1. IntroductionThe efficient numerical simulation of complex multiphysics systems is ubiquitous in Computational Science and Engineering. Although a wide range of methods exists to tackle specific problems, they often lack versatility and adaptability, especially when the modelling is of increasing complexity as in real-world applications.
Infinite-dimensional port-Hamiltonian systems (pHs) have been first introduced in [1] using the language of differential geometry. They provide a powerful tool to model complex multiphysics open systems (whether or not being linear) for control purpose. A wide range of physical systems has been written within this formalism, see e.g. [2] [3] [4]. This twenty-year-old framework [5] enjoys nice properties, such as the relevant physical meaning of the variables, a useful underlying linear structure (namely Stokes-Dirac structure) which encodes the power balance satisfied by the Hamiltonian (often chosen as energy), and last but not least: the interconnection of multiple pHs remains a pHs. This allows “modular” modelling of complex multiphysics systems.
Since then, many researchers have developed numerical methods to discretize these systems in a structure-preserving manner, hence keeping the advantages of the infinite-dimensional pHs. Such methods aim at constructing approximate finite-dimensional pHs at the discrete level. Our aim is to show the versatility of PFEM thanks to a new unified framework and to introduce the ongoing project SCRIMP together with companion Jupyter notebooks [6]. Each particular example discussed here has been treated previously [7] [8] [9] [10]. Here, the existence of an underlying common structure for many pHs is highlighted. Obtaining such a general scheme for infinite-dimensional pHs is of major importance for control purposes [11], and for coupling atomic elements into a more complex system with the guarantee of well-preserved energy exchanges between subsystems [12].
The first proposed structure-preserving scheme for pHs dates back to [13], where the authors proposed a mixed finite element spatial discretization for hyperbolic systems of conservation laws. Pseudo-spectral methods relying on higher-order global polynomial approximations were studied in [14]. Unfortunately, this method seems to be limited to the one-dimensional case. A finite difference method with staggered grids was developed in [15] for two-dimensional domains, but complex geometries are then difficult to tackle. Weak formulations leading to Galerkin numerical approximations began to be explored in the past few years. In [16] the prototypical example of hyperbolic systems of two conservation laws has been discretized by a weak formulation. However, the construction of the necessary power-preserving mappings is not straightforward on arbitrary meshes. All these methods require ad hoc implementations and are usually restricted to particular cases of pHs. Furthermore, since they do not rely on well-established and versatile numerical libraries, using such techniques remains confined within a small community of experts. We refer the reader for a more complete overview of structure-preserving discretization for pHs to [5] [17] and the references therein.
Thanks to [8] it has become clear that there exists a deep relation between structure-preserving discretization of pHs and Mixed Finite Element Method (MFEM). Indeed velocity-stress formulations for the wave dynamics [18] and elastodynamics problems are of Hamiltonian type and their mixed discretization preserves this structure for closed systems. This leads to the intuition that the MFEM may be used to discretize the underlying geometric structure of pHs in a unified way, even for open systems, translating the infinite-dimensional Stokes-Dirac structure into a finite Dirac structure. A first successful application has already been achieved in [19] for discretizing the 1-D Navier-Stokes equations for reactive flows formulated in a port-Hamiltonian formulation. The discretization strategy relies on the partitioned structure of the problem and for this reason, goes under the name of Partitioned Finite Element Method (PFEM). This method proves nice convergence properties, see e.g. [20] for a recent proof on the wave equation, that does not require the fulfilment of the usual inf-sup condition for MFEM, generalizing the results cited in ( [21], Remark 6).
It has to be pointed out that the core idea of PFEM, i.e. performing integration by parts on a partition of the weak formulation of the system of equations, has already been proposed for closed hyperbolic systems in [21]. Therein, the formulations are called either primal-dual or dual-primal, depending on the chosen partition of the system.
The major difference between MFEM and PFEM relies on the choice of the test functions in the weak formulation, hence on the finite element form functions. Indeed, in PFEM, they never carry homogeneous boundary conditions. In e.g. ( [22], Section 7.1), it is shown that for a Dirichlet control, test functions are taken in the kernel of the Dirichlet trace. As already mentioned, in [21], the proposed primal-dual and dual-primal discretizations are then suitable for the structure-preserving discretization of closed systems. Nevertheless, by keeping these homogeneous conditions in the test functions, not only the application of Dirichlet control is difficult, but the definition of the Neumann observation, necessary for the discrete power balance, would be more complex. PFEM aims at easing the mimicking of the continuous power balance at the discrete level, by relaxing the test functions used in [21]. To the best of our knowledge, this relaxation has not been investigated yet in the case of boundary controlled wave-like systems, and this probably comes from the fact that MFEM has been first developed for elliptic systems. Indeed, in this case, such a boundary condition is mandatory for well-posedness (especially to obtain the ellipticity in the kernel condition ( [22], Eq. (5.1.7)), while PFEM is made for evolution systems, and more especially for pHs.
In our opinion, the driving forces of PFEM are threefold: first, PFEM takes collocated boundary controls and observations into account in a simple manner; secondly, PFEM is structure-preserving, meaning in particular that the discrete power balance perfectly mimics the continuous one; thirdly, the implementation of PFEM only relies on existing finite element libraries, such as FEniCS [23], selected in the ongoing project SCRIMP for its robustness and efficiency. Last but not least, the pHs point of view allows us to separate axioms of physics (such as conservation laws) from constitutive laws and equations of state (such as Hooke’s law and ideal gas law). PFEM is based on this separation, providing the possibility to tackle parabolic or nonlinear systems, at the price of solving a finite-dimensional port-Hamiltonian Differential Algebraic Equation (pHDAE). In the particular case of linear hyperbolic PDE, as shown in Section 2, the constitutive laws can be easily (i.e. without matrix inversions) taken into account in order to recover an Ordinary Differential Equation (ODE).
PFEM could also be named e.g. extended MFEM or relaxed MFEM. Since only evolution systems are considered (not necessarily of hyperbolic type, see e.g. Section 3.4), relaxed conditions for the selection of the test functions hold, hence for the finite elements as well. We choose to follow the terminology introduced in [8] and widely used since then. Furthermore, it emphasizes the pH formulation of the initial system to discretize.
Main contributions
We first aim at presenting the strategy of the structure-preserving discretization PFEM, in a new unified abstract framework, allowing for an easy application to a wide class of boundary controlled partial differential equations. Then, in order to show the versatility of our approach, we successively apply PFEM to the boundary controlled wave equation, the boundary controlled Mindlin plate model, and the boundary controlled heat equation with a thermodynamically well-founded Hamiltonian (namely the internal energy, instead of the quadratic functional commonly used). Taking advantage of the strong underlying structure, we finally describe a unified object-oriented implementation of these models via PFEM. Companion interactive Jupyter notebooks [6] are discussed to illustrate our methodology. For the purpose of this manuscript, smooth functions are selected for the physical parameters of the problem under considerations. Nothing prevents from choosing less regular coefficients. This will of course affect the global convergence of the underlying finite elements [20].
Structure of the manuscript
The manuscript is organized as follows. In Section 2 the abstract pHs framework is introduced, with a particular focus on both hyperbolic and parabolic linear systems of partial differential equations. In Section 3 the general structure-preserving discretization is presented, and then specialized on the two cases previously mentioned. In Section 4 the ongoing environment SCRIMP is described in detail. In Section 5 the three companion interactive Jupyter notebooks [6] are thoroughly explained. Conclusions and perspectives are finally drawn in Section 6.
2. Definition of the General FrameworkIn this section, we introduce an abstract class of pHs and their underlying geometric structure: the Dirac structure for the finite-dimensional case and the Stokes-Dirac structure for the infinite-dimensional case. For the infinite-dimensional case it is shown how hyperbolic and parabolic systems easily fit into this framework.
2.1. Finite Dimensional Port-Hamiltonian SystemsState representation
Let us begin with a classical definition of a pHs in finite dimension. Consider the time-invariant dynamical system [24]:
( d x d t = ( J ( x ) − R ( x ) ) ∇ H ( x ) + B ( x ) u , y = B ( x ) Τ ∇ H ( x ) , (1)
where H ( x ) : X ≃ ℝ n → ℝ , the Hamiltonian, is a real-valued function of the vector of energy variables x , bounded from below. Matrix-valued functions J ( x ) (the structure operator) and R ( x ) (the dissipative or resistive operator) are skew-symmetric and symmetric positive semi-definite respectively. The control u ∈ ℝ m is applied thanks to the matrix-valued control function B ( x ) of size n × m . Variable y ∈ ℝ m is the power conjugated output to the input.
Such a system is called a port-Hamiltonian system, as it arises from the Hamiltonian modelling of a physical system and it interacts with the environment via the input u and the output y , included in the formulation. The vector ∇ H ( x ) is made of the co-energy variables.
Due to the structural properties of J ( x ) and R ( x ) , the port-Hamiltonian system enjoys the nice following power balance:
d H d t = − ( ∇ H ( x ) ) Τ R ( x ) ∇ H ( x ) + u Τ y ≤ u Τ y , (2)
meaning that R ( x ) accounts for dissipation, and that the input-output product corresponds to the power supplied to (or taken from) the system, through the control u .
Flow-effort representation
Consider two finite-dimensional vector spaces E = F ≃ ℝ n . The elements of F are called flows, while the elements of E are called efforts. Those are port variables and their combination gives the power flowing inside the system. The space B : = F × E is called the bond space of power variables. Therefore, identifying E as the dual of F, the power is defined as 〈 e , f 〉 : = e ( f ) .
Definition 1 ( [25], Def. 1.1.1) Given the finite-dimensional space F and its dual E with respect to the inner product 〈 ⋅ , ⋅ 〉 F : F × F → ℝ , consider the symmetric bilinear form:
〈 〈 ( f 1 , e 1 ) , ( f 2 , e 2 ) 〉 〉 : = 〈 e 1 , f 2 〉 + 〈 e 2 , f 1 〉 , where ( f i , e i ) ∈ B , i = 1,2.
A Dirac structure on B : = F × E is a subspace D ⊂ B , which is maximally isotropic under 〈 〈 ⋅ , ⋅ 〉 〉 . Equivalently, a Dirac structure on B : = F × E is a subspace D ⊂ B which equals its orthogonal companion with respect to 〈 〈 ⋅ , ⋅ 〉 〉 , i.e. D = D [ ⊥ ] , where:
D [ ⊥ ] : = { ( f , e ) ∈ B | 〈 〈 ( f , e ) , ( f ˜ , e ˜ ) 〉 〉 = 0, ∀ ( f ˜ , e ˜ ) ∈ D } .
The connection between the concept of Dirac structure and pHs in its canonical form (1) is achieved by considering the following ports:
· the storage ports ( f x , e x ) : = ( d x d t , ∇ H ( x ) ) ∈ ℝ n × ℝ n , made of the storage flow f x (time-derivative of the energy variables) and storage effort e x (the co-energy variables);
· the resistive (or dissipative) ports ( f R , e R ) ∈ ℝ k × ℝ k , made of the resistive (or dissipative) flow f x and resisitive (or dissipative) effort e R ;
· the interconnection ports ( f u , e u ) : = ( − y , u ) ∈ ℝ m × ℝ m , made of the interconnection flow f u and interconnection effort e u .
Assuming that the matrix R ( x ) has constant rank, from classical matrix factorizations there exist matrices G (not necessarily square, of size k × n ) and K symmetric positive semi-definite of size k × k such that R = G Τ K G . These notations at hand, the pHs (1) rewrites:
( f x ( x ) f R ( x ) f u ( x ) ) = [ J ( x ) − G ( x ) Τ B ( x ) G ( x ) 0 0 − B ( x ) Τ 0 0 ] ︸ J e ( e x ( x ) e R ( x ) e u ( x ) ) , (3)
together with the (dissipative or resistive) constitutive relation:
e R ( x ) = K ( x ) f R ( x ) . (4)
It is clear that the extended structure operator J e appearing in (3) is skew-symmetric of size ( n + k + m ) × ( n + k + m ) . Its graph is a Dirac structure with respect to the Euclidean inner product, as a kernel representation, see [24]. Hence, it comes:
〈 e x ( x ) , f x ( x ) 〉 ℝ n + 〈 e R ( x ) , f R ( x ) 〉 ℝ k + 〈 e u ( x ) , f u ( x ) 〉 ℝ m = 0.
Noting that d H d t = 〈 e x ( x ) , f x ( x ) 〉 ℝ n (by definition of the storage port) leads to:
d H d t = − 〈 e R ( x ) , f R ( x ) 〉 ℝ k − 〈 e u ( x ) , f u ( x ) 〉 ℝ m ,
and thanks to the symmetry of R , (4) gives:
d H d t = − 〈 f R ( x ) , K ( x ) f R ( x ) 〉 ℝ m − 〈 e u ( x ) , f u ( x ) 〉 ℝ m .
Finally, from (3) and the definition of the storage and interconnection ports, the power balance (2) is recovered.
The relation between (1) and (3)-(4) can be understood as follows: the power balance (2) is encoded in the Dirac structure (obtained from the extended structure operator J e ) together with the resistive constitutive relation.
Remark 1. The canonical Euclidean inner product has been used here, but other inner products are allowed to take into account mass matrices (symmetric positive definite) on the left-hand side of (1), (3), and (4). This is crucial after the spatial discretization procedure. This corresponds to a kernel representation of Dirac structure [24].
System 1 is a pHs in canonical form. Recently, finite-dimensional differential algebraic port-Hamiltonian systems (pHDAE) have been introduced both for linear [26] and nonlinear systems [27]. This enriched description shares not only all the crucial features of ordinary pHs, but also easily accounts for algebraic constraints, time-dependent transformations and explicit dependence on time in the Hamiltonian. The application of the proposed discretization method naturally leads to pHDAEs. Indeed, a constitutive relation between f x : = d x d t and e x : = ∇ H ( x ) is needed to be well-defined. But PFEM takes into account constitutive relations apart from (3) as constraints. However, as shown later in Sections 3.2 and 3.3, the method simplifies in the case, for instance, of linear hyperbolic systems.
2.2. Infinite-Dimensional Port-Hamiltonian SystemsIn this section an infinite-dimensional generalization of pHs is presented. For sake of readability, the (Stokes-) Dirac structure is first defined, and secondly, infinite-dimensional pHs are then described in both hyperbolic and parabolic cases. A more general framework can be designed, but this goes beyond the aim of this present work.
Structure operator
As to avoid functional difficulties, the analogue of the extended structure operator will not be written as in (3). More precisely, the control operator will not be included in an extended structure unbounded operator, but given apart. The Stokes-Dirac will be then obtained thanks to a structure operator related to the boundary control operator through an abstract Green formula. However, like in finite dimension, the aim is to establish a link between flow and effort variables. Most importantly, the underlying Stokes-Dirac structure must encode the power balance of the dynamical system under study.
Consider a Lipschitz domain Ω ⊂ ℝ d , d ∈ { 1,2,3 } , and the relation:
( f 1 f 2 ) = [ 0 − L * L 0 ] ( e 1 e 2 ) , e 1 ∈ H L , e 2 ∈ H − L * , (5)
with:
H L : { e 1 ∈ L 2 ( Ω , A ) | L e 1 ∈ L 2 ( Ω , B ) } , H − L * : { e 2 ∈ L 2 ( Ω , B ) | − L * e 2 ∈ L 2 ( Ω , A ) } .
By L 2 ( Ω , X ) we denote the space of square integrable X -valued functions. Symbol A , B denote either the space of scalars ℝ , vectors ℝ d , symmetric tensors ℝ sym d × d = : S or a Cartesian product of those, depending on the particular example. The operator L : H L → L 2 ( Ω , B ) is a generic differential, and therefore linear but unbounded, operator. The notation L * : H − L * → L 2 ( Ω , A ) denotes the formal adjoint of L , defined by the relation:
〈 L e 1 , e 2 〉 L 2 ( Ω , B ) = 〈 e 1 , L * e 2 〉 L 2 ( Ω , A ) , e 1 ∈ C 0 ∞ ( Ω , A ) , e 2 ∈ C 0 ∞ ( Ω , B ) . (6)
Of course, for (5) to be well-defined, constitutive relations are needed. Only physical laws will be taken into account when constructing the above relation on concrete examples. As pointed out in the introduction, PFEM aims at both preserving this relation and discretizing constitutive laws to close the system.
Remark 2. One can be confused by the lack of evolution in time in (5). However, this emphasizes an important paradigm in the proposed point of view: this relation translates the time-independent geometric structure of the pHs as an equation with differential operator (ill-posed on its own), while constitutive relations will bring back the time dependency of the problem. In particular, some flows must be the time derivative of the energy variables.
Throughout the paper, 〈 ⋅ , ⋅ 〉 X denotes the inner product of the Hilbert space X. Definition (6) is analogous to Definition 5.80 in [28]. In Section 3.2, the operator L is the gradient, denoted by grad, and its formal adjoint is minus the divergence, denoted by -div, from the so-called Green’s formula (integration by parts). In Section 3.3, the operator L contains both grad and Grad. This latter corresponds to the symmetric part of the gradient and represents the deformation tensor in continuum mechanics:
Grad ( e ) : = 1 2 ( ∇ e + ∇ Τ e ) , e ∈ C ∞ ( ℝ d ) .
The formal adjoint of Grad is minus the tensor divergence -Div. For a tensor field E : Ω → M : = ℝ d × d , with components e i j , the divergence is a vector, defined columnwise as:
Div ( E ) : = ( ∑ i = 1 d ∂ x i e i j ) j = 1, ⋯ , d .
Finally, in Section 3.4, L is made of the gradient and the identity operator.
Stokes-Dirac structure
Definition 1 still remains valid in infinite dimension. Nevertheless, as stated above, the structure operator in (5) is not extended to include the control operator. Hence an additional assumption has to be made for [ 0 − L * L 0 ] to define a Dirac structure in relation with a pHs coming from boundary control of partial differential equations. In other words, a Stokes-Dirac structure requires the specification of boundary variables in order to express a general power conservation property for open physical systems. This assumption is based on the so-called Stokes’ theorem (also known as the divergence theorem, Gauss’s theorem or Ostrogradsky’s theorem) and its corollaries, as the Green’s formula.
Assumption 1 (Abstract Green’s formula) The operator L is assumed to satisfy the abstract Green’s formula:
〈 L e 1 , e 2 〉 L 2 ( Ω , B ) − 〈 e 1 , L * e 2 〉 L 2 ( Ω , A ) = 〈 Γ 0 e 1 , Γ ⊥ e 2 〉 V ∂ , ( V ∂ ) ′ , ∀ e 1 ∈ H L , e 2 ∈ H − L * , (7)
where the right-hand side is the duality bracket at the boundary, on a well-suited boundary functional space V ∂ for some trace operators Γ 0 , Γ ⊥ . From now on, this duality bracket will be denoted by 〈 ⋅ , ⋅ 〉 ∂ Ω with a slight abuse of notation.
Remark 3. This abstract formula is well-known in the boundary control systems theory, see e.g. ( [29], Chapter 10).
Remark 4. In practice, Equation (7) dictates the causalities, i.e. the possible choices for the boundary control u ∂ and the boundary observation y ∂ , via the equality 〈 Γ 0 e 1 , Γ ⊥ e 2 〉 ∂ Ω = 〈 u ∂ , y ∂ 〉 ∂ Ω (with a slight abuse of notation for the right-hand side to make sense). Of course, the admissible causalities are also related to the well-posedness of the system under study, and in particular to the definitions of the boundary functional spaces.
For sake of simplicity, a focus on the two following causalities will be made. Let the boundary variables associated to system (5) be defined by:
e ∂ = Γ ⊥ e 2 ∈ ( V ∂ ) ′ , f ∂ = − Γ 0 e 1 ∈ V ∂ , (8)
or the other way:
e ∂ = Γ 0 e 1 ∈ V ∂ , f ∂ = − Γ ⊥ e 2 ∈ ( V ∂ ) ′ . (9)
In light of (7), systems:
( f 1 f 2 ) = [ 0 − L * L 0 ] ( e 1 e 2 ) , ( e ∂ f ∂ ) = [ 0 Γ ⊥ − Γ 0 0 ] ( e 1 e 2 ) , (10)
and:
( f 1 f 2 ) = [ 0 − L * L 0 ] ( e 1 e 2 ) , ( e ∂ f ∂ ) = [ Γ 0 0 0 − Γ ⊥ ] ( e 1 e 2 ) , (11)
define Stokes-Dirac structures with respect to the bilinear pairing:
〈 〈 ( f 1 1 , f 2 1 , f ∂ 1 , e 1 1 , e 2 1 , e ∂ 1 ) , ( f 1 2 , f 2 2 , f ∂ 2 , e 1 2 , e 2 2 , e ∂ 2 ) 〉 〉 = 〈 f 1 1 , e 1 2 〉 L 2 ( Ω , A ) + 〈 f 2 1 , e 2 2 〉 L 2 ( Ω , B ) + 〈 f 1 2 , e 1 1 〉 L 2 ( Ω , A ) + 〈 f 2 2 , e 2 1 〉 L 2 ( Ω , B ) + 〈 f ∂ 1 , e ∂ 2 〉 ∂ Ω + 〈 f ∂ 2 , e ∂ 1 〉 ∂ Ω .
Obviously, for systems (10) and (11) to be well-defined, constitutive relations are needed.
In the remaining part of this section, only (10) will be treated in details. The other canonical causality (11), figuring in Section 3.4, straightforwardly follows with the same strategy.
Hyperbolic systems
In the hyperbolic case, both flows represent the dynamics of the independent energy variables α 1 , α 2 . The Hamiltonian is a generic functional of these variables H = H ( α 1 , α 2 ) . The co-energy variables are by definition the variational derivatives (see e.g. [1] ) of H with respect to the energy variables:
f 1 = ∂ t α 1 , e 1 : = δ α 1 H , f 2 = ∂ t α 2 , e 2 : = δ α 2 H . (12)
Then system (10) possesses the equivalent state representation:
( ∂ t α 1 ∂ t α 2 ) = [ 0 − L * L 0 ] ( δ α 1 H δ α 2 H ) , ( u ∂ y ∂ ) = [ 0 Γ ⊥ Γ 0 0 ] ( δ α 1 H δ α 2 H ) . (13)
It holds u ∂ = e ∂ , y ∂ = − f ∂ . The power balance is naturally embedded in the Stokes-Dirac structure defined by (10):
d d t H ( α 1 , α 2 ) = 〈 y ∂ , u ∂ 〉 ∂ Ω . (14)
Linear hyperbolic systems
The system is linear when the Hamiltonian has the form:
H ( α 1 , α 2 ) = 1 2 〈 α 1 , Q 1 α 1 〉 L 2 ( Ω , A ) + 1 2 〈 α 2 , Q 2 α 2 〉 L 2 ( Ω , B ) ,
where Q 1 : L 2 ( Ω , A ) → L 2 ( Ω , A ) , Q 2 : L 2 ( Ω , B ) → L 2 ( Ω , B ) are positive symmetric operators, bounded from below and above:
m 1 I A ≼ Q 1 ≼ M 1 I A , m 2 I B ≼ Q 2 ≼ M 2 I B ,
m 1 > 0 , m 2 > 0 , M 1 > 0 , M 2 > 0,
with I A and I B the identity operators in L 2 ( Ω , A ) and L 2 ( Ω , B ) respectively. In this case, the co-energy variables are given by:
e 1 : = δ α 1 H = Q 1 α 1 , e 2 : = δ α 2 H = Q 2 α 2 . (15)
Since Q 1 , Q 2 are positive and bounded from below and above, it is possible to invert them to obtain:
α 1 = Q 1 − 1 e 1 = : M 1 e 1 , α 2 = Q 2 − 1 e 2 = : M 2 e 2 , (16)
giving rise to the co-energy formulation. The Hamiltonian is rewritten as:
H ( e 1 , e 2 ) = 1 2 〈 e 1 , M 1 e 1 〉 L 2 ( Ω , A ) + 1 2 〈 e 2 , M 2 e 2 〉 L 2 ( Ω , B ) (17)
and a linear hyperbolic pHs (10) can be expressed as:
[ M 1 0 0 M 2 ] ( ∂ t e 1 ∂ t e 2 ) = [ 0 − L * L 0 ] ( e 1 e 2 ) , ( u ∂ y ∂ ) = [ 0 Γ ⊥ Γ 0 0 ] ( e 1 e 2 ) . (18)
In this particular case, the constitutive relations needed for system (10) to be well-defined are given by (15), and then directly included in (18). In Sections 3.2 and 3.3, it will be shown that PFEM leads directly to a finite-dimensional pHs of the form (1) with R ( x ) = 0 . This simplification considerably facilitates the solution in time, as (1) is an Ordinary Differential Equation (ODE). Indeed, some dissipation phenomena will induce a differential-algebraic structure on the problem, for example dissipative effects due to an unbounded (e.g. elliptic) operator (see Remark 5).
Parabolic systems
In this case, the first flow f 1 still represents a dynamics ∂ t α 1 of the energy variable α 1 . The Hamiltonian then reads H = H ( α 1 ) , and its variational derivative gives the co-energy variable e 1 = δ α 1 H .
The second flow f 2 represents an extra flow related to the effort variable e 2 appearing in the dynamics of the energy variable α 1 . The relation is given implicitly by a mapping G as G ( f 2 , e 2 ) = 0 . Then, pHs (10) of parabolic type is expressed as:
( ∂ t α 1 f 2 ) = [ 0 − L * L 0 ] ( δ α 1 H e 2 ) , ( u ∂ y ∂ ) = [ 0 Γ ⊥ Γ 0 0 ] ( δ α 1 H e 2 ) , G ( f 2 , e 2 ) = 0. (19)
In Section 3.4, an example of a parabolic-type pHs (11) is studied. It will be shown that the PFEM structure-preserving discretization of such a system naturally leads to a finite-dimensional pHDAE. Again, the power balance is naturally embedded in the Stokes-Dirac structure defined by (10):
d d t H ( α 1 ) = − 〈 f 2 , e 2 〉 L 2 ( Ω ) + 〈 y ∂ , u ∂ 〉 ∂ Ω . (20)
In practice, this becomes explicit with the constitutive relation G ( f 2 , e 2 ) = 0 as it will be seen in Section 3.4 (and more generally in [9] [10] ). Note that this latter relation has to be accurately discretized to ensure that the discretized power balance mimics the continuous one.
Remark 5. By adding resistive port(s), dissipation(s) can easily be taken into account (both internal or at the boundary), as done in the finite-dimensional case via R ( x ) playing the role of a output feedback gain matrix. In this case, the system becomes a parabolic system, the dissipative constitutive relation being represented by G . See [30] [31] for a detailed discussion about structure-preserving discretization of dissipative systems.
3. The Partitioned Finite Element Method (PFEM)We are now in a position to introduce a general methodology to discretize infinite-dimensional pHs in a structure-preserving manner. The main contribution in this section is the application of PFEM to a general abstract class of pHs, unifying the previously published results. This generality is notably of particular interest for the development of a well-structured software for the numerical simulations of physics-based models. The power balances (14), (20) are deeply linked to a linear underlying Stokes-Dirac. The main idea of PFEM is to mimic this structure, in order to obtain a discretized copy of these power balances as (2). This systematically translates the Stokes-Dirac structure into a finite-dimensional Dirac structure. The compatible discretization, with respect to this Dirac structure, of the constitutive relations allows to mimic the continuous power-balance. This method goes under the name Partitioned Finite Element Method (PFEM), and was originally presented in [8]. The procedure is a natural extension of MFEM to pHs and boils down to these three simple steps:
1) System (5) is written in weak form;
2) The integration by parts (7) is carried out on a partition of the system (5) to make the appropriate boundary control appear;
3) A Galerkin method is employed to obtain a finite-dimensional system. For the approximation basis, the finite element method is used here but spectral methods can be chosen as well.
This strategy of structured discretization in order to mimic the continuous power balance at the discrete level has been addressed for closed abstract linear hyperbolic systems in [21]. This pioneering work already proposed the key point of PFEM: the integration by parts on a partition of the weak formulation of the system. The author called the obtained systems primal-dual or dual-primal formulation, depending on which line is integrated by parts. In the port-Hamiltonian formalism, systems are opened with control and observation. It appears that [21] admits PFEM as a generalization for structure-preserving space discretization. The choice of a control in the pHs community is called a causality, and primal-dual or dual-primal correspond in this work to the canonical causalities (10) and (11) respectively.
3.1. General StrategyConsider smooth test functions v 1 and v 2 and the weak form of (5):
〈 v 1 , f 1 〉 L 2 ( Ω , A ) = 〈 v 1 , − L * e 2 〉 L 2 ( Ω , A ) , 〈 v 2 , f 2 〉 L 2 ( Ω , B ) = 〈 v 2 , L e 1 〉 L 2 ( Ω , B ) . (21)
Next the integration by parts is performed either on the first or on the second line (the system is partitioned), depending on the causality.
Integration by parts of the term 〈 v 1 , − L * e 2 〉 L 2 ( Ω , A )
In this case case, using (7), it is obtained:
− 〈 v 1 , L * e 2 〉 L 2 ( Ω , A ) = − 〈 L v 1 , e 2 〉 L 2 ( Ω , B ) + 〈 Γ 0 v 1 , Γ ⊥ e 2 〉 ∂ Ω .
The boundary variable e ∂ : = Γ ⊥ e 2 in (10) explicitly appears. Then the equation defining the corresponding f ∂ : = Γ 0 e 1 is put into weak form to obtain the final system for all smooth test functions v 1 , v 2 , and v ∂ :
〈 v 1 , f 1 〉 L 2 ( Ω , A ) = − 〈 L v 1 , e 2 〉 L 2 ( Ω , B ) + 〈 Γ 0 v 1 , e ∂ 〉 ∂ Ω , 〈 v 2 , f 2 〉 L 2 ( Ω , B ) = 〈 v 2 , L e 1 〉 L 2 ( Ω , B ) , 〈 v ∂ , f ∂ 〉 ∂ Ω = − 〈 v ∂ , Γ 0 e 1 〉 ∂ Ω . (22)
Now, a Galerkin discretization is introduced. Test, energy and co-energy functions with the same subscript are discretized using the same basis, for all t ≥ 0 :
□ 1 ( t , x ) ≈ □ 1 d ( t , x ) : = ∑ i = 1 N 1 φ 1 i ( x ) □ 1 i ( t ) , ∀ x ∈ Ω , □ 2 ( t , x ) ≈ □ 2 d ( t , x ) : = ∑ i = 1 N 2 φ 2 i ( x ) □ 2 i ( t ) , ∀ x ∈ Ω , □ ∂ ( t , s ) ≈ □ ∂ d ( t , s ) : = ∑ i = 1 N ∂ ψ ∂ i ( s ) □ ∂ i ( t ) , ∀ s ∈ ∂ Ω , (23)
where □ stands for v, f, and e and φ 1 i ∈ H L , φ 2 i ∈ L 2 ( Ω , B ) , and ψ ∂ i ∈ L 2 ( ∂ Ω , ℝ m ) .
Remark 6. In general, a discretization in the same basis of either ( f 1 , e 1 ) and ( f 2 , e 2 ) or ( f 1 , f 2 ) and ( e 1 , e 2 ) (as done in [16] ) must be performed. The former is our choice since it directly leads to square mass matrices, while the latter may be more appropriate when dealing for instance with Maxwell’s equations for electromagnetics, see [32] and references therein for details on the difficulties that may then occur.
Then plugging the approximations into (22), it is computed:
[ M 1 0 0 0 M 2 0 0 0 M ∂ ] ( f 1 f 2 f ∂ ) = [ 0 − D L Τ B ⊥ D L 0 0 − B ⊥ Τ 0 0 ] ( e 1 e 2 e ∂ ) , (24)
where vectors f 1 , f 2 , e 1 , e 2 , f ∂ , and e ∂ are given by the column-wise concatenation of the respective degrees of freedom of f 1 d , f 2 d , e 1 d , e 2 d , f ∂ d , and e ∂ d , and where the matrices are defined as follows:
M 1 i j = 〈 φ 1 i , φ 1 j 〉 L 2 ( Ω , A ) , M 2 m n = 〈 φ 2 m , φ 2 n 〉 L 2 ( Ω , B ) , M ∂ l k = 〈 ψ ∂ l , ψ ∂ k 〉 ∂ Ω , D L m i = 〈 φ 2 m , L φ 1 i 〉 L 2 ( Ω , B ) , B ⊥ i k = 〈 Γ 0 φ 1 i , ψ ∂ k 〉 ∂ Ω , (25)
where 1 ≤ i , j ≤ N 1 , 1 ≤ m , n ≤ N 2 , and 1 ≤ l , k ≤ N ∂ . System (24) is a kernel representation of a Dirac structure as in (3) (see Remark 1).
Remark 7. Note that matrices D L and B ⊥ are not square.
The discrete Hamiltonian is naturally defined as the continuous one evaluated in the discrete energy variables. As done in Section 2, it is easier to distinguish the linear hyperbolic from the parabolic case.
Hyperbolic case
In this setting, the flows f i , i = 1,2 , are given by the time derivative of the energy variables α i . Hence, the discretization of these energy variables is given by:
α i d ( t , x ) = α i d ( 0, x ) + ∫ 0 t f i d ( s , x ) d s , i = 1,2.
The discrete Hamiltonian is then defined by H d ( α _ 1 , α _ 2 ) : = H ( α 1 d , α 2 d ) , where α _ 1 and α _ 2 are the column-wise concatenation of the time varying coefficients of α 1 d and α 2 d in their respective basis.
Definition 2. The discretization of the constitutive relations is said to be compatible if and only if:
∇ H d = ( ∇ α _ 1 H d ∇ α _ 2 H d ) = [ M 1 0 0 M 2 ] ( e 1 e 2 ) .
Proposition 1. If the discretisation of the constitutive relations is compatible, the discrete power balance reads at the discrete level:
d d t H d = − ( e ∂ ) Τ M ∂ f ∂ ,
which perfectly mimics the continuous identity.
Proof 1. A straightforward computation gives:
d d t H d = ( ∇ α _ 1 H d ) Τ α ˙ _ 1 + ( ∇ α _ 2 H d ) Τ α ˙ _ 2 = ( M 1 e 1 ) Τ f 1 + ( M 2 e 2 ) Τ f 2 = ( e 1 ) Τ M 1 f 1 + ( e 2 ) Τ M 2 f 2 = − ( e ∂ ) Τ M ∂ f ∂ ,
where the symmetry of the mass matrices and the Dirac structure have been used.
Remark 8. In the special case of linear hyperbolic systems, it has been seen that the co-energy formulation allows to take the constitutive relations into account directly in the differential equations. Applying PFEM to (18) then leads to an ODE, and the constitutive relations are then automatically discretized in a compatible manner.
Parabolic case
In this setting, only the flow f 1 is the time derivative of the energy variable α 1 . This energy variable is discretized as in the hyperbolic case. The discrete Hamiltonian is then defined by H d ( α _ 1 ) : = H ( α 1 d ) .
Definition 3. The discretization of the constitutive relation is said to be compatible if and only if: ∇ H d = M 1 e 1 .
Proposition 2. If the discretisation of the constitutive relations is compatible, the discrete power balance reads at the discrete level:
d d t H d = − e 2 Τ M 2 f 2 − e ∂ Τ M ∂ f ∂ ,
which perfectly mimics the continuous identity.
Proof 2. The proof can be derived similarly as in the hyperbolic case.
Remark 9. Of course, an accurate discretization of the implicit constitutive relation G ( f 2 , e 2 ) = 0 is also required to conclude. This will be illustrated in Section 3.4.
Integration by parts of the term 〈 v 2 , L e 1 〉 L 2 ( Ω , B )
Using (7), it comes:
〈 v 2 , L e 1 〉 L 2 ( Ω , B ) = 〈 L * v 2 , e 1 〉 L 2 ( Ω , A ) + 〈 Γ ⊥ v 2 , Γ 0 e 1 〉 ∂ Ω .
Now the boundary variable e ∂ : = Γ 0 e 1 explicitly appears, i.e. the causality considered in (11). The weak formulation then reads:
〈 v 1 , f 1 〉 L 2 ( Ω , A ) = − 〈 v 1 , L * e 2 〉 L 2 ( Ω , A ) , 〈 v 2 , f 2 〉 L 2 ( Ω , B ) = 〈 L * v 2 , e 1 〉 L 2 ( Ω , A ) − 〈 Γ ⊥ v 2 , e ∂ 〉 ∂ Ω , 〈 v ∂ , f ∂ 〉 ∂ Ω = 〈 v ∂ , Γ ⊥ e 2 〉 ∂ Ω . (26)
Plugging the approximations (23) into (26), this time with φ 1 i ∈ L 2 ( Ω , A ) , φ 2 i ∈ H − L * , and ψ ∂ i ∈ L 2 ( ∂ Ω , ℝ m ) , gives the following kernel representation of a finite-dimensional Dirac structure:
[ M 1 0 0 0 M 2 0 0 0 M ∂ ] ( f 1 f 2 f ∂ ) = [ 0 D − L * 0 − D − L * Τ 0 B 0 0 − B 0 Τ 0 ] ( e 1 e 2 e ∂ ) , (27)
where the matrices D − L * and B 0 are defined by:
D − L * i m = 〈 φ 1 i , − L ∗ φ 2 m 〉 L 2 ( Ω , A ) , B 0 m k = 〈 Γ ⊥ φ 2 m , ψ ∂ k 〉 ∂ Ω . (28)
The power balances proven above still hold true with this causality, where the role played by e ∂ and f ∂ have been switched.
In the sequel, this methodology is applied to the wave equation, the Mindlin-Reissner plate model and the heat equation. These models have been chosen to demonstrate the versatility of our methodology. The wave equation is the prototype of linear hyperbolic systems, and the first example treated by PFEM [8]. The Mindlin model combines wave dynamics and plane elastodynamics, and requires the introduction of tensor-valued variables. Finally, the heat equation is the prototype of parabolic systems, and leads to a pHs with intrinsic algebraic constraint, namely, to a pHDAE.
3.2. The Wave EquationThe wave equation is a well-known model, used as the first example of linear hyperbolic systems in many lecture notes and books. This work is no exception to the rule. However, to account for more realistic physics, let us consider the heterogeneous and anisotropic multidimensional wave equation. The equation reads (see [33] ):
ρ ∂ 2 w ∂ t 2 = div ( T grad ( w ) ) , ( x , t ) ∈ Ω × [ 0, t end ] , Ω ⊂ ℝ N , (29)
where ρ is the mass density (bounded from above and below), T is the tensorial Young modulus (symmetric and positive definite) and w is the deflection from the equilibrium.
Let us denote α p : = ρ ∂ w ∂ t the linear momentum and α q : = grad ( w ) the
strain, as energy variables. Hence the Hamiltonian is given as the total energy (summing kinetic and potential energies) by:
H = 1 2 ∫ Ω { α p 2 ρ + α q ⋅ ( T α q ) } d Ω . (30)
The co-energy variables are by definition the variational derivatives of H with respect to the energy variables, i.e.:
e p : = δ α p H = α p ρ = w t , e q : = δ α q H = T α q = T grad ( w ) , (31)
the velocity and stress. With these notations, Equation (29) rewrites:
∂ ∂ t ( α p α q ) = [ 0 div grad 0 ] ( e p e q ) , (32)
together with the constitutive relations given in (31).
Let us denote:
e 1 : = e p , e 2 : = e q , M 1 : = ρ , M 2 : = T − 1 ,
L : = grad , Γ ⊥ : = γ ⊥ = γ 0 ⋅ n , Γ 0 : = γ 0 ,
where γ 0 is the Dirichlet trace operator. Then the Hamiltonian (30) rewrites as (17). The wave equation (29) with Neumann boundary control u ∂ = γ ⊥ ( e q ) is given by (18).
The application of PFEM directly gives:
[ M ρ 0 0 M T − 1 ] d d t ( e 1 e 2 ) = [ 0 − D grad Τ D grad 0 ] ( e 1 e 2 ) + [ B ⊥ 0 ] u ∂ M ∂ y ∂ = [ B ⊥ Τ 0 ] ( e 1 e 2 ) , (33)
where:
M ρ i j : = 〈 φ 1 i , ρ φ 1 j 〉 L 2 ( Ω , ℝ ) , M T − 1 k l : = 〈 φ 2 k , T − 1 φ 2 l 〉 L 2 ( Ω , ℝ N ) ,
are the discretizations of the operators � 1 and � 2 respectively.
3.3. The Mindlin Plate ModelThe Mindlin model is a generalization to the 2D case of the Timoshenko beam model and is expressed by a system of two coupled PDEs (see [34] ):
( ρ b ∂ 2 w ∂ t 2 = div ( q ) + f , ( x , t ) ∈ Ω × [ 0, t end ] , Ω ⊂ ℝ 2 , ρ b 3 12 ∂ 2 θ ∂ t 2 = q + Div ( M ) + t , (34)
where ρ is the mass density, b the plate thickness, w the vertical displacement, θ = ( θ x , θ y ) Τ collects the deflection of the cross section along axes x and y respectively. The fields f , τ represent distributed forces and torques. Variables M , q represent the momenta tensor and the shear stress. Hooke’s law relates those to the curvature tensor and shear deformation vector:
M : = D b K ∈ S , q : = D s γ , K : = Grad ( θ ) ∈ S , γ : = grad ( w ) − θ .
D s : = E Y b K sh 2 ( 1 + ν ) is the shear rigidity coefficient, where E Y is the Young modulus, ν is the Poisson modulus, K sh is the shear correction factor. Tensor D b is the bending stiffness:
D b ( ⋅ ) = E Y b 3 12 ( 1 − ν 2 ) [ ( 1 − ν ) ( ⋅ ) + ν Tr ( ⋅ ) ] . (35)
An appropriate selection of the energy variables is the following [7] [35]:
α w = ρ b ∂ t w , α θ : = ρ b 3 12 ∂ t θ , A κ : = K , α γ : = γ . (36)
The Hamiltonian H (total energy) is expressed in terms of energy variables as:
H = 1 2 ∫ Ω { 1 ρ b α w 2 + 12 ρ b 3 ‖ α θ ‖ 2 + A κ : ( D b A κ ) + D s ‖ α γ ‖ 2 } Ω , (37)
where A : B denotes the tensor contraction. The co-energy variables are defined as follows:
e w : = δ α w H = ∂ t w , e θ : = δ α θ H = ∂ t θ , E κ : = δ A κ H = M , e γ : = δ α γ H = q . (38)
System (34) is then expressed in port-Hamiltonian form as [7] (forces and torques have been omitted for simplicity):
∂ ∂ t ( α w α θ A κ α γ ) = [ 0 0 0 div 0 0 Div I 2 × 2 0 Grad 0 0 grad − I 2 × 2 0 0 ] ( e w e θ E κ e γ ) . (39)
By applying the divergence theorem, the energy rate is expressed as the duality product of the boundary variables:
d H d t = 〈 ∂ t α w , e w 〉 L 2 ( Ω , ℝ ) + 〈 ∂ t α θ , e θ 〉 L 2 ( Ω , ℝ 2 ) + 〈 ∂ t A κ , E κ 〉 L 2 ( Ω , S ) + 〈 ∂ t α γ , e γ 〉 L 2 ( Ω , ℝ 2 ) , = 〈 γ 0 e w , γ ⊥ e γ 〉 ∂ Ω + 〈 γ 0 e θ , γ ⊥ E κ 〉 ∂ Ω = 〈 y ∂ ,1 , u ∂ ,1 〉 ∂ Ω + 〈 y ∂ ,2 , u ∂ ,2 〉 ∂ Ω , (40)
where:
u ∂ ,1 = γ ⊥ e γ , u ∂ ,2 = γ ⊥ E κ , y ∂ ,1 = γ 0 e w , y ∂ ,2 = γ 0 e θ .
The traces γ 0 u = u | ∂ Ω , γ ⊥ U = U ⋅ n | ∂ Ω correspond to the Dirichlet trace for ℝ d vectors and to the normal trace for ℝ d × d tensors. The mass operators are given by:
M 1 = [ ρ b 0 0 ρ b 3 12 ] , M 2 = [ D b − 1 0 0 D s − 1 ] . (41)
The L , Γ 0 , and Γ ⊥ operators are:
L = [ 0 Grad grad − I 2 × 2 ] , Γ 0 = [ γ 0 0 0 γ 0 ] , Γ ⊥ = [ 0 γ ⊥ γ ⊥ 0 ] . (42)
Introducing the approximations for the test and co-energy variables:
Δ w = ∑ i = 1 N w ϕ w i Δ w i , Δ θ = ∑ i = 1 N θ ϕ θ i Δ θ i , Δ κ = ∑ i = 1 N κ Φ κ i Δ κ i , Δ γ = ∑ i = 1 N γ ϕ γ i Δ γ i , (43)
where Δ = { v , e } , and for the boundary controls:
□ ∂ ,1 = ∑ i = 1 N ∂ ,1 ψ ∂ ,1 i □ ∂ ,1 i , □ ∂ ,2 = ∑ i = 1 N ∂ ,2 ψ ∂ ,2 i □ ∂ ,2 i , □ = { v , u , y } , ψ ∂ ,1 i ∈ ℝ , ψ ∂ ,2 i ∈ ℝ 2 , (44)
PFEM can be applied to obtain:
Diag [ M ρ h M I θ M D b − 1 M D s − 1 ] ( e ˙ w e ˙ θ e ˙ κ e ˙ γ ) = [ 0 0 0 − D grad Τ 0 0 − D Grad Τ − D 0 Τ 0 D Grad 0 0 D grad D 0 0 0 ] ( e w e θ e κ e γ ) + [ B ⊥ , w 0 0 B ⊥ , θ 0 0 0 0 ] ( u ∂ ,1 u ∂ ,2 ) , Diag [ M ∂ ,1 M ∂ ,2 ] ( y ∂ ,1 y ∂ ,2 ) = [ B ⊥ , w Τ 0 0 0 0 B ⊥ , θ Τ 0 0 ] ( e w e θ e κ e γ ) . (45)
The notation Diag denotes a block diagonal matrix. The mass matrices M ρ h , M I θ , M C b , M C s are computed as:
M ρ h i j = 〈 ϕ w i , ρ h ϕ w j 〉 L 2 ( Ω ) , M I θ m n = 〈 ϕ κ m , I θ ϕ κ n 〉 L 2 ( Ω , ℝ 2 ) , M D b − 1 p q = 〈 Φ κ p , D b − 1 Φ κ q 〉 L 2 ( Ω , ℝ sym 2 × 2 ) , M D s − 1 r s = 〈 ϕ γ r , D s − 1 ϕ γ s 〉 L 2 ( Ω , ℝ 2 ) , (46)
where i , j ∈ { 1, N w } , m , n ∈ { 1, N θ } , p , q ∈ { 1, N κ } , r , s ∈ { 1, N γ } . Matrices D grad , D Grad , D 0 assume the form:
D grad r j = 〈 ϕ γ r , grad ϕ w j 〉 L 2 ( Ω , ℝ 2 ) , D Grad p n = 〈 Φ κ p , Grad ϕ θ n 〉 L 2 ( Ω , ℝ sym 2 × 2 ) , D 0 r n = − 〈 ϕ γ r , ϕ θ n 〉 L 2 ( Ω , ℝ 2 ) . (47)
Matrices B w , B θ , M ∂ ,1 , M ∂ ,2 are computed as:
B ⊥ , w i g = 〈 γ 0 ϕ w i , ψ ∂ ,1 g 〉 ∂ Ω , B ⊥ , θ m g = 〈 γ 0 ϕ θ m , ψ ∂ ,2 g 〉 ∂ Ω , M ∂ ,1 f g = 〈 ψ ∂ ,1 f , ψ ∂ ,1 g 〉 L 2 ( ∂ Ω , ℝ m ) , M ∂ l k = 〈 ψ ∂ ,2 l , ψ ∂ ,2 k 〉 L 2 ( ∂ Ω , ℝ m ) , f , g ∈ { 1, N ∂ ,1 } , l , k ∈ { 1, N ∂ ,2 } (48)
The discrete Hamiltonian is then computed as:
H d = 1 2 e w Τ M ρ h e w + 1 2 e θ Τ M I θ e θ + 1 2 e κ Τ M D b − 1 e κ + 1 2 e γ Τ M D s − 1 e γ . (49)
From system (45) the discrete energy rate is readily obtained:
d H d d t = y ∂ ,1 Τ M ∂ ,1 u ∂ ,1 + y ∂ ,2 Τ M ∂ ,2 u ∂ ,2 . (50)
The discrete energy rate then mimics its infinite-dimensional counterpart.
Remark 10. Equivalently a purely mixed formulation can be obtained by integrating by parts the third and fourth lines of (39). In this case, the system of equations gathers together a plane elasticity problem [36] and a wave equation in mixed form. Conforming finite elements for the plane elasticity system on simplicial meshes have been constructed in [37]. The simpler PEERS elements based on a weak symmetry formulation have been proposed in [38]. The PEERS elements have been used in [39] to construct a stable locking-free mixed formulation for the static Mindlin problem.
3.4. The Heat EquationThe heat equation is the simplest example of parabolic system. Instead of rewriting the well-known PDE as a pHs, a direct pHs modelling is presented, as done in [9] [10]. The model is constructed in order to keep apart thermodynamical principles from equations of state. Indeed, the pHs formalism allows to modify the latter, by keeping the structure of the former.
Let Ω ⊂ ℝ N be a bounded open connected set. Assume that this domain models a rigid body: its volume does not change over time and no chemical reaction is to be found. Let us denote: ρ the mass density, u the internal energy density, J Q the heat flux, T the local temperature, β : = 1 T the reciprocal temperature, s the entropy density, J S : = β J Q the entropy flux, C V : = ( d u d T ) V the isochoric heat capacity.
The first law of thermodynamic reads:
ρ ∂ u ∂ t = − div ( J Q ) . (51)
Under the hypothesis of an inert rigid solid, Gibbs formula reads d u = T d s , giving:
∂ u ∂ t = T ∂ s ∂ t . (52)
Defining σ : = grad ( β ) J Q , and seeingu as a function of the entropy density s, Gibbs formula (52) gives:
ρ ∂ s ∂ t = − div ( J S ) + σ . (53)
Then σ is the irreversible entropy production.
In this work, the following constitutive equations of state will be assumed:
· The rigid body is at room temperature: the Dulong-Petit model is supposed to be satisfied, i.e. u = C V T , with time-invariant C V ;
· The thermal conduction is given by Fourier’s law, with a symmetric positive tensor λ : J Q = − λ grad ( T ) .
Thanks to (51) and the equations of state, we easily recover the classical PDE for the temperature T: ρ C V ∂ T ∂ t = div ( λ grad ( T ) ) .
The “L2-energy” ( ∫ Ω ρ C V T 2 d Ω ) 1 2 is commonly used as Hamiltonian for the heat equation. However, it lacks of a thermodynamical meaning. The internal energy would be more accurate for this physical problem, even though it will rise some difficulties. Nevertheless, the pHs formalism allows dealing with it, and PFEM proves to be powerful enough to discretize the system in a structure-preserving manner even for this choice of Hamiltonian.
Let the internal energy be seen as a functional of the local entropy as energy variable: α s : = ρ s , then:
H : = ∫ Ω ρ u ( α s ) d Ω .
The co-energy variable is given by e s : = δ α s H = d ρ u d ρ s = T , the local temperature. Denoting e S : = J S , one can introduce a new flow variable f S such that:
( ∂ α s ∂ t f S ) = [ 0 − div − grad 0 ] ( e s e S ) + ( σ 0 ) .
Obviously, f S = − grad ( e s ) . In order to get a formally skew-symmetric operator, let us also introduce an entropy port ( f σ , e σ ) , such that e σ = − σ . Then:
( ∂ α s ∂ t f S f σ ) = [ 0 − div − 1 − grad 0 0 1 0 0 ] ( e s e S e σ ) . (54)
Remark 11. As surprising as it can be, in this setting, Fourier’s law appears to be stated in a nonlinear way: e s e S − λ f S = 0 . This comes from the necessity to express the constitutive relations in function of the flows and efforts appearing in the equation defining the Stokes-Dirac structure.
Remark 12. Two variables have been added to obtain (54), but only one equation naturally appears: f σ = e s . Thus, another equation is needed to close the system: G ( f 2 , e 2 ) = 0 . Here, it is given by the definition of the irreversible entropy production σ : = grad ( β ) J Q , rewritten in the flows and efforts variables. This leads to the nonlinear constitutive relation: f S e S + f σ e σ = 0 .
Remark 13. Usually the system energy is taken to be ( ∫ Ω ρ C V T 2 d Ω ) 1 2 , that gives rise to the well-known linear diffusive system. At first glance, our approach may be surprising since it leads to a lossless nonlinear differential-algebraic system. There is indeed a major advantage in doing so, in view of the modelling and discretization of complex systems by interconnection of several pHs. For instance, if one wants to interconnect both thermal and mechanical processes, for the energy exchanges to be consistent, physics must be coherent. When dissipation occurs, through e.g. friction or viscosity, the kinetic energy is converted into internal energy. Hence, for a physically meaningful system, the internal energy is indeed the one to consider.
Let us define:
f 1 : = ∂ α s ∂ t , f 2 : = ( f S f σ ) , e 1 : = e s , e 2 : = ( e S e σ ) ,
and:
L : = ( − grad 1 ) , Γ ⊥ : = ( − γ ⊥ 0 ) , Γ 0 : = γ 0 .
Then, the heat Equation (54) with boundary control e ∂ : = u ∂ = Γ 0 e 1 = γ 0 ( e s ) and boundary observation f ∂ : = − y ∂ = Γ ⊥ e 2 = γ ⊥ ( e S ) rewrites under the form (11). Thus PFEM will be applied with an integration by parts on the second line in this strategy, leading to (27), which rewrites with the current variables:
Diag [ M s M S M σ M ∂ ] ( d d t α _ s f S f σ − y ∂ ) = [ 0 D − div − M σ 0 − D − div Τ 0 0 B 0 M σ 0 0 0 0 − B 0 Τ 0 0 ] ( e s e S e σ u ∂ ) , (55)
where:
M s i j : = 〈 φ 1 i , φ 1 j 〉 L 2 ( ∂ Ω , ℝ ) , M S k l : = 〈 φ S k , φ S l 〉 L 2 ( Ω , ℝ N ) , M σ k ˜ l ˜ : = 〈 φ σ k ˜ , φ σ l ˜ 〉 L 2 ( Ω , ℝ N ) , B 0 k m : = − 〈 Γ ⊥ φ S k , ψ ∂ m 〉 L 2 ( ∂ Ω , ℝ ) ,
with φ 2 k : = ( φ S k 0 ) if 1 ≤ k ≤ N S , and φ 2 k : = ( 0 φ σ k − N S ) if N S + 1 ≤ k ≤ N S + N σ .
To be compatible, the discretizations of the constitutive relations are given as follows:
· Dulong-Petit model reads: M s α _ s = M ρ C V e s , with M ρ C V i j : = 〈 φ 1 i , ρ C V φ 1 j 〉 L 2 ( Ω , ℝ ) ;
· Following Remark 11, Fourier’s law reads: Λ f S = M e s e S with Λ i j : = 〈 φ S i , λ φ S j 〉 L 2 ( Ω , ℝ N ) and M e s i j : = 〈 φ S i , e s φ S j 〉 L 2 ( Ω , ℝ N ) ;
· The constitutive law coming from the introduction of the irreversible entropy production, as explained in Remark 12, is taken into account by:
( e S ) Τ M S f S + ( e σ ) Τ M σ f σ = 0. (56)
Remark 14. In Fourier’s law, the mass matrix M e s depends on the co-energy variable e s . This will rise difficulties for the numerical solution in time.
To conclude, the structure-preserving property can be appreciated in the following result.
Proposition 3. Let H d ( α _ s ) : = H ( α s d ) be the discrete Hamiltonian, where α s d is the discretization of the energy variable in the basis φ 1 . It holds:
d d t H d = u ∂ Τ M ∂ y ∂ ,
that is the first law of thermodynamics at the discrete level.
Proof 3. Thanks to the compatible discretization of the Dulong-Petit model, Proposition 2 gives:
d d t H d = − e 2 Τ M 2 f 2 − e ∂ Τ M ∂ f ∂ .
By definition of f 2 , e 2 , M 2 , e ∂ , and f ∂ one computes:
d d t H d = − e 2 Τ M 2 f 2 − e ∂ Τ M ∂ f ∂ , = − ( e S e σ ) Τ [ M S 0 0 M σ ] ( f S f σ ) + u ∂ Τ M ∂ y ∂ , = u ∂ Τ M ∂ y ∂ ,
thanks to the constitutive relation (56) coming from the irreversible entropy production.
Remark 15. Fourier’s law does not contribute to the power balance of the internal energy. Nevertheless, such a constitutive relation is needed for the problem to be well-defined.
Remark 16. The methodology detailed so far is certainly not limited to the previous three examples. Indeed higher-order differential [40], curl operator for Maxwell’s equations [32], nonlinear system [41], and different Hamiltonian choices can be handled as well. For instance, in the case of the heat equation, the entropy or the classical L2-norm of the temperature can be alternatively considered as Hamiltonian functional [9] [10] . In addition, mixed boundary conditions can be incorporated either by introducing Lagrange multipliers or by employing a virtual domain decomposition method [42]. As already mentioned, dissipation (both in the domain and on the boundary) can also be considered in this strategy [30] [31]. Hence a very wide class of (nonlinear) multiphysics systems can be discretized in a structure-preserving manner (with well-represented exchanges of energy between the subsystems).
In the next section we present an ongoing project which has been initiated to prove the efficiency of the PFEM methodology, leveraging well-established and robust software tools for the finite element discretization of partial differential equations and time integration.