Parallel eciency of monolithic and xed-strain solution strategies for poroelasticity problems Denis Anuprienko

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Parallel eﬃciency of monolithic and ﬁxed-strain

solution strategies for poroelasticity problems

Denis Anuprienko

Nuclear Safety Institute RAS, Moscow 115191, Russian Federation

anuprienko@ibrae.ac.ru

Abstract. Poroelasticity is an example of coupled processes which are

crucial for many applications including safety assessment of radioactive

waste repositories. Numerical solution of poroelasticity problems dis-

cretized with ﬁnite volume – virtual element scheme leads to systems of

algebraic equations, which may be solved simultaneously or iteratively.

In this work, parallel scalability of the monolithic strategy and of the

ﬁxed-strain splitting strategy is examined, which depends mostly on lin-

ear solver performance. It was expected that splitting strategy would

show better scalability due to better performance of a black-box linear

solver on systems with simpler structure. However, this is not always the

case.

Keywords: poroelasticity, multiphysics, splitting, linear solvers, paral-

lel eﬃciency

1 Poroelasticity problem

Modeling of coupled physical processes is important in many engineering ap-

plications, such as safety assessment of radioactive waste repositories. It is ac-

knowledged that complex thermo-hydro-mechanical-chemical (THMC) processes

should be taken into account in such modeling [1]. Software package GeRa (Ge-

omigration of Radionuclides) [2][3] which is developed by INM RAS and Nuclear

Safety Institute RAS already has some coupled modeling capabilities [4][5] and

is now moving toward hydromechanical processes. Poroelasticity is the simplest

example of such processes.

Numerical solution of coupled problems is a computationally expensive task.

Arising discrete systems require eﬃcient solution strategies, and parallel com-

putations are a necessity. In this work, two solution strategies are tested in their

scalability when a black-box linear solver is used.

1.1 Mathematical formulation

This work is restricted to the simplest case of elastic media ﬁlled with water only.

Following theory introduced by Biot [6], the following equations are considered:

sstor

∂h

∂t +∇ · q+α∇ · ∂u

∂t =Q, (1)

arXiv:2210.06206v2 [math.NA] 18 Oct 2022

2 D. Anuprienko

∇ · (σ−αP I) = f.(2)

Equation (1) represents water mass conservation taking into account porous

medium deformation. Equation (2) represents mechanical equilibrium in porous

medium in presence of water pressure and external forces. Here his water head,

sstor is the speciﬁc storage coeﬃcient, Qis speciﬁc sink and source term, σis

the stress tensor, fis the external force vector, water pressure Pis related to

water head has P=ρg (h−z); αis the Biot coeﬃcient, which is equal to 1 in

this work.

The following constitutive relationships complete the equations: Darcy law

q=−K∇h(3)

and generalized Hooke’s law:

σ=Cε=C∇u+ (∇u)T

2.(4)

Here qis the water ﬂux, Kis the hydraulic conductivity tensor, a 3×3 s.p.d.

matrix, Cis the stiﬀness tensor, εis the strain tensor and uis the displacement

vector.

Water head hand solid displacement uare the primary variables.

The system is closed with initial and boundary conditions. The following

boundary conditions are available:

–speciﬁed head hor normal ﬂux q·nfor ﬂow;

–speciﬁed displacement u, traction σ·nor roller boundary condition with

zero normal displacement for mechanics.

1.2 Discretization

Subsurface ﬂow modeling is a well-established technology in GeRa and uses the

ﬁnite volume method (FVM). Choice of the discretization method for elasticity

was guided by following criteria: (a) applicability on general grids, (b) ability

to work with arbitrary tensor C, (c) suﬃcient history of application in mul-

tiphysics. Criterion (a) makes use of traditional ﬁnite element method (FEM)

problematic, since meshes for subsurface domains can contain cells which are

general polyhedra. While FVM for geomechanics exists [7][8] and is applied in

poroelastic case [10] and more complex ones [9], it is still somewhat new and

not so widely used option. Recent developments include FVM scheme achieving

improved robustness by avoiding decoupling into subproblems and introducing

stable approximation of vector ﬂuxes [11].

For discretization of elasticity equations in GeRa the virtual element method

(VEM) [12] was ultimately chosen. VEM, applied to elasticity equation [13],

can handle cells which are non-convex and degenerate, its simplest version uses

only nodal unknowns and is similar to FEM with piece-wise linear functions. An

important feature of VEM is existence of FVM-VEM scheme for poroelasticity

Parallel eﬃciency for poroelasticity 3

with proved properties [14]. A drawback of this scheme is the use of simplest

FVM option, the linear two-point ﬂux approximation (TPFA) which is incon-

sistent in general case. In this work, TPFA is replaced with a multi-point ﬂux

approximation, MPFA-O scheme [15]. This scheme gives reasonable solutions on

a wider class of grids, capturing media anisotropy, but is not monotone which is

important for more complex physical processes.

Discretization in time uses ﬁrst-order backward Euler scheme, which results

in a system of linear equations at each time step.

2 Solution strategies for discrete systems

The system of discrete equations has the form

AFAF M

AMF AM·h

u=bF

bM,(5)

where subscripts Fand Mdenote parts related to ﬂow and mechanics subprob-

lems, respectively. Here hand udenote vectors of discrete unknowns on a given

time step. The system matrix has block form with square block AFrepresenting

FVM discretization of equation (1), square block AMrepresenting VEM dis-

cretization of equation (2) and oﬀ-diagonal blocks AF M and AMF representing

coupling terms discretized with VEM (example of matrix Ais depicted at ﬁg-

ure 1). Right-hand side terms bFand bMcontain contributions from boundary

conditions, source and force terms and previous time step values.

Fig. 1. Matrix pattern for a 4×4×4 cubic grid

In multiphysics applications, diﬀerent approaches to solution of coupled prob-

lems exist.

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Paralleleciencyofmonolithicandxed-strainsolutionstrategiesforporoelasticityproblemsDenisAnuprienkoNuclearSafetyInstituteRAS,Moscow115191,RussianFederationanuprienko@ibrae.ac.ruAbstract.Poroelasticityisanexampleofcoupledprocesseswhicharecrucialformanyapplicationsincludingsafetyassessmentofradioacti...

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Parallel eciency of monolithic and xed-strain solution strategies for poroelasticity problems Denis Anuprienko

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