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From basic science to translation, modern biomedical research demands computational models which integrate several interacting physical systems. This paper describes the infrastructural framework for a generic multi- physics integration implemented in the software CHeart, a finite-element code for biomedical research. To generalize the coupling of physics systems, we introduce a framework in which the geometric and operator relationships between the constituent systems are rigorously defined. We then introduce the notion of topological interfaces and define specific operators encompassing many common model coupling requirements. These interfaces enable the evaluation of weak form integrals between mesh subregions of arbitrary finite element bases orders, types, and spatial dimensions. Equation maps are introduced which provide abstract representations of the individual physics systems that can be automatically combined to permit a monolithic matrix assembly. Flexible solution strategies for the resulting coupled systems are implemented, permitting fine–tuning of solution updates during fixed point iterations, and sub-grouping where several problems are being solved together. Partitioning of coupled mesh domains for optimal load balancing is also supported, taking into account the per-processor cost of the entire coupled problem within the graph problem. The demonstration of the performance is illustrated through important real-world multi-physics problems relevant to cardiac physiology.
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2016 Society for Industrial and Applied Mathematics
Vol . 38 , N o . 3, p p . C1 5 0– C1 7 8
Abstract. Fr om ba s i c s c ien c e t o tran s l a t ion , mo de r n b i ome d i c al res e a r ch d e m a nds co m p uta-
tional models which integrate several interacting physical systems. This paper describes the infra-
structural framework for generic multiphysics integration implemented in the software CHeart,a
finite-element code for biomedical research. To generalize the coupling of physics systems, we intro-
duce a framework in which the geometric and operator relationships between the constituent systems
are rigorously defined. We then introduce the notion of topological interfaces and define specific
operators encompassing many common model coupling requirements. These interfaces enable the
evaluation of weak form integrals between mesh subregions of arbitrary finite-element bases’ orders,
type s, and spatial dimensions. Equation maps are intro duced wh ich pr ovide abstra ct representations
of the individual physics systems that can be automatically combined to permit a monolithic matrix
assembly. Flexible solution strategies for the resulting coupled systems are implemented, permitting
fine-tuning of solution updates during fixed point iterations, and subgrouping where several prob-
lems are being solved together. Partitioning of coupled mesh domains for optimal load balancing is
also supported, taking into account the per-processor cost of the entire coupled problem within the
graph problem. The demonstration of the p erformance is illustrated through important real-world
multiphysics problem s relevant to cardiac physi ology.
Key words. cardiac modeling, multiphysics, parallel computing, coupled problems, modeling
AMS subject classifications. 92-04, 92-08, 74F10, 74S05, 76M10, 68W10, 92C30
DOI. 10.1137/15M1014097
1. Introduction. Computational modeling of biophysical phenomena has be-
come a common approach for investigating problems that arise in medical sciences.
From subcellular processes [22, 86] to whole-organ function [65, 81], mathematical
models continue to demonstrate their capacity to predict physiology, provide added
insight, and address questions of both scientific and clinical interest. The demand
for solving computational models has led to the development of a variety of soft-
ware packages [82, 83, 43, 74, 59, 24, 52, 73, 47, 71, 10, 75]. The success of many
of these tools stems from their ability to oer ecient and scalable computation
for specific physics problems. These packages widely employ finite-element methods
Submitted to the journal’s Software and High-Performance Computing section March 27, 2015;
accepted for publication (in revised form) March 28, 2016; published electronically May 26, 2016.
This work was supported by the British Heart Foundation (NH/11/5/29058), the Engineering and
Physical Sciences Research Council (EP/G0075727/2), and the Europ ean Commission funded eu-
Heart project (FP7-ICT-2007-224495:euHeart). It was also supported by the Wellcome Trust-EPSRC
Centre of Excellence in Medical Engineering (WT 088641/Z/09/Z) and the NIHR Biomedical Re-
search Centre at Guy’s and St.Thomas’ NHS Foundation Trust and KCL. The views expressed are
those of the authors and not necessarily those of the NHS, the NIHR, or the DoH.
http: // s/s isc/38-3/M101409.html
Department of Biomedical Engineering, King’s College London, London, UK (,,,,,
Mathematics Institute of Computational Science and Engineering, EPFL, Lausanne, Switzerland
§Department of Biomedical Engineering, King’s College London, London, UK, and Faculty of
Engineering, University of Auckland, Auckland, New Zealand (
Corresponding author. Department of Biomedical Engineering, King’s College London, London,
UK (
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Fig. 1.1.An example of finite-eleme nt meshes used for mult iphysics simulations. Illustrated
is the coupling between overlapping domains (hexahedral (1) and tetrahedral meshes (5, 6) for solid
mechanics, monodomain, and poroelasticity problems) and over partially overlapping domains (en-
docardial (2) and cavity mesh (3) for fluid-structure interaction (FSI) problems and vascular mesh
(4) for the coronary flow model).
(FEMs) for numerical approximation of dierential equations and have introduced a
range of techniques that span the computational spectrum. From support for low
order h-refined techniques [82, 83], to high order p-FEMs [43], to varied mixed meth-
ods [24, 10, 33] and h-adaptive techniques [73, 52], diverse approaches for problem
solving have been enabled. A number of software packages also utilize mesh parti-
tioning (such as ParMETIS [44], Scotch [72], or Zoltan [19]) and parallel linear solver
libraries (such as Trilinos [34] and PETSc [5, 4, 6]) for ecient multicore paralleliza-
tion. These strategies for scability are necessary to address the computational chal-
lenges presented by many mathematical models (e.g., hemodynamics [74, 43, 52, 24]
and electrophysiology [59, 82, 83, 73]).
Despite these advances, the seemingly simple task of multiphysics modeling by in-
tegrating existing computational models remains challenging. For a general coupling
between arbitrary problems, many aspects of the individual problem—including nu-
merical discretization and solution schemes, disparate solution domains, and data ac-
cessibility issues—have to be formalized, presenting new implementational challenges.
For instance, in the context of cardiac modeling a variety of numerical techniques are
employed (see Figure 1.1), spanning from the one dimensional (1D) network mani-
folds used in vascular flow modeling [79], to curvilinear high order elements used in
mechanics [61], to extremely refined tetrahedral meshes used in electrophysiology [62].
The resulting equations also require the ability to solve large scale linear systems as
well as highly nonlinear systems. The varied nature of these physical models leads to
many dierent forms of coupling, where integrated models may require equating kine-
matics/force (as found in fluid-solid interaction), mass transport/force (as found in
poromechanics or multilevel transport), or strain/tension (as found in electromechan-
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ics) between models on heterogeneous domains. The permutation of these factors
has made the generalized coupling of multiphysics systems a significant challenge to
the scientific community. While the integration of specific physical systems has been
explored in the literature [24, 52, 33], there is still much that can be gained from
a focused examination of the fundamental design that consolidates a multiphysics
In this paper we describe an infrastructural framework for generalizing the integra-
tion of physical phenomena within CHeart, a multiphysics software engine developed
at King’s College London. To date, it has been used to solve a variety of problems in
physiological modeling, including the Navier–Stokes equations for fluid flow [17], non-
linear solid and porous mechanics [13, 32, 31, 3, 2], scalar transport equations [60, 41],
elastic wave mechanics [49], pressure Poisson for approximating pressure from Navier–
Stokes using known velocities [48, 20], and electrophysiology [84]. These essential
single-physics building blocks provide the components for assembly and linking to
form multiphysics models. This is achieved through a software infrastructure using
what we call object-sets that act as generic sets to store dierent core FEM objects
that may be constructed, used, or shared by any multiphysics problem. This strat-
egy is used to manage bases, meshes and variables, topological interfaces, equation
abstraction through equation mappings, and a series of general physics couplings to
flexibly interlink heterogeneous physical models. These processes enable partitioned
coupling, monolithic coupling, or a mixed approach for dierent components of the
physical systems allowing for a multitude of possible solution strategies. The infra-
structural organization outlined has enabled the interlinking of a variety of models,
including fluid-solid coupling [67, 68, 70, 56, 18, 57, 58], vascular-porous flow [50, 51],
and electromechanics [84].
The remainder of the paper will describe a general strategy for multiphysics inte-
gration and demonstrate its application. In section 2 we outline the physics systems
which provide core simulation components for multiphysics integration. The general
object-set approach, construction of function and variable spaces as well as the intro-
duction of topological interfaces, is provided in section 3. A series of single-physics
coupling strategies is described in section 4. The generic construction and handling
of these dierent systems is introduced in section 5 with particular emphasis on their
abstraction through the use of equation mappings. In section 6, we propose a graph
parallelization strategy which enables ecient multicore processing. These techniques
are then validated and demonstrated on a series of multiphysics systems relevant to
cardiac physiology (see theonlinesupplement (M101409 01.pdf [local/web 4.23MB])),
with a specific example for fluid-structure interaction (FSI) provided in section 8.2.
2. Core single-physics module. In the finite-element framework the govern-
ing equations for physical systems are provided in standard weak form. In this section
we introduce a generic notation used for defining operators, or functionals, for dif-
ferent physical systems. These functionals provide the building blocks for integrated
multiphysics systems.
Dierent physical systems and their weak form functionals vary in the terms,
function spaces, number of variables, and their inclusion of dierent fields. Here we
consider each problem as a specialization of a general weak functional, f:Vh(h)×
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Tab l e 2 . 1
Summary of available options for dierent physics systems and physics-based operations.
Physics-based operators & operations
ODE/DAE systems Adaptive time stepping, implicit/explicit schemes (Euler, RK4, etc.)
Lumped parameter cardiovascular models (incl. [77, 85])
Var i ou s c e l l mo d e l s (i n c l. t e n Tu ss che r 2 0 06 [8 0 ] , Lu o Ru d y 1 9 91 [5 3 ] )
Dynamic loading of precompiled cell models
Solid mechanics Incompressible, penalty, weak penalty, nearly incompressible formulations
Arbitrary mixed formulations: displacement, velocity, pressure
Arbitrary constitutive laws (isotropic, orthotropic, anisotropic laws)
Summable constitutive laws, i.e., σ(x)=!kwk(x)σk(x)
Scalar transport Static/transient, inclusion/exclusion of advection and/or reaction
Optional SUPG stabilization
Solve as linear or nonlinear system
Fluid mechanics Stokes and Navier–Stokes, static and transient
Eulerian and arbitrary Lagrangian–Eulerian (ALE) formulations
Conservative and noncon servative ALE formulations
General mixed formulations
SUPG formulation for all forms
1D fluid flow Static/transient linear/nonlinear formulations
Area-flow or area-velocity formulations
Riemann invariant or compatibility-based b oundary condition formulations
Simplification to Poiseuille network flow (compliant and rigid vessel wall)
Accommodates arbitrary wall constitutive mo dels
Primitive variable or Windkessel-type boundary conditions
Pressure Poisson Stokes and Navier–Stokes, static and transient
Elastic wave equation Linear and nonlinear formulations
Norms & measures Compute L2,H1norms/seminorms of variable expressions
Compute general inner-product expressions
Projections Compute L2,H1,andHdiv projection operators
Parameter e stim atio n Reduced-order unscented Kalman filter
Support for arbitrary spatially varying parameters
where vhis an n-dimensional variable (which may consist of one or more state vari-
able(s)), yhis the nth-dimensional test function, Vh(h)andYh(h)aretheset
of trial and test spaces, respectively, and hRdis the FEM domain description.
are specific functionals of the variable, vh, which depend on the underlying physical
system. In a single-physics example, the functional fwould be used to find the
solution vhin the weak form statement: find vhVh(h)suchthat
(2.2) f(vh,yh;
h)=0 yhYh(h).
As the variability between physics arises predominantly in the aand cfunctionals, this
abstract definition of the weak form problem enables generic handling of the core FEM
procedures. In the remainder of this section we introduce the specific core physics
currently implemented in CHeart. A summary of these modules is also provided in
Table 2.1.
2.1. Scalar transport. The conduction of electrical current, movement of me-
tabolites, and conduction of simple flow potential problems are a few examples of
a wide group of transport processes. Their simulation can be achieved through the
solution of the transient scalar advection-diusion-reaction equation or some simpli-
fication of it (see Table 2.1). As a result, scalar transport equations encompass a
broad range of problems ranging from the most simplified form, Laplace, to the full
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transport system. In the case of a transient advection-diusion-reaction equation
discretized using the backward Euler scheme, the operators in (2.1) have the form
a(vh) := 1
h+s, c(vh) := Dvk
where vk
hVh=Vhis the unknown scalar value at the kth time step, δtis the time
step interval, uis an advective velocity field, λis a linear reaction coecient field, D
is a tensor diusion coecient field, and sis a source/sink field.
2.2. Solid mechanics. Solid mechanics in cardiac applications is solved us-
ing large strain finite deformation theory [55, 9, 7] within quasi-static or transient
frameworks. The specific formulations can vary depending on whether the material is
modeled as compressible, incompressible, or nearly incompressible [32] as well as the
specific selection of constitutive laws (see Table 2.1). As an example, a quasi-static
incompressible material can be modeled by (2.1) using the aand coperators
h) := (ρb,det |uh+I|1),c(uh,p
h) := (σ(uh)phI,0),
where (uh,p
h)Uh×Wh=Vhare displacement and pressure, ρand bdenote mate-
rial density and body force, respectively, σ(uh) is the deviatoric Cauchy stress tensor
which may be defined as an arbitrary sum of core constitutive laws (see Table 2.1)
scaled by a weighting field, and Iis the identity tensor. Currently, a collection of
isotropic (i.e., neo-Hookean, Mooney–Rivlin [9]), general anisotropic constitutive laws
(i.e., Costa [14], Guccione [30], and Holzapfel–Ogden [36]) and viscoelastic models are
implemented and can be selectively applied tothetotalorthedeviatoriccomponent
of the strain tensor. Extensions to the existing constitutive laws are achieved by the
generalized implementation of the stress tensor, enabling straightforward inclusion of
new materials.
2.3. Fluid mechanics. Fluid mechanics can be solved for a variety of problems
in the heart (see Table 2.1), ranging from steady Stokes flow in small coronary vessels
to the more complex arbitrary Lagrangian–Eulerian (ALE) form of Navier–Stokes
equations [35, 40, 37, 64, 25, 66, 70] in the cardiac heart chambers. As an example, in
the nonconservative form of the Navier–Stokes equations discretized using backward
Euler, the operators aand cof (2.1) are written as
where (vk
h)Vh×Wh=Vhare the fluid velocity/pressure at the kth time step
with step-size δt;ρ, µ are the density and viscosity; and σ(vh) is the stress operator
(see Table 2.1). By using various bases, Vh×Whmay be defined for a wide range of
inf-sup stable spaces (see section 3.4), such as PκPκ1Taylor–Hood elements [11],
QκPκ1Nicolaides–Boland [8] elements, and Crouzeix–Raviart [16] spaces as well
as others. Time step handling, in general, is consistent with that shown in [70] for
ALE formulations. Further, as stabilization is often required for practical problems,
the test and trial spaces may bedierent,enablinggeneralintroductionofstreamline
upwinding Petrov–Galerkin (SUPG) methods [38].
2.4. Darcy flow. Darcy’s law in combination with conservation of mass is used
to model the flow of a fluid through a permeable porous medium [15, 60, 13]. In the
steady form, the operators aand cof (2.1) are written as
h) := (µK1vh,·vhs),c(vh,p
h) := (ph,0),
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where (vh,p
h)Vh×Wh=Vhare the Darcy velocity and pore pressure, respec-
tively, µis the dynamic viscosity, Kis the permeability tensor, and sis a volumetric
flow rate source or sink. Currently, this implementation relies on inf-sup stable spaces
Vhand Wh.
2.5. 1D fluid flow. The 1D fluid flow equations in an elastic vessel can be
obtained from the incompressible Navier–Stokes equations written in cylindrical co-
ordinates by assuming axisymmetric flow and by absorbing the dependence on the
radial coordinate into a lumped parameter [39, 26, 76]. This results in an equation
set defined by
h) := $Qk
h) := $α(Qk
wdA, Qk
where (Qk
h)Vh×Wh=Vhis the fluid volumetric flow rate/vessel cross-
sectional area at the kth time step with step-size δt,κis a friction coecient, cw
is the local pulse wave velocity, and αis the momentum correction factor which
depends on the assumed fluid velocity profile. At junctions where branching occurs,
the above equations are further augmented by a set of coupling conditions, i.e., mass
conservation and the Bernoulli equation [76].
2.6. ODE and DAE systems. Various systems of ordinary dierential equa-
tions (ODEs) and dierential algebraic equations (DAEs) ranging from 0DWindkessel
models [85, 78, 1] to tissue-level cell models [23, 53, 80] can be described using the
equation v=g(v), where gis known as the rate function. The solution of such prob-
lems can be obtained using explicit, semi-implicit, or implicit time integration with
constant time steps or adaptive time stepping. For example, the standard explicit
Euler scheme can be written in the form of (2.1) using the aand coperators,
a(vh) := 1
h),c(vh) := 0,
where vk
hVh=Vhis the state variable, and δk
tis the step-size at the kth time
step. The domain of the problem is normally either a point or a series of points which
eectively decouples the solution to individual nodes or modes (i.e., for a system with
nvariables, vh[L(h)]n).
In practice, directly incorporating a wide spectrum of ODE and DAE models into
the code base requires frequent editing and recompilation. Flexibility is achieved in
CHeart through dynamic loading of user-defined rate functions g.Thisallows,for
example, a direct C-code export from online model repositories such as CellML1to
be integrated without global recompilation.
3. FEM weak form construction. The core physics-based operators intro-
duced in the previous section rely on a series of routines common to all FEM codes.
The additional challenge of multiphysics comes with the ecient reuse, integration,
and cross-communication of fields, domains, and weak form functionals. Critical to
this is the formation of object-sets—representing a collection of core objects—that may
be reused freely. An illustrated example of how dierent object-sets within CHeart
interact together is shown in Figure 3.1.
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(line search)
Section 3.5
Section 3.1
Section 3.2
Section 5.3
Section 5.5
Section 3.3
Section 2
Section 3.4
Linear fluid
Cubic solid
solid domain
Fluid pressure
Fluid velocity
Fluid Space
Domain velocity
Boundary Space
Lagrange multiplier
Solid displacement
Solid pressure
fluid domain
Fig. 3.1.Illustration relating various object-sets used by CHeart to solve physics problems.
In this case, we consider the coupled fluid-solid interaction (see Results). The fluid is solved on
atetrahedralgridusingP2P1Ta ylo r–H ood e le m ent s a nd t he AL E N avi e r–S t ok e s for m ul a tio n
introduced in section 2.3.ThesolidissolvedonahexahedralcurvilineargridusingQ3Q2Tayl or–
Hood elements and the solid mechanics formulation from section 2.2.Object-setBis constructed
using five basis object types with varying order shape and dimension. These are used with five
topology objects comprising T,andtheinterfaceIobject-set is composed of two identity and two
injective interfaces. Vis then built from eight variables used in the four core problems feeding into
the monolithic solver.
In this section we review the core routines implemented to facilitate multiphysics
integration. As bas is interpolation schemes may be widely dierent, we formulate a
basis object-set for which we developed a single strategy for computation and interpo-
lation of dierent basis functions. These basis tools form the foundation for the gen-
eration of topologies—meshes with specified interpolation—which are integrated into
a topology object-set forming flexible structures to house meshesofdierent order,
size, dimension, etc. To enable cross-communication between fields and topologies, we
introduce a general construct of interfaces that allows the formulation of functionals
which relate variables defined on dierent topologies. With these tools, we construct
variable object-sets, which may be used in the general functionals introduced above
to assemble weak form statements.
3.1. Basis generation. The core physics modules presented in section 2 re-
quire a range of dierent interpolation functions that may rely on dierent refer-
ence elements, interpolation orders, and expansion types. This is handled by con-
struction of a basis object-set Bwith each individual basis object, B, being defined
as B={τ,{ϕ1,...,ϕ
K},Q}.Hereτdenotes the master (or reference) element,
K}is a set of K-basis functions used to construct finite-dimensional func-
tion spaces, and Q={(wp,ξp),p=1,...,N
p}defines the quadrature rule via the list
of weights wpR+and points ξpτ. The full list of available options is detailed in
Table 3.1. All bases dened for a sp ecic problem constitute the basis ob j e c t - s e t B.
A wide range of basis types, including modal [43], nodal [28, 88], Crouzeix–Raviart
[16], and bubble functions [28], can be written using the cardinal expansion. If Mis
the total number of unique polynomial terms of B,d=dim(τ), and Kis the number
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Tab l e 3 . 1
Summary of available element/basis specifications in CHeart.Optionsareavailableinany
combinati on (ex cl ud in g poi nt e le me nt ty pes which cannot s up port i nt er pol at io n).
Element/basis specifications
Element types point, line, quadrilateral, triangle, tetrahedral, hexahedral, prism
Basis types nodal, continuous/discontinuous, order (0–5)
modal, continuous/discontinuous, order (1–15)
user specified
Quadrature schemes Gauss–Legendre, integrable order (0–60)
Keast–Lyness, integrable order (0–10)
Raviart element types, which are supplied using a simple text file containing basis coecients and
powers (i.e., αand β).
of functions in the basis B,thenϕi(i=1,...,K) and its derivatives can be computed
α(i, j)
α(i, j)
D(βkj ,n
where ξ=(ξ1,...,ξ
d)Tis a local coordinate in τ,αRK×Mis the sparse coe-
cient matrix, βRd×Mis the corresponding powers in the cardinal expansion, and
D(p, n)=(n1
k=0 (pk),D(p, 0) = 1. As shown in (3.1), the derivatives can be auto-
matically evaluated and refactored as a new sparse cardinal expansion with adjusted
coecients. Here we note that for nodal Lagrange and modal basis functions the
number of terms in the cardinal expansion is equivalent to the number of basis func-
tions (i.e., K=M). However, in the case of bubble or Crouzeix–Raviart elements,
this is not the case.
The coecient matrix αmay either be user-supplied or constructed algorithmi-
cally. For example, for nodal Lagrange polynomials, α=ATis the inverse transpose
of the Vandermonde matrix (i.e., A(i, j)=(d
k)β(k,j)with ξ1,...,ξKτbeing
nodal coordinates).
Furthermore, tensor-based expansions can be written in a similar form,
α(γ(i, k),j)ξβ(j)
α(γ(i, k),j)D(β(j),n
where αRK×K1/d and βRK1/d are the 1D sparse expansions, and γRK×d
is an index array defining the corresponding 1D basis. For ecient computation the
expansion for each basis function is written as a sum of the nonzero components of
Due to the arbitrary degree of the basis, a range of quadrature schemes is required.
Schemes specifically for triangular and tetrahedral elements, such as those by Lyness
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and Jespersen [54] and Keast [46], were integrated based on the source code provided
by [63]. Higher order Gauss–Legendre schemes are also available based on finding the
roots of the Jacobi polynomials [43], and using tensor product expansion/collapsed
expansions to integrate up to 60th order. In addition to these built-in bases, CHeart
supports arbitrary user-defined bases and/or quadrature schemes to be provided via
a simple text file.
3.2. Topologies. Similar to the outlined basis object-set B,anobject-setof
topologies Tis constructed. A topology object TTin CHeart is constructed
using a specific basis BBand provides the mapping between mesh element indices
and global basis function (node or mode) indices used to evaluate fields over elements.
Specifically, the topology T={E,B,N
N}is comprised of NEand NN, denoting
the total number of elements and global basis functions, respectively, Bbeing the
basis used to perform field evaluations, and ENNE×Kthe element mapping such
that N=E(e, k)forsome e{1,...,N
3.3. Interfaces. By default, each topology TTis defined as a singular entity
with an unknown relation to others in the object-set. However, in many cases, such
as mixed formulations or multiphysics interactions, it is desirable for a well-defined
relationship to exist between topologies, enabling the transfer of data from one to the
other. This may be accomplished by defining an interface object IijI(with I
being the interface object-set) which eectively maps element indices and local element
coordinates from Tito Tj,Ti,T
jT. Each interface object Iij={Ti,T
j,ei,j ,Ii,j }
contains components to construct a mapping. Letting NE,i and NE,j be the numbers
of elements, and τiand τjthe master elements in Tiand Tj, respectively, then ei,j
NNE,i defines the relationship between element indices in the two topologies, and
Ii,j RNE,i×Kthe relationship between local coordinates. The mapping can be fully
described by the mapping ζij:{1,...,N
(3.2) ζij(e, ξ)=)ei,j (e),
Ii,j (e, m)ϕi,m(ξ)*=(e,ξ),
for some (e,ξ){1,...,N
E,j}×τjand (e, ξ){1,...,N
ζijenables interpolation and evaluation of fields and field derivatives as will be
discussed in section 3.4.
In CHeart all interfaces are assumed to be null mappings by default, i.e., no
mapping exists between topologies. Various kinds of interfaces may, subsequently, be
defined. These are outlined below and illustrated in Figure 3.2.
Identity interface. The most commonly used interface in FEMs is the identity
interface, given in Definition 3.1 and shown in Figure 3.2a, which simply returns the
same element and local element coordinates. Here, the associated basis of each topol-
ogy must be compatible in that their respective master elements must be geometrically
equivalent and their number of elements identical.
Definition 3.1 (identity interface). Let Ti,T
jTwith NE,i =NE,j =NE
and τi=τj=τ.AninterfaceIijIis called an identity interface if for any
(e, ξ){1,...,N
E}×τthe mapping function ζij(e, ξ)=(e, ξ),whereζijis
defined by (3.2).
This is often employed for classic mixed-element formulations, such as those used
in fluid mechanics or solid mechanics formulations, where velocity/displacement share
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Fig. 3.2.Types o f i nt e r fa ce s s up po rted in CHeart.(a) Identity interfaces between linear,
quadratic, and cubic quadrilateral meshes. (b) Injective interface. (c) Injective interface via an
intermediary. (d) An injectiv e interface between nonconforming meshe s through a conforming in-
termediary. (e) Adegenerateinterfacebetweena1Dvesselterminalandtheboundaryfaceofthe
the same mesh as pressure variables but use dierent interpolations (i.e., dierent
Topology groups. As topologies linked by an identity interface share many
commonalities, it is often useful to define topology groups (see Definition 3.2), which
link these objects together. This is particularly convenient, as more complex interfaces
may then be defined for one member of the topology group and used for all other group
Definition 3.2 (topology group). Let TgT;thenTgis a topology group if
there exists an identity interface IijIfor every Ti,TjT
referred to as the topology group of Tiif TiT
Injective interface. In multiphysics simulations the interconnectivity of topolo-
gies is often more complex. For this injective interface, mappings may be introduced
which provide a continuous map between element/local coordinate data and topolo-
gies. This may be on a portion of a domain, as in FSI where exchange between the
fluid and solid occurs on an interface. Definition 3.3 specifies the details of injective
interfaces which are also illustrated graphically in Figure 3.2b–d.
Definition 3.3 (injective interface). Let Ti,T
cal led injective if for every (e, ξ){1,...,N
E,i}×τithere exists a unique (e,ξ)
E,j}×τjsuch that ζij(e, ξ)=(e,ξ),whereζijis defined by (3.2).
This type of interface provides a mapping for all element/local coordinate pairings
from Tiinto some or all pairings from Tj. As a result, the injective interface allows for
the embedding of refined grids or lesser dimensional manifolds into another topology.
This formulation enables a wide range of associations between topologies. Injective
maps can be used to couple subsets of two topologies where only a portion of each is
interfaced by introducing an intermediary topology which is injective to both over the
desired subsets (see Figure 3.2c). This need arises in fluid-solid coupling, for example,
where the interaction between fluid and solid domains occurs on some portion of the
domain boundaries (see section 8.2). Further, injective interfaces can be easily ex-
tended to fully nonconforming grids by introducing an additional polygonal topology
nested within the intersections of the interfacing topologies (see Figure 3.2d).
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An important property of injective interfaces is their transferability to the associ-
ated topology groups, which facilitates data transfer between single-physics problems
employing mixed interpolation spaces. As shown in Lemma 3.4, it is clear that an
injective interface Iij={Ti,T
j,ei,j ,Ii,j }can be extended to any TkT
g(Ti) using
the mapping ζij.
Lemma 3.4. Suppose Ti,T
g(i)and IijIis injective. Then, there is an
implicit mapping IkjIthat is injective.
Proof. Since Ti,T
g(i), then ζki(e, ξ)=(e, ξ)andIkiis an identity in-
terface. The mapping for Ikjcan therefore be constructed as ζkj(e, ξ)=ζij
ζki(e, ξ)=ζij(e, ξ) for all (e, ξ){1,...,N
Ek}×τk. Since the interface Iijis
an injective interface, Ikjis also an injective interface.
An important subset of identity and injective interfaces includes those which
are complete (see Definition 3.5). By definition, all identity interfaces are complete.
However, in general, an injective interface enables linking one topology Tito a subset
of another Tj.Acompleteinterfaceensuresthatthissubsetextendstotheentirety
of Tj. This is particularly important for FEM weak form statements which must be
held over a defined domain which may require interfaces to other variables.
Definition 3.5 (complete interface). An injective interface IijIis called
complet e if ζijis bijective.
If an injective interface IijIis defined between two topologies Ti,T
an inverse mapping can be implicitly evaluated and constructed (see Definition 3.6).
However, explicitly constructing Ijiin the form shown in (3.2) is not, in general,
straightforward. Instead, the inverse of an injective interface is computed procedurally
by CHeart based on the provided injective interface.
Definition 3.6 (inverse interface). For an inj ect i v e interfa ce IijIwe may
implicitly define an inverse interface Ijithat for some (e,ξ){1,...,N
there exists a unique (e, ξ){1,...,N
E,i}×τisuch that ζij(e, ξ)(e,ξ)=(0,0).
Degenerate interface. In certain cases, the requirements of injective interfaces
or their inverses can be too restrictive, requiring the construction of degenerate inter-
faces (see Definition 3.7), where a topology may be mapped nonuniquely. This case
arises when reduced-dimensional formulations are coupled to full-dimensional prob-
lems, such as 1D blood vessel flow coupled to three-dimensional (3D) fluid flow (see
Figure 3.2e). Clearly, the reverse of this mapping is ill-posed and can only be obtained
in simplified instances.
Definition 3.7 (degenerate interface). An interface Iij={Ti,T
j,ei,j ,Ii,j }
is called degenerate if for some (e1,ξ1),(e2,ξ2){1,...,N
E,i}×τithere exists a
single pair (e,ξ){1,...,N
E,j}×τjsuch that ζij(e1,ξ2)=(e,ξ),ζij(e2,ξ2)=
An object-set of well-defined interfaces plays a crucial role in reusability of
a single-physics code within coupled problems, as it enables the abstraction of
interpolation-specific implementations as well as cross-talk between dierent com-
putational domains.
3.4. Variables and function spaces. Using the basis, topology, and interface
structures outlined, it is possible to formally construct interpolated variables as well
as discrete function spaces. As with previous structures, an object-set of variables V
may be defined, enabling the construction or calculation of numerous quantities that
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are either required for solving physical systems or desired as outputs. Each variable
object VVis defined as V={TV,CV},whereTV={EV,B
N}Tis the
topology, and CVRn×NNthe coecient matrix, so that a finite-element variable
can be computed as
(3.3) ˆvh(e, ξ)=
CV(l, EV(e, k))ϕk(ξ)el,(e, ξ){1,...,N
where {ϕ1,...ϕ
K}BV,e1,...,enRnare the unit base vectors for Rn.Herewe
add a hat ), distinguishing the variable from those introduced in section 2, as it is
principally a function of the element index and master element coordinate. All fields
in CHeart, including the mesh coordinates, are specified in this way, enabling the use
of any topology defined for these fields. In the case of spatial domains, this flexibility
allows for the construction of standard simplex and hexahedral domains as well as
the use of curvilinear grids. We assume each domain variable D={CD,T
is well constructed (i.e., continuous, nonoverlapping, locally invertible), enabling the
construction of a computational domain,
(3.4) h:= +xRd,,x=ˆx(e, ξ)(e, ξ){1,...,N
where the definition of ˆx follows from (3.3) using the appropriate domain topology
TD(and its components) as well as the scaling array CD.
However, unlike standard variables, domains have additional properties enabling
them to represent dierent spatial reference frames, for example, Eulerian/Lagrangian
[55] or ALE [21]. This is accomplished by allowing the domain hto be considered
as a function of time h(t) where the spatial positions of xhare summed with
an additional variable V={TV,CV}Vto give the current x(t)h(t), i.e.,
x(t)=ˆx(e, ξ,t)=
l=1 !KD
CD(l, ED(e, k)"ϕD,k(ξ)+
CV(l, EV(e, k))$
where {ϕm,1,...,ϕ
m,Km}Bmfor m={D, V }.WenotethatIDVis assumed to
be an identity interface to enable summing.
Vari a b l e- s p a c e pai r i n g. While (3.3) outlines the nth-dimensional vector func-
tion ˆvhas a function of local element index and local coordinates, it is desirable
in FEMs to instead define vh:
hRnas a function of the spatial domain h.
Whether this definition is possible depends on if the variable-space pairing (ˆvh,h)
is well defined.
Definition 3.8. Avariable-spacepairing(ˆvh,h)(defined in (3.3) and (3.4))
is well defined if the spatial domain his well constructed (i.e., continuous, non-
overlapping, locally invertible), and there exists a topology TrTand interfaces
rVI,whereIrDis complete and IrVis an identity or injective inter-
face. In this case, Tris called a root topology.
The stipulations of Definition 3.8 ensure that for each xhthere exists a unique
corresponding pair (e, ξ){1,...,N
Er}×τr. The existence of a root topology Tras
outlined implies that the composition
(3.5) vh(x)=ˆvhζrV(e, ξ),x=ˆx ζrD(e, ξ)
is well defined for all xh. As a result, Trprovides a topology for which vhand
hcan be related. The completeness requirement for Irensures that each xhas
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acorrespondingvh;however,IrVneed not be complete, allowing for conditions or
weak equations on subsets of a variable field (i.e., use of constraints along boundaries
or interior regions).
Discrete function spaces. Using the definition of discrete variables as shown
in (3.3), and the concept of variable-space pairings introduced in Definition 3.8 and
(3.5), it is natural to define the discrete function spaces introduced in (2.1) as
Vh(h)=.vh[L2(h)]n,,vh(x)=ˆvhζrV(e, ξ),x=ˆx ζrD(e, ξ)
(e, ξ){1,...,N
The properties of the discrete space Vh(h), such as the smoothness and order, depend
on the underlying connectivity of TVgiven by EV, the choice of basis, as well as the
spatial domain over which the space is defined.
3.5. Expressions. Often during a modeling process, it is desirable to conduct
algebraic computations with the variables defined over the finite-element meshes at
runtime. This functionality is implemented in CHeart through an object-set of plain-
text mathematical expressions provided at problem specification time. The expres-
sions can take variable pointers, time, and other expressions as inputs and produce
outputs that can be assigned to set boundary conditions, material and source fields,
and other variables as part of partitioned solver strategies and postprocessing. The
parsing of expressions is implemented using an internal bytecode-compiler.2Based
on an algebraic expression provided by the user, the internal compiler decomposes
the string into its component operators which are then stored as a sequence of simple
bytecode instructions. This conversion occurs only once, when the problem is set up,
and so in this way the expressions can be evaluated at key points in the computation
phase of execution with a minimal amount of runtime overhead. The user can specify
both scalar and vector expressions that contain standard algebraic and trigonomet-
ric functions as well as logical and conditional operators, modulo max/min and data
3.6. General weak form systems. Multiphysics systems are comprised of mul-
tiple weak form operators which act on dierent variables. These operators, outlined
for specific physics in section 2, are constructed as needed and built into an object-set
F,whereeachoperatorfFis defined as in (2.1).
Remark 3.9. In the case of Lagrangian or arbitrary Lagrangian–Eulerian (ALE)
domains, functionals can be defined as
h(t)) = !h(t)
enabling definition of the weak form on a specific reference frame. In the case of
conservative ALE formulations [25, 70],
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is used with functionals a0,a1,anda2.
Remark 3.10. Boundary conditions may be introduced to operators fFas
discussed in section 4.2.
Each operator fFmay be combined with others to form multiphysics systems.
This is done by defining the summed operator fF,
(3.6) f(vh,yh) := &
fk(vh,k,yh,k ;
where fkand h,k are the kth operator/domain and (vh,k ,yh,k) its respective variables
(where vh,k vhand yh,k yh). Importantly, the operator fand its variables,
vh,yh, do not reside on a single domain. This enables the merger of operators which
span dierent h,k,asoftenoccursinmultiphysics systems. Each operator fdenotes
aspecicsystemforwhichwelooktofindvhVhso that
(3.7) f(vh,yh)=0 yhYh,
where Vh,Yhdenote the complete trial/test spaces.
4. Physics coupling. In sections 2 and 3 we introduced the formulation of
single-physics operators as well as their construction through a series of object-sets.
To further enable flexible multiphysics integration, this section discusses a series of
coupling strategies developed to formally link operators (as shown in (3.7)) which
represent multiphysics weak form systems. Here we define a coupling as a functional
relationship between single-physics systems within a problem, the examples of which
can be found in Table 4.1. While this may occur in numerous instances, here we
identify three principal mechanisms:
(1) functional dependence of existing operators on evolving variables from an-
other problem,
(2) functional dependence of boundary conditions on evolving variables from an-
other problem, and
(3) new coupling operators which eectively link problems together.
These approaches are all made possible using dierent couplings methods.
4.1. Coupling via existing operators. As full access to all variables in V
is available to any problem, the inclusion of variables through existing operators is
straightforward. For example, the diusion coecient in a scalar transport problem
may be linked to any variable in V, including one which is solved for within a dierent
system (e.g., as is the case in strongly coupled electromechanics). This enables the
automatic coupling of systems. Further, as dierent solve strategies may be applied as
discussed in section 5.5, CHeart enables this coupling to be either explicit or handled
through a fixed point iteration.
4.2. Coupling via boundary conditions. Each operator (see section 3.6) may
be linked to any number of boundary conditions. Steady or time-dependent Dirichlet
or Neumann boundary conditions may be applied to dierent portions of the domain
boundary (typically a series of user-defined nodes or element faces).
CHeart boundary conditions may be specified using variables as well as numeric
expressions, so that the solution of one problem may be provided as the boundary
data for another problem. This is often required in partitioned ALE Navier–Stokes
where the domain motion can be computed independently from the flow while the two
problems are coupled along portions of the domain boundary. This type of coupling is
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Tab l e 4 . 1
Summary of available coupling options for dierent styles of multiphysics.
Physics coupling operators/options
Simple coupling Coupling via share d variables
Coupling via algebraic expressions
Coupling via b oundary conditions
Lagrange coupling Coupling one or m ore variables to a constraint
Coupling variable or variable rate, i.e., Du=uor Du=u/t
Coupling all components or norm al components, i.e., L(u)=uor L(u)=u·n
User defined constraint expressions available
Support for constraints on 1D, 2D, and 3D manifolds or volumes
Volume coupling Linear coupling through reaction terms
Support for 2D, 3D
also common in FSI, where the motion of the interface between fluid and solid domains
is set as a Dirichlet condition and the traction on fluid is applied as a Neumann-type
condition to the solid.
4.3. Coupling via expressions. While the integration of multiple physics us-
ing variables provides a powerful tool for coupling systems, this is not always sucient
as the coupling may require algebraic or dierential manipulation of a variable. This
feature is enabled in CHeart via expression strings (see section 3.4) which may be
provided by users to define the specific mathematical form of coupling.
In many cases, such as with boundary conditions, expressions and variables may
be used interchangeably. In addition, CHeart allows the casting of expressions to
variables, which enables the coupling of certain operations which would otherwise
need to be hardcoded.
4.4. Monolithic integration via coupling operators or Lagrange multi-
pliers. Multiphysics integration often necessitates the inclusion of additional terms
or constraints in the system. In some cases, as in partitioned approaches, this merely
involves adding a source or augmenting boundary conditions. In other approaches,
the coupling may be imposed via cou pl ing o perat ors ,orbytheintroductionofLa-
grange multipliers. In these cases, the additional elements are considered as separate
physical problems which augment traditional single-physics examples. The modular
implementation of these couplin g problems in CHeart promotes code reuse in various
types of multiphysics problems.
Coupling operators. In CHeart coupling operators provide a link between
variables from distinct physics problems through the addition of new terms or opera-
tors into the weak form. Considering a set of variables (u1,...,um)Uh
which have been introduced for a set of physics problems, the general form of ain
equation (2.1) for coupling operators is given by
a(u1,...,um) := $m
γ1kL1kuk, ...,
γmkLmk uk%,
where γik is a scalar coecient detailing the coupling between uiand ukand Lik
the corresponding linear operator (see Table 4.1). As a result, the weak form defined
for uiis augmented with 0m
k=1 γikLik uk,whichmaybetailoredtoaddressaspecic
type of coupling. One classical example of the use of coupling operators is in Boussi-
nesq convection, where the Navier–Stokes equations are coupled to advection-diusion
through the advective velocity field and a temperature-based forcing term.
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Lagrange multipliers. Lagrange multipliers are typically used for the addition
of new constraint equations which may augment a single-physics system or link mul-
tiple systems. Specifically, an additional variable λMhis added to the set of
variables (u1,...,um) in order to weakly impose over some domain or manifold, h,
that the variables satisfy
where Lkand Dkare the kth linear algebraic and dierential operators, respectively
(see Table 4.1), wkis the kth scalar weight, and sis some provided source. The
additional constraint terms can be included using the operator aof (2.1) defined as
a(u1,...,um,λ) := $w1L1(λ),...,w
where (u1,...,um,λ)Uh
using multipliers and may also be used to construct projection operators for iterative
fixed point schemes. Examples of this approach are found in the imposition of complex
physical constraints as well as fluid-solid coupling.
5. Assembly, matrix operations, and building. Ecient and flexible ma-
trix assembly operations are a principal enabler of multiphysics integration in finite-
element modeling and a key engine driving its computational performance. The core
CHeart library provides automated functionalities and standardized interfaces to en-
able rapid and extensible developments. The process by which the topology group,
equation map, and variable are used to build and populate sparse matrices is described
5.1. FEM element assembly. FEM element assembly of residuals, matri-
ces, and other objects (see section 3) is usually handled by an element loop over
amesh. Inthecode,CHeart contains a single element loop routine, using a series of
procedure pointers—Setup,LocalElementCalculate,LocalElementAssemble,and
CleanUp procedures—to control the tasks executed. These procedure pointers may
be linked (or nullified) before execution of the element loop and altered accord-
ing to the desired assembly task (i.e., residual evaluation, Jacobian construction,
norm calculations, etc.). Setup and CleanUp procedures are commonly used to ex-
tract tasks which need not be repeated during the element loop—such as dynami-
cally allocating/deallocating memory or constructing/destroying access shortcuts to
data required for FEM assembly—to optimize computational performance. The
Calculation and Set routines are established to do tasks which may vary by ele-
ment, with the Calculation step involving the computation of a physics-dependent
element-level matrix or residual, for example. The Set step is a low-level, abstracted
task used to incorporate element-level data into global matrix structures.
Beyond the standard steps followed in FEM assembly, the presence of merged
functionals and interface mappings necessitates important extensions. As the weak
form functional for a physical system may be a sum of multiple physical systems
(see (3.6)), the general assembly occurs over each individual functional fkf,for
which we define a symbolic variable dependence using equation maps, E(fk). Cru-
cially, this equation map contains a root topology Tr, over which element-level proce-
dures may be computed.
Since integral evaluations of weak form equations (see (2.1)) are computed over
Tr, all other topologies used as part of the problem governed by fkmust be evaluated
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on Tr. This list may be automatically compiled by recording the usage of various
topologies as problems are defined by the user. The completeness of Trensures the
looping process covers the indicated spatial domain.
In addition, the corresponding basis functions and their gradients of all involved
terms must be evaluated on Tr.IfIrkfor TkTis an identity interface (see
Definition 3.1), no mapping is required, and pre-evaluations of the basis functions
(and their derivatives) can be used directly. However, when Irkis a more complex
mapping, such as an injective interface, then the basis on Tkmust be mapped onto
Tr(see (3.6)). Users can elect to either execute these computations on the fly or have
the program precompute these quantities into fast access cached lists.
Algorithm (FEM LOOP)
for all fkf
call SetupProcedure
for all eTrE(fk)
for all Ti{Tk}
Compute needed ϕ,rϕ, . . . at Qrfor all ϕBi
end for
call LocalElementCalculateProcedure
call LocalElementAssembleProcedure
end for
call CleanUpProcedure
end for
5.2. Equation mappings. The translation of weak form operators finto dis-
crete finite-element equivalents may be achieved via the use of equation maps (Defini-
tion 5.1). An equation map is an abstraction of the dependence of weak form operators
on their spaces, which completely encapsulates the row-column variable mappings
necessary to construct the local element tensors. Considering (2.2), the equation
mapping defines whether f(v
h=(0,...,vh,i ,...,0)and
h=(0,...,yh,j ,...,0)withvh,i Vh
iand yh,j Yh
j, that is, whether there are
operators in fwhich pair the ith and jth trial and test spaces.
Definition 5.1 (equation map). Let {Vh
m}and {Yh
n}be well-
defined function spaces on h.Then,iffis a functional form as shown in (2.2) on
the discrete finite-element spaces, the equation map E(f)of fis defined for each row
variable i{1,...,n}by Ei(f)where
Ei(f) := {j{1,...,m}|f(vh,yh;
h)̸=0 for some (vh,yh)Vh
denoting the symbolic nonzero dependence of terms in fon {Vh
Remark 5.2. In Definition 5.1, the equation map E(f) implies that the operator,
h(with topology TDT). Hence, there is
a root topology, Tr, which links all topologies of variables in E(f). That is, IrDis
complete and IrKis nonnull for each TkE(f).
Encoding the operators of f,theequationmapE(f) enables the generalization
of many lower-level routines and operations which require knowledge of the weak
form structure, such as linearized system assembly, merger, etc. In addition to the
symbolic dependence outlined, the equation map E(f)offalso labels the structure
of each term as identity,trace,orfull. This relates to the pairing of vector variables,
which, due to operators, may share components in dierent ways (see Definition 5.3).
For example, the standard inner-product of two vector variables may be classified as
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Fluid mechanics
Solid mechanics
Lagrange multiplier coupling Monolithic cou
led s
Fig. 5.1.Illustration of three separate equation maps—incompressible Navier–Stokes with ve-
locity and pressure variables (v,p),incompressiblesolidmechanicswithdisplacementandpressure
variables (u,q),andLagrangemultipliercoupling,λ,between(v,u)—combined into a single equa-
tion map to form the monolithically coupled system used in FSI; see section 8.2.
having trace structure, while the standard inner-product of two scalar variables is a
full structure operator.
Definition 5.3 (equation map structure). Let E(f)be the equation map of the
operator fgiven in Definition 5.1.ConsideringthevariablepairingVj×Yi,forsome
jand yhYh
itakes vh:
hRsand yh:
Letting vn,k
jdenote the nth basis function in the kth component and, similarly,
letting ym,p
idenote the mth basis function in the pth component, the structure
of the variable pairing is defined as
(Identity) if f(vn,k
h)̸=0for some n=m, k =p,
(Trace) if f(vn,k
h)̸=0for some n, m, k =p,
(Full) if f(vn,k
h)̸=0for some n, k , m, p,
where the least generic structure applicable is selected for the specific functional f.
While the equation mapping of an operator enables the construction of a generic
template for FEM calculations, its dependence on a single domain, h, is limiting,
particularly in multiphysics contexts where multiple domains are linked together. As
aresult,themultiphysicsoperatorfis enabled through the formation of equation
map sets where the final structure is the union of each equation map automatically
evaluated from the individual maps provided, i.e., E(f) := 1kE(fk). An example of
this process is illustrated in Figure 5.1. In these cases, operations are done on each
individual equation map, E(fk), and their results are combined to emulate the final
mapping, E(f).
5.3. Matrix assembly and monolithic integration. The final weak form
system shown in (3.7) may be solved in a variety of ways. This is achieved, in part,
by allowing single-physics or other operators to be integrated into a single monolithic
problem which may be solved implicitly. Importantly, not all operators of the total
problem fneed be monolithic, thereby allowing partitioned solving of some/all physics
For problems with monolithic integration, their implicit solution requires the for-
mulation of a Jacobian matrix (or some approximation to it) which may be inverted
or iteratively solved to give the solution or a solution update (as in Newton–Raphson
methods [57, 32]). Similar to other structures, we construct an object-set Mof ma-
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trices. Each matrix, MM, is assembled on an underlying equation mapping E(f)
which may be composed of a simple physical system or a more complex multiphysics
system spanning multiple domains and variables. From the equation map definition,
we seek to assemble Min compressed sparse row (CSR) format [27] by determining
the connectivity between a series of rows and columns. In this sense, the matrix,
(5.1) M={Vr,Vc,E},
is a graph with row/column vertices (Vrand Vc)andedges(E)whichprovidetheir
connectivity. In the case of structurally simple equation maps—such as that which
arises from scalar Laplace—the matrix construction process is straightforward. In
this case, if the solution variable is U={U,T}Vwith NNnodes/modes, then by
computing the inverse mapping of the topology array ET(the node-to-element
map), E1(n)={e{1,...,N
E}| n=E(e, k),for some k{1,...,K}},n
N}one can eectively determine the interdependence of nodes/modes on
elements. Subsequently, the inverse map can be used to directly compute the matrix
graph M={Vr,Vc,E}.HereVr=Vcis defined so that, for n{1,...,N
Vr(U, n)=nand
N}| m=E(e, k)forsomek{1,...,K}and eE1(n)}.
Remark 5.4. In the case of discontinuous Galerkin assembly, ˜
E1is defined and
used in place of E1in (5.2) with ˜
k{1,...,K},and elN(e)},n {1,...,N
N},and N(e)denotingthesetof
neighboring element indices of eand eitself.
In a multiphysics environment, this approach must be adapted to deal with ad-
ditional (potentially multidimensional) variables, multiple equation maps (i.e., f=
{fk}), and complex interfaces. However, the construction of more complex linear
systems can be eciently managed using both the equation mapping E(f) as well as
the underlying interface mappings, I.E(f)providesdetailsofhowthesetofi-row
variables, Y={Y1,...,Y
i}V, as well as j-column variables, V={V1,...,V
is interrelated. Using this data, we may construct a variable-matrix mapping for both
rows and colums. That is, for any Y={C,T}Yand n{1,...N
row mapping Vr(Y,n)=R, R {1,...,N
Vr},whereNVris the total number of rows
(and similarly for the column mapping). We note that, in this case, Ris uniquely
paired to Yand n. This definition eectively orders how nodes/modes associated
with specific variable spaces are mapped to the linear system.
Remark 5.5. Construction of Vrand Vcmappings follows element clustering,
where all variables are ordered according to elements, or variable clustering, where
each variable is ordered sequentially, depending on the parallelization strategy.
Subsequently, assembly is managed by constructing the matrix connectivity, E,as
seen in Definition 5.6. Here, the connection between row and column variables is ex-
tracted by first operating on each equation mapping E(fk) resulting from functionals
fkf.ThisprovidesalinktotheroottopologyTrkover which all row and column
variables of E(fk) are related. Each element of the root topology Trkthen acts as
the common point, enabling mapping through the defined interfaces to determine the
corresponding elements in the topology used by all variables in the mapping. By col-
lating these nodes/modes, it is then possible to construct the collective connectivity,
E, as given in Definition 5.6 and illustrated in Figure 5.2.
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Fig. 5.2.An il lustration of mon olit hic matrix assemb ly procedure. Top left: T wo topologies
(quadrilateral and triangle types) with node numbers indicated. The equation maps for the individual
systems are E(f1(U)) and E(f2(V)).Whenthetwosystemsarecoupled,thecouplingoperator
provides the additional functionals, f3,andequationmaps,E(f3),linkingthevariables. Following
the formation of the monolithic E(f),thematrixconnectivityEcan be dete rm in ed thro