A numerical method for finitestrain mechanochemistry with localised chemical reactions treated using a Nitsche approach
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Abstract
In this paper, a novel finiteelement based method for finitestrain mechanochemistry with moving reaction fronts, which separate the chemically transformed and the untransformed phases, is proposed. The reaction front cuts through the finite elements and moves independently of the finiteelement mesh, thereby removing the necessity for remeshing. The proposed method solves the coupled mechanicsdiffusion–reaction problem. In the mechanical part of the problem, the force equilibrium and the displacement continuity conditions at the reaction front are enforced weakly using a Nitschelike method. The formulation is applicable to the case of large deformations and arbitrary constitutive behaviour, and is also consistent with the minimisation of the total potential energy.
Keywords
Chemomechanical processes Finite strains Moving interfaces Nitsche method Cut elements Variational consistency1 Introduction
Chemical reactions, such as oxidation or lithiation, in solid bodies lead to large volumetric expansions of materials and thereby lead to the emergence of mechanical stresses, which, in turn, affect the rates of the chemical reactions. This, for example, has been experimentally observed for the chemical reaction of silicon lithiation [40]. Chemical reactions in solids can be either volumetric or localised at a surface (a chemical reaction front) inside a solid body. In recent years, there has been an emergence of models that describe mechanochemistry of volumetric reactions, e.g. [16, 18], and localised reactions, e.g. [4, 6, 7, 8, 14, 43]. In this paper, the focus is on the latter.
From the physical point of view, in the case of localised reactions, the reaction front moves due to the consumption of the diffusive reactant, which is supplied to the reaction front by a diffusion process. The velocity of the front is also affected by the stresses, which, in turn, emerge due to transformation of the material as the front moves. This problem is similar to the classical stressinduced phase transformations in solids, where the phase boundaries move due to the configurational (driving) force (e.g. [1, 10, 21, 33] and references therein) equal to the jump of the normal component of the Eshelby stress tensor at the phase boundary. A few years ago it was shown that, in the case of a localised chemical reaction, the configurational force is the normal component of a chemical affinity tensor, which is equal to the combination of the chemical potential tensors of the reaction constituents, which, in turn, are equal to the Eshelby stress tensors divided by reference mass densities [6, 7, 8]. Both cases, phase transformations and chemical reactions, can be handled computationally in a similar way.
There are two major approaches to formulating the problem with the localised reactions. The first approach is the phasefield method, e.g. [37] in application to mechanochemistry and [31, 32] in application to the classical phase transformations. In this approach, an auxiliary field quantity is introduced, which takes distinct values inside the phases and smoothly changes between the phases. An additional partial differential equation (PDE) governs the evolution of this field. Thus, the interface^{1} between the phases is of a finite thickness.
In the second approach, the reaction front is prescribed to be infinitely thin (“sharp”), i.e. curve in 2D or surface in 3D. The boundary conditions for the mechanical and/or diffusion equations are enforced at the moving reaction front, the velocity of which depends on the solutions of the equations. This paper focuses on “sharp” interface approach.
It is noteworthy to mention that there are also “hybrid” approaches, such as [44], where the mechanodiffusion with damage was considered, and where both mechanics and diffusion were solved in the entire computational domain; however, an infinitely thin interface was introduced and different material properties were used on the opposite sides of the interface. The kinetics of the interface was governed by the diffusion flux and by the stresses.
One established way of solving mechanochemical and phase transformation problems with sharp interfaces is the boundary integral method, e.g. [15, 35, 36] in application to modelling the formation of precipitates. Although term “mechanochemistry” was not used there, the problem in question was tracking the movement of phase boundaries influenced by both mechanics and diffusion, while the diffusion took place in the “untransformed” material. This approach, however, has only been formulated for linear elasticity and quasisteady state diffusion. Moreover, it is difficult to handle topological changes during phase transformations within this approach.
Another popular computational approach to solving mechanochemical problems is the finiteelement method (FEM). The major problem with the standard FEM is the requirement for the reaction front to coincide with the element edges/faces in 2D/3D. Therefore, the geometry should be remeshed each time the reaction front moves, e.g. [9]. This leads to accumulation of the numerical error and excessive computations. To avoid this, there has been a significant effort to create computational approaches that avoid remeshing.
An established way of modelling mechanochemistry and phase transformations using FEM without remeshing is the combination of the extended finiteelement method (XFEM) to solve the mechanical problem and the levelset method to move the interface, e.g. [45, 46] in application to phase transformations and [5, 47] in application to mechanochemistry. In this case, FEM is modified such that the interface cuts through the elements and moves independently of the mesh. However, at the moment, to the best knowledge of the authors, XFEM and levelset combination has only been formulated for and applied to linear elastic problems in the context of phase transformation problems.
In [22], the application of isogeometric FEM was proposed for linear elastic mechanochemistry. In this approach, the reaction front still coincides with the element edges/faces; however, when the front moves, the elements are simply distorted, i.e. stretched or compressed, without changing topology or connectivity of the mesh. Distortion of the elements is utilised due to the nature of the isogeometric method, where higher aspect ratios of the elements compared to the standard FEM can be handled. The main advantage of this method is the welldefined normal to the reaction front at any point, due to the description of the front by Bsplines. This property is important for calculation of the velocity of the front and moving the interface.
A computational approach for finitestrain mechanochemistry with the reaction fronts that are nonconforming to the finiteelement mesh was first proposed in [27, 28], where the interface motion was handled using the levelset method, while the nonconforming interface was handled by the enhanced gradient FEM in the mechanical and the diffusion problems. The enhanced gradient method avoids remeshing; however, treatment of the elements that are intersected by the interface is relatively complicated. As in XFEM, special finiteelement basis functions are constructed for these elements. These basis functions are piecewise polynomial and can be discontinuous; additional degrees of freedom are found from the interface conditions.
The aim of this paper is to develop a numerical method for finitestrain mechanochemistry that considers interfaces that are nonconforming to the finiteelement mesh, avoids remeshing and is relatively simple to implement, i.e. which does not rely on special finiteelement basis functions. The major difficulty for such approach is the correct enforcement of the interface conditions (displacement and traction continuity). One possible approach is the Nitsche method, which is a way of weakly imposing boundary/interface conditions for linear PDEs, e.g. [11, 12, 30] in application to mechanics with linear elastic constitutive laws. However, an extension of the Nitsche method to nonlinear PDEs is nontrivial and has been less studied. One such extension, which, as explained below, has key differences to previous formulations, is proposed in this paper.
The first attempt of extension of the Nitsche method along with the discontinuous Galerkin method to largedeformation mechanics has been reported in [29]. As the Nitsche penalty terms in the weak form contain arbitrary multipliers, in [29], the first PiolaKirchhoff tensor was used. This, however, is not consistent with the minimisation of the total potential energy, as will be shown in this paper. In [38], it was shown that when the weak form is obtained by finding the saddle point of the total potential energy, the fourthorder acoustic tensor (derivative of the first PiolaKirchhoff tensor by the deformation gradient tensor) appears in the penalty term multiplier.
In this paper, the weak form of the finitestrain mechanical problem is obtained by minimising the total potential energy, which includes the Nitschelike penalty terms. As the interface can cut a small volume fraction of an element in mechanochemical problems, in this paper, additional interelement stabilisation term, similar to one presented in [2, 12, 34], is also introduced. The weak form of the coupled problem is then derived and discretised using finiteelement formulation. Furthermore, in this paper, an algorithm for moving the points of the reaction front in the mechanochemical problem is presented. The method is illustrated with 2D numerical examples to show its numerical robustness in application to finitestrain mechanochemical problems with moving reaction fronts.
2 Notation
3 Mechanochemical problem formulation
In this section, a general mechanochemical problem formulation is presented, in which movement of a chemical reaction front takes place. No specific assumptions about constitutive relations are made below. However, for numerical examples, specific functional dependencies were chosen and are presented in Sect. 5.1.
The solution of the mechanochemical problem consists in finding two timedependent field quantities and one timedependent curve (in a 2D setting) or surface (in a 3D setting). The first unknown field quantity is the displacement of all material points of a body. The second unknown field quantity is the concentration of the reactant. The last unknown is the configuration/position of reaction front. All these quantities are timedependent, since the reaction front moves due to a chemical reaction.
3.1 Configurations and kinematics
The solid body is split into two domains: the chemically transformed and the untransformed phases, which are denoted as \(B_+\) and \(B_\), respectively. Both phases undergo mechanical deformation. Therefore, there are three configurations: the current configuration, the reference configuration of the chemically untransformed material and the reference configuration of the chemically transformed material. The kinematics description used in this paper follows [6, 7, 8].
To perform numerical calculations, it is convenient to work only with one reference configuration. In this paper, the reference configuration of the untransformed material (\(B_\)) is chosen and, in the rest of the text, is denoted as “the reference configuration”. The reference configuration of \(B_+\) is referred to as “the chemically transformed configuration”.
In the current configuration, domains of \(B_+\) and \(B_\) are denoted as \(\omega _+\) and \(\omega _\), respectively. These domains are separated by the reaction front \(\gamma _*\). The position vector in the current configuration is denoted as \(\vec {x} \). Mappings of \(\omega _+\) and \(\omega _\) onto the reference configuration are denoted as \(\varOmega _+\) and \(\varOmega _\), respectively. These domains are separated by the reaction front \(\varGamma _*\). The normal to the reaction front is defined as the outer normal to \(\varOmega _+\) and is denoted as \(\vec {N} _*\). The position vector in the reference configuration is denoted as \(\vec {X} \). Mappings of \(\omega _+\) and \(\omega _\) onto the chemically transformed configuration are denoted as \(\varOmega _+'\) and \(\varOmega _'\), respectively.
Quantities (deformation gradient, stress, displacement, etc.) corresponding to phases \(B_+\) and \(B_\) are denoted with subscripts “\(+\)” and “−”, respectively. To shorten the description in the rest of the text, subscript “±” is used on some occasions to combine equations corresponding to \(B_+\) and \(B_\) into one equation.
3.2 Mechanics
3.2.1 Boundary/interface conditions in the reference configuration
3.3 Diffusion
For the purpose of this paper, the diffusion is assumed to be quasistationary. This assumption is motivated by the fact that in most cases, the rate of the diffusion is much higher than the rate of the chemical reaction [17, 42] and the diffusion process can be assumed to take place at the thermodynamic equilibrium.
3.4 Chemical reaction
The chemical reaction takes place at the reaction front. The diffusive component reacts with the untransformed material, which leads to the formation of a new transformed material. Thereby, the position of the reaction front changes. The kinetics of the reaction front is stress and concentrationdependent and varies for different mechanochemical models. There are models where an expression for the velocity of the reaction front is simply postulated. Recently, an expression for the thermodynamic driving force in the case of a localised chemical reaction has been derived [6, 7, 8]. This allows choosing an expression for the velocity as a function of this force.
It can be seen that the balance of momentum equation, Eq. (5), the mass balance equation, Eq. (15), and the velocity equation, Eq. (20), form a coupled system of PDEs with respect to unknown variables \(\vec {u} \), c and \(\vec {X} _*\). In general, there are two independent coupling mechanisms: via the constitutive equations and via the moving boundary. The first one results from the concentration entering the constitutive equation for the stresses, Eq. (6), and from the stresses entering the constitutive equation for the diffusion flux, Eq. (16). The second coupling mechanism is the dependence of solutions of (5) and (15), \(\vec {u} \) and c, respectively, on the position of the reaction front, which, in turn, depends on both fields. Thus, even if the first mechanism is not present, the system of PDEs is still coupled. Further discussion of the mechanochemical coupling can be found in [26].
4 Numerical method
There are two general ways of dealing with the time stepping for the reaction front movement—explicit and implicit methods. The explicit scheme suggests that the position of the reaction front is known at a given time step. This allows solving the mechanical and the diffusion problems for displacement and concentration fields, calculating the reaction front velocity using these fields and, subsequently, finding the new position of the front using the velocity. In contrast, within the implicit scheme, the position of the front is unknown at a given time step. Therefore, all equations (mechanical, diffusion and front velocity/position) must be solved simultaneously with respect to the unknown displacement, concentration and position of the front. In this work, the explicit time stepping scheme for the interface movement is used to avoid an increase in complexity of numerical implementation (coding) associated with the implicit scheme and, in particular, evaluation of Jacobian components that correspond to degrees of freedom representing the front.
The description of the numerical method is split into three major parts: the mechanics, the diffusion and the movement of the reaction front. As presented in the problem formulation, the mechanical problem and the diffusion problem are fully coupled. Therefore, in the general case, the finiteelement formulation for the fully coupled problem must be formulated. However, for clarity of the presentation of the method, the decoupled case is presented first: the mechanical problem (for the case when stresses do not depend on concentration), followed by the diffusion problem (for the case when the deformation and the stresses are already known). Afterwards, the finiteelement formulation for the fully coupled problem is presented.
From an implementational point of view, when the concentration does not enter the constitutive law for stresses, the mechanical and the diffusion problems can be solved sequentially within a particular time step. This means that the mechanical problem is solved first, followed by the solution of the diffusion problem using known stresses, followed by the calculation of a new position of the reaction front. However, when the concentration enters the constitutive law for stresses, the mechanical and the diffusion problems must be solved as one system of coupled nonlinear equations within a particular time step. After the solution is found, the reaction front is moved.
4.1 Mechanics
At first, the weak form of the mechanical problem is formulated. The idea follows [30], where the potential energy with an interface penalty term is formulated and the variation of the potential energy is performed; however, in this paper, the finitestrain case is considered.
4.1.1 Variation of the potential energy and the weak form
Weak form (33) is similar to the weak form of [38, 39], however, with different interface stabilisation terms. In [38, 39], the stabilised discontinuous Galerkin (DG) method was proposed for large deformations, while in [3], this approach was extended to include models of damage and debonding.
4.1.2 Transformation between configurations for some quantities
4.1.3 Consistency with the strong form
4.1.4 Finiteelement formulation
The finiteelement mesh covers the entire volume of the body \(\varOmega = \varOmega _+ \cup \varOmega _\) and is arbitrary with respect to interface \(\varGamma _*\), i.e. the interface cuts through elements, as illustrated in Fig. 2a. Without loss of generality, the mesh is considered to be conforming to the external boundary of the body, \(\partial \varOmega \). Obviously, it is also possible to formulate the method in the case when the external boundary is also nonconforming to the mesh. However, since the reference configuration is considered and the external boundary does not move in this configuration, in most cases, there will be no practical purpose of creating a mesh that is nonconforming to the external boundary.
Although the above finiteelement formulation follows relatively standard notation accepted in numerical analysis, it should be emphasised that when integrals in (33) are calculated for \(a\left( \vec {u} _+^h, \vec {u} _^h, \vec {\varphi } _+^h, \vec {\varphi } _^h \right) \), the domains, over which the integration is performed, have discretised piecewise linear boundaries. Thus, for the finiteelement formulation, domains of integration \(\varOmega _\pm \), \(\varGamma _*\) and \(\varGamma _\mathrm {T}\) are replaced by \(\varOmega _\pm ^h\), \(\varGamma _*^h\) and \(\varGamma _\mathrm {T}^h\), respectively. Interiors of \(\varOmega _+^h\) and \(\varOmega _^h\) do not overlap and \(\varGamma _*^h = \varOmega _+^h \cap \varOmega _^h\). This is usually taken as obvious; however, in this paper, the distinction is made to separate \(\varGamma _*\) and \(\varGamma _*^h\). The rules, according to which \(\varGamma _*^h\) is constructed, are discussed in Sect. 4.4.1.
4.1.5 Assembling finiteelement equations
As seen from system (44), the actual number of nonlinear equations is \(3\left( N_\mathrm {b}+N_\mathrm {c}N_\mathrm {a}\right) \). Thus, the number of nonlinear equations is increased by \(3\left( N_\mathrm {b}N_\mathrm {a}\right) \), compared to the standard FEM problem (where an interface might be present, but is conforming to the mesh). Thus, compared to the standard FEM, in this approach, number of DOFs, calculation of which is nontrivial, is doubled for all nodes that belong to the elements that are cut by the interface.
As seen from the first equation of system (44), when equations corresponding to \(\vec {U} ^+_i\) are assembled, \(\vec {\varphi } _^h = \vec {0} \) everywhere. Analogously, the second equation of system (44) indicates that when equations corresponding to \(\vec {U} ^_i\) are assembled, \(\vec {\varphi } _+^h = \vec {0} \) everywhere.
There is another possible interpretation of the above finiteelement formulation, which is as follows. There are two different meshes, which consist of sets of elements \(\mathcal {T}_+\) and \(\mathcal {T}_\). These meshes overlap over a set of elements \(\mathcal {T}_+ \cap \mathcal {T}_\). There are two different solutions, \(\vec {U} ^+_i\) and \(\vec {U} ^_i\), each defined on the corresponding mesh. Thus, there are 3 DOFs per node of each mesh. To assemble the system of equations for the mechanical problem, at first, the test function defined on the first mesh, \(\vec {\varphi } _+^h\), takes all possible values, while \(\vec {\varphi } _^h = \vec {0} \); afterwards, the test function defined on the second mesh, \(\vec {\varphi } _^h\), takes all possible values, while \(\vec {\varphi } _+^h = \vec {0} \). Such interpretation is illustrated schematically in Fig. 2b.
4.1.6 Stabilisation term
When interface \(\varGamma _*\) cuts through elements \(\mathcal {T}\), it is possible that some elements are partitioned into highly unequal area fractions, as illustrated in Fig. 2c. This can create a numerical problem—the system of equations can become illconditioned. Therefore, in Nitschelike methods, additional stabilisation terms are introduced in such cases, e.g. [2, 12, 34].
In Eq. (45), \(\varvec{F} _\pm \) is stabilised across the interelement boundary. This choice is rather arbitrary and follows analogous stabilisation term from [2]. For the mechanical problem it is possible to use \(\varvec{P} _\pm \) instead of \(\varvec{F} _\pm \) in Eq. (45), i.e. stabilise stresses instead of deformation gradients; however, this choice of stabilisation term is not investigated in this paper.
4.2 Diffusion
4.2.1 Finiteelement formulation
The calculation of integrals in b requires \(f_*\), which is a function of V, which, in turn, depends on \(\varvec{\sigma } _+\), \(\varvec{F} _+\) and \(W_+\). These quantities result from the solution of the mechanical problem. Moreover, in general case, \(\vec {j} \) depends on \(\varvec{\sigma } _+\) and \(\varvec{F} _+\). Therefore, the above problem formulation is valid for the case when \(\varvec{\sigma } _+\), \(\varvec{F} _+\) and \(W_+\) are already known, i.e. the mechanical problem can be solved first within a particular time step, otherwise, the coupled problem should be solved, as presented in Sect. 4.3. Evaluation of V in the finiteelement representation is discussed in Sect. 4.4.2.
Since in the case of diffusion problem, integrals in (48) are also evaluated in cut elements, similar interelement stabilisation term as in the mechanical problem, Sect. 4.1.6, can be introduced. However, such term was not used in the numerical examples of this paper, as no issues related to the conditionality of the problem were observed.
4.2.2 Assembling finiteelement equations
4.3 Coupled system
4.4 Movement of the reaction front
Reaction front \(\varGamma _*\) is a curve in a 2D setting or a surface in a 3D setting. In the finiteelement formulation, reaction front \(\varGamma _*\) is represented by a piecewise linear continuous curve/surface \(\varGamma _*^h\), which crosses the edges of finite elements. Within each element \(\varGamma _*^h\) is strictly linear. There are two additional requirements imposed on \(\varGamma _*^h\), as presented below. It should be noted that the method can be generalised to the case when \(\varGamma _*^h\) is piecewise polynomial; however, for clarity of the presentation, the piecewise linear case is selected here.
The reaction front is moved using an explicit time stepping, i.e. for a certain position of \(\varGamma _*^h\), the mechanical and the diffusion problems are solved, the current velocities are obtained and the points of the reaction front are moved using these velocities. The general scheme, which starts from the discretisation of a continuous interface, is illustrated in Fig. 3 and further details are provided in subsections below.
4.4.1 Intersection points and surface normals

set \(\mathcal {I}\) is finite;

each element \(E \in \mathcal {T}_*\) contains only a single linear segment of \(\varGamma _*^h\).
The schematic illustration of the effect of these two requirements is demonstrated in Fig. 4, where \(\varGamma _*^h\) does not cover all intersection points of \(\varGamma _*\) and the mesh. In Fig. 4c, the effect of the first constraint is shown—the configuration of the interface shown in black colour is not allowed, as number of coinciding points of the interface and the element edges must be finite. The resulting discretised interface, after forbidden points are excluded, is shown in red colour. In Fig. 4b, the effect of the second constraint is shown—there cannot be two segments of the interface within one element. In Fig. 4a, the typical case when both constraints forbid a certain configuration of the interface is shown. Finally, in Fig. 4d, an interesting effect emerging from these constraints is shown—in some cases, a single interface curve can be separated into several curves, some of which can form closed loops.
It should be mentioned that the rules for discretising the interface are somewhat restrictive in this paper. From the mathematical point of view, it is perfectly allowable that a part of the interface coincides with a side of an element. The only reason for introducing the restriction, which forbids this, as shown in Fig. 4c, is the simplicity and the efficiency of the coding. In practice, occurrence of such case should be extremely rare, as in almost all cases the interface will fall inside the elements. However, from the programming point of view, accounting for such case requires creation of additional structure types for interface segments and coding additional conditional checks. Therefore, from the practical point of view, it is reasonable to exclude the case of Fig. 4c. Moreover, it should also be noted that, since in real applications the interface is expected to be “smooth” (on this occasion, this term is not used in strict mathematical sense, but implies small differences in orientation of line segments), occurrence of a case similar to the ones illustrated in Fig. 4 would mean that the background mesh is too coarse and thus the solution is imprecise.
If an intersection point of \(\varGamma _*^h\) and the mesh is closer than \(\zeta \) to a node, then this intersection point is moved to the location of the node. Here, \(\zeta \) is a very small numerical parameter, which is introduced to avoid a very short segment of the interface influencing the calculation of normals. In the numerical examples of this paper \(\zeta = 10^{6}\) was taken.
For each element \(E \in \mathcal {T}_*\), the normal to \(\varGamma _*^h\) is defined as \(\vec {N} _*^h\) and is the outer normal to \(\varOmega _+^h\). Thus, \(\vec {N} _*^h\) corresponds to \(\vec {N} _*\). Since \(\varGamma _*^h\) is linear within each element, \(\vec {N} _*^h\) is constant within each element.
4.4.2 Points’ velocities
The velocities of the reaction front are calculated at each point \(\vec {P} \in \mathcal {I}\) according to Eq. (20). This requires the calculation of c, \(\varvec{\sigma } _\pm \), \(\varvec{F} _\pm \) and \(W_\pm \) at point \(\vec {P} \). Obviously, some kind of interelement averaging and interpolation can be used to obtain the values of these quantities at point \(\vec {P} \).
In the case of linear finiteelements, concentration c at point \(\vec {P} \) can easily be obtained by linearly interpolating nodal values of \(c^h\) at the element edge, to which point \(\vec {P} \) belongs. Quantities \(\varvec{\sigma } _\pm \), \(\varvec{F} _\pm \) and \(W_\pm \) are defined inside the elements. In this case of linear finite elements, \(\varvec{\sigma } _\pm \), \(\varvec{F} _\pm \) and \(W_\pm \) are constant within an element; therefore, the values at point \(\vec {P} \) can be obtained by simple averaging of the values at neighbouring to \(\vec {P} \) elements \(E_j \in \mathcal {T}_*\). Such averaging scheme is the simplest; a more elaborate interelement stress/energies averaging procedure might improve the overall accuracy of the method.
It should be noted that in this paper, velocity depending on Open image in new window is taken, as presented in Sect. 5.1. Therefore, in the numerical examples of this paper, interelement averaging is applied directly to term Open image in new window .
In the case when the interface crosses the boundary of the body, the intersection points of the interface and the edges of the elements that belong to the boundary of the body, \(\vec {P} \in \mathcal {I} \cap \partial \varOmega ^h\), may have fewer neighbouring elements than the rest of the intersection points. In the 2D case, these intersection points may have only one neighbouring element, while other intersection points have at least two. Therefore, due to a reduced number of available neighbouring elements to perform interelement averaging/interpolation of quantities such as \(\varvec{\sigma } _\pm \) and \(\varvec{F} _\pm \), the accuracy of the calculated velocities at these intersection points may be reduced.
To mitigate the issue of the influence of nonaveraged/noninterpolated quantities on the velocity of the interface, velocities at points \(\vec {P} \in \mathcal {I} \cap \partial \varOmega ^h\) are obtained by extrapolation from velocities at neighbouring intersection points that do not belong to \(\partial \varOmega ^h\). For the extrapolation, V is represented as a function on curve/surface \(\varGamma _*^h\). It should be noted that in the numerical examples of this paper, such extrapolation (using 4 neighbouring points) is applied to quantity q, which is introduced in Sect. 5.1, and not to the velocity.
4.4.3 Moving points and finding new configuration of the interface
Depending on the orientation of the interface and the velocities of the points, it may happen that neither of points \(\vec {P} ^\Delta \in \varOmega ^h\) belongs to the boundary of the body, \(\partial \varOmega ^h\), while \(\varGamma _*^h \cap \partial \varOmega ^h \ne \emptyset \). In this case, extrapolation is used to extend curve/surface \(\hat{\varGamma }_*^h\) upto \(\partial \varOmega ^h\). The 2D case of such extrapolation is shown in Fig. 5c, d. For each end point of \(\hat{\varGamma }_*^h\), a set of N points \(\vec {P} ^\Delta \), which are closest to the end point of \(\hat{\varGamma }_*^h\), are taken and a linear fit for these points is performed. The intersection point of the obtained line and a line parallel to \(\partial \varOmega ^h\) and distanced from it by h is found. Curve \(\hat{\varGamma }_*^h\) is extended to include this intersection point using a linear segment. In the numerical examples of this paper, \(N=4\) was taken.

the set of intersection points of \(\tilde{\varGamma }_*^h\) and element edges is finite;

each element, which is crossed by \(\tilde{\varGamma }_*^h\), contains only a single linear segment of \(\tilde{\varGamma }_*^h\).
5 Numerical examples
Since the purpose of this paper is the presentation of a numerical method, units are omitted for all the quantities, which is common in numerical analysis.
The unit square geometry is selected. The geometry is meshed using linear finite elements composing a structured mesh. All elements of the mesh are isosceles right triangles with side lengths of h. The example of finiteelement mesh will be shown later.
5.1 Constitutive relations for numerical examples
In the numerical examples of this paper, the body is considered to be at a constant temperature. Thus, the temperature only influences the values of physical parameters and does not enter equations explicitly.
In the numerical examples, bulk moduli \(K_+ = K_ = 10\) and shear moduli \(G_+ = G_ = 2\) were taken. Diffusion parameters \(D = s = 1\) and reaction parameters \(p_1 = p_2 = p_3 = p_4 = 1\) were taken. Parameter \(\alpha \) was taken to be different on different edges to take into account different boundary conditions, as described in Sect. 5.4. Chemical expansion ratio was taken as \(g = 1.1\). In the NewtonRaphson method, absolute tolerances for the \(\ell ^\infty \)norms of the function and of the change of the solution were taken to be \(10^{11}\). Numerical parameters \(\lambda \) and \(\kappa \) were varied and are provided below.
5.2 Example of a mechanical problem with stationary nonconforming interface
The purpose of this section is to demonstrate that the mechanical part of the problem for the case of stationary (fixed) interface provides a reliable solution. This is shown by demonstrating the expected convergence with respect to the mesh size and by comparing the nonconforming interface problem with conforming interface problem resolved via standard FEM.
5.2.1 Case 1: flat interface
Since the number of DOFs is doubled for the nodes that belong to the elements, which are intersected by the interface (i.e. the intersected elements are doubled), the full Cauchy stress tensor in the intersected elements was averaged first and the pressure and the von Mises stress were calculated afterwards. This explains the nonphysical stress values in the intersected elements in Fig. 7. Thus, when the nonconforming mesh is used, a somewhat more elaborate stress visualisation technique might be beneficial, where parts of the intersected elements are coloured according to the stresses in the corresponding phases.
Furthermore, the influence of numerical parameters \(\lambda \) and \(\kappa \) on the numerical error was investigated for the the nonconforming interface FEM problem. In Fig. 9a, the influence of \(\lambda \) on the error is shown. For \(\lambda < 10^{1.5}\), the solution sometimes did not converge, hence there are some gaps in the plot. Moreover, for small \(\lambda \), the error somewhat oscillates and levels off for large \(\lambda \). In Fig. 9b, the influence of \(\kappa \) on the error is shown. Obviously, the introduction of the stabilisation term somewhat increases the error; however, it can be seen that for small \(\kappa \) this increase is negligible, while such stabilisation term can be important for cases when the interface cuts a small area fraction in some elements.
5.2.2 Case 2: curved interface
In this example, the interface was taken to be parabolic curve \(X_2 = 11/17  \left( X_11/2\right) ^2\), while the materials below and above the interface were taken to be the transformed and the untransformed materials, respectively. This example is less trivial than the previous example and shows the main advantage of the method—an ability to handle curved interfaces nonconforming to the mesh. Mesh sizes \(h = 1/N\), \(N \in \left\{ 32, 64, 128 \right\} \) were used.
In Fig. 12a–c, parameter \(\lambda \) was varied and it can be observed that for large \(\lambda \), its influence on the convergence rate is negligible. Parameter \(\lambda \) should only be large enough for the solution to converge, as was discussed in Sect. 5.2.1. Parameter \(\kappa \) influences convergence rate in a nonlinear way. As seen in Fig. 12d–f, up to \(\kappa = 10^{1}\) the convergence rate is still around the theoretical convergence value, which is 2; however, for \(\kappa = 1\), the convergence rate drops to 1. This obviously happens due to overconstrained interelement jump of the deformation gradient in the intersected elements, as prescribed in Eq. (45). The main advantage of the interelement stabilisation term is the regularisation of the solution for certain values of \(\kappa \), which can be seen in Fig. 12d–f, where for \(\kappa = 10^{2}\) and \(10^{1}\) the convergence rate becomes more spatially homogeneous than for \(\kappa = 0\) (absence of the interelement stabilisation term).
It can be concluded that there are optimal values of numerical parameters \(\lambda \) and \(\kappa \). Parameter \(\lambda \) is the coefficient in front of the interface stabilisation term, which is usually introduced in Nitschelike methods, and should be relatively large, although extremely large values of \(\lambda \) lead to the large norm of the function in the NewtonRaphson method, which makes it difficult to determine when the convergence is reached. Parameter \(\kappa \) controls the interelement stabilisation and should be relatively small, however, very small values lead to insufficient stabilisation and hence illconditionality of the problem.
5.3 Testing the interface movement scheme
The full mechanochemical problem relies on the scheme of the interface movement. Therefore, it is important to test this scheme as a separate component of the full method.
The expected secondorder convergence of the error of the radius of the circular interface was obtained (the plot is not shown here for brevity). This error was calculated as the \(\ell ^1\)norm of the vectorcolumn containing all intersection points, divided by the number of points. This illustrates the expected performance of the interface discretisation and movement method.
5.4 Example of a coupled problem with moving nonconforming interface in a constrained 2D body
The initial position of the interface was taken to be line \(X_2 = 2/17\). The materials below and above the interface were taken to be the transformed and the untransformed materials, respectively. The movement of the interface was simulated from \(t_0 = 0\) to \(t_\mathrm {e} = 2.5\).
Mesh sizes \(h = 1/N\), \(N \in \left\{ 4, 8, 16, 32, 64 \right\} \) were used. Time steps \(\Delta t = 1/N_\mathrm {t}\), \(N_\mathrm {t} \in \lbrace 8,16,32,64,128,256,512 \rbrace \) were used. Numerical parameters \(\lambda = 10^4\) and \(\kappa = 10^{3}\) were used based on the examples from the previous section.
5.4.1 Interface movement and stresses
The configuration of the interface at different times is illustrated in Fig. 13a. It can be seen that the initially flat interface acquires certain curvature, as the central part of the interface has lower velocity than the edges. Moreover, as the interface approaches the top boundary, the right edge of the interface slows down in comparison to the left edge. In Fig. 13b, the velocities of 4 interface points are shown as functions of time. It can be seen that there is a general decrease in the velocity of the interface as it moves. Initially, point \(X_1 = 7/8\) has higher velocity than points \(X_1 = 3/8\) and \(X_1 = 5/8\); however, its velocity drops below the velocities of central points as the interface approaches the top boundary. The oscillations of the velocity in Fig. 13b are discussed further.
The reason for the progressive curvature of the interface is the stressaffected kinetics of the reaction front. As it can be seen in Fig. 14, there is a buildup of the hydrostatic as well as the deviatoric stresses in the centreright part of the computational region. Higher stresses lead to the decrease of the normal velocity according to the chosen kinetic law (56). The stresses, in turn, develop due to propagation of the reaction front, i.e. creation of the transformed material, which has intrinsic volumetric expansion in comparison to the untransformed material. Moreover, the inclination of the top boundary (fixed displacements) creates nonsymmetric stress distribution which slows down the righthand side of the reaction front more than the lefthand side. The current configurations at the initial and the final positions of the reaction front are shown in Fig. 15.
In Fig. 13b, there are small, although visible, oscillations of the velocities of interface points related to the points moving from one interelement boundary to another. Since the interface normal velocity depends on the stress (as well as the deformation gradients and the strain energy densities), the reason for these oscillations is the discontinuous stress field (as well as \(\varvec{F} \) and W) as given by the finiteelement solution with piecewise linear test functions. As the point jumps from one interelement boundary to another, different elements become involved in calculation/averaging of the velocity at the intersection point, hence, there is a jump in the velocity. These oscillations can be significantly reduced by using a more elaborate interelement stress averaging, higherorder test functions or stress calculation procedures with an improved accuracy, for example [24, 25], but this is outside of the scope of this paper.
5.4.2 Concentration of the reactant
5.4.3 Convergence
As discussed in previous subsections, the nonconforming interface FEM has the same convergence rate as the standard FEM with respect to the mesh size for the case of fixed interface. In the mechanochemical problems, the interface moves in time, therefore, it is also necessary to demonstrate that the coupled problem converges with respect to the mesh size and the time step. Obviously, since solution \(\vec {u} _\pm ^h\) depends on the configuration of the interface, the error in the configuration of the interface affects the solution. Hence, the convergence rate for the solution cannot be higher than the convergence rate for the configuration of the interface with respect to the mesh size and the time step.
Since the points of the interface are moved using the explicit scheme similar to the forward Euler method, as described in Sect. 4.4.3, the convergence rate of the interface position and consequently of solution \(\vec {u} _\pm ^h\) with respect to the time step should be 1. Moreover, as this problem involves the explicit time integration, while the interface is spatially discretised, the existence of a stability condition, similar to the Courant–Friedrichs–Lewy (CFL) condition, can be expected, i.e. the time step must be decreased with the decrease of the spatial step, otherwise the interface configuration becomes numerically unstable. Loss of stability was observed in the numerical simulations when relatively large time steps were used; however, the nonlinear nature of the problem makes it difficult to derive an analytical expression for the stability condition and this has to be a subject of a separate study.
There is also another limitation that arises in the full mechanochemical problem with stressdependent velocity of the reaction front. In this case, the interface velocity and consequently its configuration are affected by the numerical error in the stresses (as well as in the deformation gradients and in the strain energy densities). It is wellknown that in FEM, the convergence rate for the stresses with respect to the mesh size is lower by 1 than the convergence rate for the displacements. Therefore, in the full mechanochemical problem with stressdependent velocity, the convergence rate for the displacements with respect to the mesh size drops by 1, following the convergence rate for the stresses. This issue is not specific to the considered method and is expected in any method dealing with the stressdependent velocity. The major potential improvement of the numerical framework is an addition of a stress calculation procedure with superior convergence rates, e.g. [24, 25]. This will affect the calculated velocities, hence the interface configuration and consequently the solution (the nodal displacements).
6 Conclusions
In this paper, a numerical method for solving finitestrain mechanochemical problems with moving sharp chemical reaction fronts separating chemically transformed and untransformed phases has been presented. The main advantage of the method is the ability to track the propagation of the front without remeshing the geometry each time step. The method is general in a sense that it does not rely on any constitutive laws, including functional dependency of the velocity on stresses/energies. Furthermore, it is applicable not only to mechanochemical problems, but to all problems where sharp interfaces move in deformable bodies due to configurational forces, such as classical phase transformations (in this case there is no reactant and the velocity does not depend on the concentration), and modelling surfaces of growth in biomechanics.
The proposed method accounts for three interdependent problems: the mechanical problem, the diffusion problem and the movement of the interface (the chemical reaction front). The method relies on using fixed finiteelement mesh and the interface that cuts through the elements. In the mechanical part of the problem, the interface conditions are enforced weakly using a Nitschelike method. One of the novelties of this work consists in derivation of the weak form of the problem from the total potential energy for the case of large deformations, thereby ensuring that the Nitschelike interface penalty terms are consistent with the energy minimisation. Furthermore, the algorithm that moves the points of the interface has been proposed.
The method was illustrated with several case studies assuming hyperelastic constitutive behaviour. The mechanical part of the problem was tested separately for the case of a stationary interface. The interface, which is nonconforming to the mesh and is handled using the Nitschelike method, was compared to the standard FEM. The expected secondorder convergence in the case of linear elements was observed.
The fully coupled mechanochemical problem capturing the propagation of a chemical reaction front in a clamped 2D body was solved to illustrate the applicability of the method. The kinetics of the chemical reaction front was chosen to be governed by the chemical affinity tensor [6, 7, 8], where the velocity of the chemical reaction front depends on stresses and elastic strain energy densities. In the computational example, the stressaffected deceleration of the reaction front was observed, with inhomogeneous stress distribution leading to the change of the curvature of the front.
Although 2D numerical examples were presented in this paper, the method is directly applicable to 3D. The main challenge for 3D applications is developing the code that builds the new position of the interface, i.e. the code that tracks correctly the intersection points of 3D element edges and the surface of the interface.
Finally, as the spatially discretised interface is moved explicitly, the existence of a CFLlike stability condition can be expected. In the numerical simulations, the loss of numerical stability for large time steps was observed (results are not included in the paper), however, the analytical derivation of this stability condition is yet to be performed and must be a subject of a separate study.
Footnotes
 1.
In this paper, “interface” and “reaction front” are used interchangeably.
 2.
As discussed earlier, in addition to the unknown deformation, the full mechanochemical problem contains the unknown concentration and the unknown position of the reaction front. However, since the explicit method of the interface movement is considered in this paper, the position of the front is known at the current time step, while the deformation and the concentration are unknown. Furthermore, in this subsection, only the mechanical part is considered. Therefore, the discrete representation of other unknowns, as well as the interface movement scheme, is introduced later in text.
 3.Column \(U^h\) can be formally defined aswhere \(\vec {u} ^h\) is the solution obtained on the mesh with size h, while \(h_\mathrm {c} = 1/N_\mathrm {c}\) is the size of the coarse mesh, nodes of which are taken as the comparison points.$$\begin{aligned} U^h_{2 N_\mathrm {c} i + 2i + 2j + k}= & {} \vec {e} _k \cdot \vec {u} ^h\left( \vec {X} _{ij}\right) , \quad \vec {X} _{ij} = \vec {e} _1 h_\mathrm {c} i + \vec {e} _2 h_\mathrm {c} j ,\\&i,j \in \left\{ 0, 1, \ldots , N_\mathrm {c} \right\} , \quad k \in \left\{ 1, 2 \right\} , \end{aligned}$$
Notes
Acknowledgements
The authors would like to express their sincere gratitude to Prof. Gunilla Kreiss (Uppsala University, Department of Information Technology) and Prof. Alexander B. Freidin (Institute for Problems in Mechanical Engineering of Russian Academy of Sciences and Peter the Great St. Petersburg Polytechnic University) for extremely helpful discussions, which facilitated the development of the method, and for invaluable comments on the final draft of the paper. The authors acknowledge the financial support from the EU Horizon 2020 project “Silicon based materials and new processing technologies for improved lithiumion batteries (Sintbat)”, number 685716.
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