}{\overset{\mathrm{S}}{\mathcal{P}}}\\ \mathit{\beta}\u03f5{\mathcal{B}}_{\square}\end{array}}{sup}\underset{\gamma \u03f5{\mathcal{G}}_{\square}}{inf}{\pi}_{\square}\left(\stackrel{-}{\mathit{g}},\mathit{v},{p}^{\mathrm{S}},\mathit{\beta},\gamma \right)\end{array}

(25)

### Weak form of the macroscale problem

Although this paper mainly focuses on the subscale problem, we choose to present the macroscale equation in order to achieve completeness. By taking the directional derivative of the the global macroscale potential, we produce

\begin{array}{l}{\u0471}_{\stackrel{-}{p}}^{\prime}=\underset{\Omega}{\int}\frac{\partial {\u0471}_{\square}}{\partial \stackrel{-}{g}}\bullet \delta \stackrel{-}{g}\mathrm{d}V-\underset{{\mathrm{\u0490}}_{\mathrm{V}}^{\mathrm{F}}}{\int}{\widehat{v}}_{n}\delta \stackrel{-}{p}\mathrm{d}S=0\end{array}

(26)

We now define the macroscale seepage as

\begin{array}{l}\stackrel{-}{\mathit{w}}=\frac{\partial {\u0471}_{\square}}{\partial \stackrel{-}{\mathit{g}}}=\frac{1}{\u2502{\Omega}_{\square}\u2502}\underset{{\Omega}_{\square}^{\mathrm{F}}}{\int}\mathit{v}\mathrm{d}V=\varphi {\u3008\mathit{v}\u3009}_{\square}\end{array}

(27)

where *ϕ* is the porosity defined as \u2502{\Omega}_{\square}^{\mathrm{F}}\u2502/\u2502{\Omega}_{\square}\u2502 and 〈•〉_{□} is the intrinsic averaging operator. We now recognize the weak form of the macroscale problem as that of finding all \stackrel{-}{p}\u03f5{\mathcal{P}}^{\mathrm{M}} such that

\begin{array}{ll}\underset{\Omega}{\int}\stackrel{-}{\mathit{w}}\bullet \delta \stackrel{-}{\mathit{g}}\mathrm{d}V-\underset{{\mathrm{\u0490}}_{\mathrm{V}}^{\mathrm{F}}}{\int}{\widehat{v}}_{n}\delta \stackrel{-}{p}\mathrm{d}S=0& \forall \delta \stackrel{-}{p}\u03f5{\mathcal{P}}^{\text{M,0}}\end{array}

(28)

where *P*^{M} and *P*^{M,0} are the trial and test spaces respectively; now pertaining to the macroscale pressure \stackrel{-}{p}.

### The RVE problem

The local (subscale) problem for a given macroscale pressure gradient \stackrel{-}{g} is produced by seeking the stationary point for variations of subscale quantities in Equation 20. The problem is stated as: Find \left(\mathit{v},{p}^{\mathrm{S}},\mathit{\beta},\gamma \right)\u03f5{\mathcal{V}}_{\square}\times {\mathcal{P}}_{\square}^{\mathrm{S}}\times {\mathcal{\beta}}_{\square}\times {\mathcal{G}}_{\square} such that

\begin{array}{lllll}{a}_{\square}\left(\mathit{v};\delta \mathit{v}\right)& -{b}_{\square}\left(\delta \mathit{v},{p}^{\mathrm{S}}\right)\phantom{\rule{2em}{0ex}}& -{c}_{\square}\left(\delta \mathit{v},\mathit{\beta}\right)& =& -{e}_{\square}\left(\delta \mathit{v},\stackrel{-}{\mathit{g}}\right)\phantom{\rule{2em}{0ex}}\end{array}

(29)

\begin{array}{llll}-{b}_{\square}\left(\mathit{v},\delta {p}^{\mathrm{S}}\right)& -{d}_{\square}\left(\delta {p}^{\mathrm{S}},\gamma \right)\phantom{\rule{2em}{0ex}}& =& \phantom{\rule{0.3em}{0ex}}0\phantom{\rule{2em}{0ex}}\end{array}

(30)

\begin{array}{lll}-{c}_{\square}\left(\mathit{v},\delta \mathit{\beta}\right)& =& \phantom{\rule{0.3em}{0ex}}0\phantom{\rule{2em}{0ex}}\end{array}

(31)

\begin{array}{lll}-{d}_{\square}\left({p}^{\mathrm{S}},\mathrm{\delta \gamma}\right)\phantom{\rule{2em}{0ex}}& =& \phantom{\rule{0.3em}{0ex}}0\phantom{\rule{2em}{0ex}}\end{array}

(32)

for all \delta \mathit{v}\u03f5{\mathcal{V}}_{\square}, \delta {p}^{\text{S}}\u03f5{\mathcal{P}}_{\square}^{\mathrm{S}}, \delta \mathit{\beta}\u03f5{\mathcal{\beta}}_{\square} and \mathrm{\delta \gamma}\u03f5{\mathcal{G}}_{\square}, where

### Homogenization of velocity and the macroscale tangent

From the definition of seepage in Equation 28, we produce the possibly non-linear relation between the seepage and the macroscale pressure gradient by differentiation as

\begin{array}{l}\stackrel{-}{\mathit{w}}=\stackrel{-}{\mathit{w}}\left\{\stackrel{-}{\mathit{g}}\right\}\Rightarrow \mathrm{d}\stackrel{-}{\mathit{w}}=\frac{\mathrm{d}\stackrel{-}{\mathit{w}}}{\mathrm{d}\stackrel{-}{\mathit{g}}}\bullet \mathrm{d}\stackrel{-}{\mathit{g}}=-\stackrel{-}{\mathit{K}}\left\{\stackrel{-}{\mathit{g}}\right\}\bullet \mathrm{d}\stackrel{-}{\mathit{g}}\end{array}

(33)

#### Remark 2.

Note that the minus sign on the positive definite permeability tensor \stackrel{-}{\mathit{K}} in Equation 32 is to ensure positive dissipation due to drag interaction between the solid and fluid phases.

From Equation 30, we see that the unit sensitivity field is given as

\begin{array}{lllll}{a}_{\prime}^{\square}\left(\mathit{v};\delta \mathit{v},\mathrm{d}\mathit{v}\right)& -{b}_{\square}\left(\delta \mathit{v},\mathrm{d}{p}^{\mathrm{S}}\right)\phantom{\rule{2em}{0ex}}& -{c}_{\square}\left(\delta \mathit{v},\mathrm{d}\mathit{\beta}\right)& =& -{e}_{\square}\left(\delta \mathit{v},{\mathit{e}}_{i}\right)\phantom{\rule{2em}{0ex}}\end{array}

(34)

\begin{array}{llll}-{b}_{\square}\left(\mathrm{d}\mathit{v},\delta {p}^{\text{S}}\right)& -{d}_{\square}\left(\delta {p}^{\text{S}},\mathrm{d}\gamma \right)\phantom{\rule{2em}{0ex}}& =& 0\phantom{\rule{2em}{0ex}}\end{array}

(35)

\begin{array}{lll}-{c}_{\square}\left(\mathrm{d}\mathit{v},\delta \mathit{\beta}\right)& =& 0\phantom{\rule{2em}{0ex}}\end{array}

(36)

\begin{array}{lll}-{d}_{\square}\left(\mathrm{d}{p}^{\mathrm{S}},\mathrm{\delta \gamma}\right)\phantom{\rule{2em}{0ex}}& =& 0\phantom{\rule{2em}{0ex}}\end{array}

(37)

for all \delta \mathit{v}\u03f5{\mathcal{V}}_{\square}, \delta {p}^{\text{S}}\u03f5{\mathcal{P}}_{\square}^{\mathrm{S}}, \delta \mathit{\beta}\u03f5{\mathcal{\beta}}_{\square} and \mathrm{\delta \gamma}\u03f5{\mathcal{G}}_{\square} where *a*^{′} is the directional derivative of *a*. Following [3], we can express an arbitrary unit pressure gradient as

\begin{array}{l}\mathrm{d}\stackrel{-}{\mathit{g}}=\underset{i=1}{\overset{{n}_{\text{dim}}}{\Sigma}}{\mathit{e}}_{i}\left[{\mathit{e}}_{i}\u2022\mathrm{d}\widehat{\mathit{g}}\right]\end{array}

(38)

From here, we make an ansatz of the response d*v* as

\begin{array}{l}\mathrm{d}\mathit{v}=\underset{i=1}{\overset{{n}_{\text{dim}}}{\Sigma}}{\mathit{v}}^{\left(i\right)}\left[{\mathit{e}}_{i}\u2022\mathrm{d}\widehat{\mathit{g}}\right]=\left(\underset{i=1}{\overset{{n}_{\text{dim}}}{\Sigma}}{\mathit{v}}^{\left(i\right)}\u2a02{\mathit{e}}_{i}\right)\u2022\mathrm{d}\widehat{\mathit{g}}\end{array}

(39)

Using the definition of seepage in Equation 28 on the above equation, we produce the relation between seepage and pressure gradient perturbations as

\begin{array}{ll}\mathrm{d}\stackrel{-}{\mathit{w}}=& \mathit{\phi}{\u3008\mathrm{d}\mathit{v}\u3009}_{\square}=\mathit{\phi}{\u3008\left(\underset{i=1}{\overset{{n}_{\text{dim}}}{\Sigma}}{\mathit{v}}^{\left(i\right)}\u2a02{\mathit{e}}_{i}\right)\bullet \mathrm{d}\stackrel{-}{\mathit{g}}\u3009}_{\square}\phantom{\rule{2em}{0ex}}\\ =& \varphi \left(\underset{i=1}{\overset{{n}_{\text{dim}}}{\Sigma}}{\u3008{\mathit{v}}^{\left(i\right)}\u3009}_{\square}\u2a02{\mathit{e}}_{i}\right)\bullet \mathrm{d}\stackrel{-}{\mathit{g}}=-\stackrel{-}{\mathit{K}}\u2022\mathrm{d}\stackrel{-}{\mathit{g}}\phantom{\rule{2em}{0ex}}\end{array}

(40)

whereby the macroscale tangent is identified.

### Bounds on effective properties for strong periodicity

According to Equation 26, an upper bound is produced by confining the function spaces {\mathcal{V}}_{\square} and {\mathcal{G}}_{\square}. Furthermore, by choosing the function space {\mathcal{V}}_{\square} in such a way that periodicity is always fulfilled, the supremum on *β* is void, producing the following inequality

\begin{array}{ll}{\psi}_{\square}\left(\stackrel{-}{\mathit{g}}\right)& =\underset{\mathit{v}\u03f5{\mathcal{V}}_{\square}}{inf}\underset{\begin{array}{c}{p}^{\mathrm{S}}\u03f5\underset{\square}{\overset{\mathrm{S}}{\mathcal{P}}}\\ \mathit{\beta}\u03f5{\mathcal{B}}_{\square}\end{array}}{sup}\underset{\gamma \square {\mathcal{G}}_{\square}}{inf}{\pi}_{\square}\left(\stackrel{-}{\mathit{g}},\mathit{v},{p}^{\mathrm{S}},\mathit{\beta},\gamma \right)\phantom{\rule{2em}{0ex}}\\ \le \underset{\mathit{v}\u03f5\underset{\square}{\overset{\prime}{\mathcal{V}}}}{inf}\underset{\begin{array}{c}{p}^{\mathrm{S}}\u03f5\underset{\square}{\overset{\mathrm{S}}{\mathcal{P}}}\end{array}}{sup}\underset{\gamma \u03f5\underset{\prime}{\overset{\square}{\mathcal{G}}}}{inf}{\pi}_{\square}\left(\stackrel{-}{\mathit{g}},\mathit{v},{p}^{\mathrm{S}},0,\gamma \right)=\underset{\square}{\overset{\text{fU}}{\psi}}\left(\stackrel{-}{\mathit{g}}\right)\phantom{\rule{2em}{0ex}}\end{array}

(41)

Equivalently, a lower bound is produced by confining the spaces {\mathcal{P}}_{\square}^{\mathrm{S}} and {\mathcal{\beta}}_{\square}

\begin{array}{ll}{\psi}_{\square}\left(\stackrel{-}{\mathit{g}}\right)& =\underset{\mathit{v}\u03f5{\mathcal{V}}_{\square}}{inf}\underset{\begin{array}{c}{p}^{\mathrm{S}}\u03f5\underset{\square}{\overset{\mathrm{S}}{\mathcal{P}}}\\ \mathit{\beta}\u03f5{\mathcal{B}}_{\square}\end{array}}{sup}\underset{\gamma \u03f5{\mathcal{G}}_{\square}}{inf}{\pi}_{\square}\left(\stackrel{-}{\mathit{g}},\mathit{v},{p}^{\mathrm{S}},\mathit{\beta},\gamma \right)\phantom{\rule{2em}{0ex}}\\ \ge \underset{\mathit{v}\u03f5{\mathcal{V}}_{\square}}{inf}\underset{\begin{array}{c}{p}^{\mathrm{S}}\u03f5{\underset{\square}{\overset{\mathrm{S}}{\mathcal{P}}}}^{\prime}\\ \mathit{\beta}\u03f5{{\mathcal{B}}_{\square}}^{\prime}\end{array}}{sup}{\pi}_{\square}\left(\stackrel{-}{\mathit{g}},\mathit{v},{p}^{\mathrm{S}},\mathit{\beta},0\right)=\underset{\square}{\overset{\mathrm{L}}{\psi}}\left(\stackrel{-}{\mathit{g}}\right)\phantom{\rule{2em}{0ex}}\end{array}

(42)

By combining Equations 37 and 38, we get

\begin{array}{l}{\psi}_{\square}^{\mathrm{L}}\left(\stackrel{-}{\mathit{g}}\right)\le {\psi}_{\square}\left(\stackrel{-}{\mathit{g}}\right)\le {\psi}_{\square}^{\mathrm{U}}\left(\stackrel{-}{\mathit{g}}\right)\end{array}

(43)

We shall now consider the special case of linear flow, defined by \u0424(\mathit{v}\u2a02\nabla )=\frac{\mu}{2}{[\mathit{v}\u2a02\nabla ]}^{\text{sym}}:{[\mathit{v}\u2a02\nabla ]}^{\text{sym}}. Assuming that *v*, *p*^{S}, *β* and *γ* satisfies Equation 30 for some \stackrel{-}{\mathit{g}}, {\psi}_{\square}\left(\phantom{\rule{0.3em}{0ex}}\stackrel{-}{\mathit{g}}\right) is rendered stationary. Thus, the stationarity condition for Equation 30 is

\begin{array}{ll}{a}_{\square}\left(\mathit{v};\delta \mathit{v}\right)=-{e}_{\square}\left(\delta \mathit{v},\stackrel{-}{\mathit{g}}\right)& \forall \delta \mathit{v}\u03f5{\mathcal{V}}_{\square}\end{array}

(44)

In the case of a linear flow, choosing *δ* *v*=*v* in the stationarity condition, Equation 40 is given as

\begin{array}{l}\underset{{\Omega}_{\square}^{\mathrm{F}}}{\int}\mu {\left[\mathit{v}\u2a02\nabla \right]}^{\text{sym}}:\phantom{\rule{0.3em}{0ex}}{[\mathit{v}\u2a02\nabla ]}^{\text{sym}}\mathrm{d}V=-\underset{{\Omega}_{\square}^{\mathrm{F}}}{\int}\mathit{v}\mathrm{d}V\bullet \nabla \stackrel{-}{p}\end{array}

(45)

Inserting the stationary point into *π*_{□} and using 41, we see that the RVE mean potential is given as

\begin{array}{ll}\frac{1}{\u2502{\Omega}_{\square}^{\mathrm{F}}\u2502}\underset{{\Omega}_{\square}^{\mathrm{F}}}{\int}& \frac{\mu}{2}{[\mathit{v}\u2a02\nabla ]}^{\text{sym}}:\phantom{\rule{0.3em}{0ex}}{[\mathit{v}\u2a02\nabla ]}^{\text{sym}}\mathrm{d}V\phantom{\rule{2em}{0ex}}\\ +\frac{1}{\u2502{\Omega}_{\square}^{\mathrm{F}}\u2502}\underset{{\Omega}_{\square}^{\mathrm{F}}}{\int}\stackrel{-}{\mathit{g}}\bullet \mathit{v}\mathrm{d}V=\frac{1}{2}\stackrel{-}{\mathit{w}}\bullet \stackrel{-}{\mathit{g}}=-\frac{1}{2}\stackrel{-}{\mathit{g}}\bullet \stackrel{-}{\mathit{K}}\bullet \stackrel{-}{\mathit{g}}\phantom{\rule{2em}{0ex}}\end{array}

(46)

Thus, by bounding *ѱ*_{□}, we have also bounded \stackrel{-}{\mathit{K}}. More specifically, we may represent Equation 39, in terms of the permeability tensor as

\begin{array}{l}\stackrel{-}{\mathit{g}}\bullet {\stackrel{-}{\mathit{K}}}^{\mathrm{L}}\bullet \stackrel{-}{\mathit{g}}\le \stackrel{-}{\mathit{g}}\u2022\stackrel{-}{\mathit{K}}\le \stackrel{-}{\mathit{g}}\le \stackrel{-}{\mathit{g}}\u2022{\stackrel{-}{\mathit{K}}}^{\mathrm{U}}\u2022\stackrel{-}{\mathit{g}}\end{array}

(47)

where

\begin{array}{l}{\psi}_{\square}^{\mathrm{L}}\left(\stackrel{-}{\mathit{g}}\right)=-\frac{1}{2}\stackrel{-}{\mathit{g}}\bullet {\stackrel{-}{\mathit{K}}}^{\mathrm{U}}\bullet \stackrel{-}{\mathit{g}}\end{array}

(48)

\begin{array}{l}{\psi}_{\square}\left(\stackrel{-}{\mathit{g}}\right)=-\frac{1}{2}\stackrel{-}{\mathit{g}}\bullet \stackrel{-}{\mathit{K}}\bullet \stackrel{-}{\mathit{g}}\end{array}

(49)

\begin{array}{l}{\mathit{\psi}}_{\square}^{\mathrm{U}}\left(\stackrel{-}{\mathit{g}}\right)=-\frac{1}{2}\stackrel{-}{\mathit{g}}\bullet {\stackrel{-}{\mathit{K}}}^{\mathrm{L}}\bullet \stackrel{-}{\mathit{g}}\end{array}

(50)

### Discretization of solutions spaces on the RVE boundary

As to the specific choice of solution spaces for the Lagrange multipliers we note that which is the most efficient depends on both the discretization and the geometry of the subscale domain. One example of a feasible discretization of the Lagrange multipliers is the one presented in [20] where the pertinent unknown functions are discretized on a mesh consisting of the union of all nodes on opposite sides of the domain. Here, however, we choose to discretize the Lagrange multipliers *β* and *γ* as global polynomials, i.e.

\begin{array}{l}{\mathcal{B}}_{\square}=\left\{\mathit{\beta}\u03f5{\mathbf{\text{R}}}^{2}:\mathit{\beta}=\underset{i=0}{\overset{{n}_{p}}{\Sigma}}{\mathit{b}}_{i}\frac{{s}^{i}}{{l}_{\square}}\right\},\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}{\mathcal{G}}_{\square}=\left\{\gamma \u03f5\mathbf{\text{R}}:\gamma =\underset{i=0}{\overset{{n}_{p}}{\Sigma}}{g}_{i}\frac{{s}^{i}}{{l}_{\square}}\right\}\end{array}

(51)

where *n*_{
p
} are the polynomial order in the respective approximation, *s* is a parameterized coordinate along {\mathrm{\u0490}}_{\square}^{+}, *b*_{
i
} and *g*_{
i
} are the respective coefficients and *l*_{□} is the side length of the RVE.

For an upper bound of the energy, we choose {\mathcal{V}}_{\square} such that the velocity is always periodic, removing the supremum on *β*.

where *𝜙*_{
i
} are basis functions for the *N*_{
v
} velocity degrees of freedom *a*_{
i
}. It should be noted that, if *𝜙*_{
i
} is represented in polynomial base, the constraint *[* *𝜙*_{
i
}*]*=0 requires approximations of order higher than 1 in the case where obstacles cross the boundary of the RVE. The reason for this is simply that the no slip condition on the obstacle surface implies zero velocity on the RVE boundary if the velocity approximation is constant or linear. For the same reason, the velocity approximation is applied patchwise between obstacles along the boundary. In practice, we use global quadratic 1D element along the boundary as shown in the example in Figure 2 and make all nodes along the boundary hang on the global element. Furthermore, we connect all nodes located on a corner, i.e. *N*_{1} is a master and *W*_{1}, *W*_{6}, *S*_{1}, *S*_{2}, *E*_{1} and *E*_{6} its slaves. Finally, we connect opposite sides, i.e *W*_{2} is a slave to *E*_{2}, *S*_{2} to *N*_{2} etc.

For a lower bound on the energy, we choose {\mathcal{P}}_{\square}^{\mathrm{S}} as

\begin{array}{l}{{\mathcal{P}}_{\square}^{\mathrm{S}}}^{\prime}=\left\{{p}^{\mathrm{S}}\u03f5{\mathcal{P}}_{\square}^{\mathrm{S}}:\phantom{\rule{2.77626pt}{0ex}}{p}^{\mathrm{S}}=0\phantom{\rule{2.77626pt}{0ex}}\text{on}\phantom{\rule{2.77626pt}{0ex}}{\mathrm{\u0490}}_{\square}^{\mathrm{F}}\right\}\subset {\mathcal{P}}_{\square}^{\mathrm{S}}\end{array}

(52)

Thus, *p*^{S} is trivially periodic.