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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2007; 69:1775–1803Published online 9 August 2006 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.1814

Numerical analysis of dissolution processes in cementitiousmaterials using discontinuous and continuous Galerkin

time integration schemes

Detlef Kuhl∗,† and Gunther Meschke

Institute for Structural Mechanics, Ruhr University Bochum, 44780 Bochum, Germany

SUMMARY

The present paper is concerned with the numerical integration of non-linear reaction–diffusion problemsby means of discontinuous and continuous Galerkin methods. The first-order semidiscrete initial valueproblem of calcium leaching of cementitious materials, based on a phenomenological dissolution model,an electrolyte diffusion model and the spatial p-finite element discretization, is used as a highly non-linear model problem. A p-finite element method is used for the spatial discretization. In the context ofdiscontinuous Galerkin methods the semidiscrete mass balance and the continuity of the primary variablesare weakly formulated within time steps and between time steps, respectively. Continuous Galerkin methodsare obtained by the strong enforcement of the continuity condition as special cases. The introduction of anatural time co-ordinate allows for the application of standard higher order temporal shape functions ofthe p-Lagrange type and the well-known Gauss–Legendre quadrature of associated time integrals. It isshown, that arbitrary order accurate integration schemes can be developed within the framework of theproposed temporal p-Galerkin methods. Selected benchmark analyses of calcium dissolution demonstratethe robustness of these methods with respect to pronounced changes of the reaction term and non-smoothchanges of Dirichlet boundary conditions. Copyright q 2006 John Wiley & Sons, Ltd.

Received 26 January 2006; Revised 15 May 2006; Accepted 15 May 2006

KEY WORDS: discontinuous Galerkin schemes; continuous Galerkin schemes; finite element method;error estimation; calcium dissolution

∗Correspondence to: Detlef Kuhl, Institute for Structural Mechanics, Ruhr University Bochum, 44780 Bochum,Germany.

†E-mail: [email protected]

Contract/grant sponsor: German National Science Foundation (DFG); contract/grant number: (SFB) 398

Copyright q 2006 John Wiley & Sons, Ltd.

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1776 D. KUHL AND G. MESCHKE

1. INTRODUCTION

1.1. Motivation

Durability of concrete structures is limited by damage caused by external loading and itsinteraction with environmentally induced deterioration mechanisms (see e.g. Reference [1]). Modelbased prognoses of the degradation of such structures are, in general, based upon coupled damagemodels, accounting for the transport of moisture, heat and aggressive substances and the variousinteractions with diffuse or localized damage. Recent progress in this emerging field of durabilitymechanics is documented e.g. in References [2–4]. Frequently, diffusion controlled degradationprocesses are characterized by a more or less pronounced reaction front moving through thestructure. Accurate numerical methods for the simulation of first-order transport processes (water,temperature, ionic species) are indispensable for successful and reliable simulation based pre-dictions of environmentally induced aging of structures. Standard time integration schemes ofthe finite difference or Newmark type (see e.g. References [5–7]) are not well suited for non-smooth Dirichlet boundary conditions and pronounced changes of source terms typically arisingin this class of parabolic differential equations (see e.g. References [8, 9]). Since according to theDahlquist theorem [10] the order of accuracy of these algorithms is restricted to two, adaptivelycontrolled Newmark schemes or alternative time integration schemes are important ingredients ofan efficient numerical strategy for the solution of multifield problems arising in durability orientedstructural analyses. In this paper discontinuous and continuous Galerkin time integration schemesare investigated in the context of the simulation of reaction–diffusion processes. In particular, thecalcium leaching of cementitious materials is used as a representative example.

1.2. Galerkin time integration schemes

Galerkin time integration schemes are based on the temporal weak formulation of the ordinarydifferential equation and finite element approximations of the state variables and the weightfunction. According to the weak and strong fulfillment of the continuity of the primary variablebetween two time steps, Galerkin methods are distinguished in their discontinuous and continuousversions, respectively. Historically, first ideas of Galerkin time integration schemes have beenpublished at the end of the 1960s. In particular, Argyris and Scharpf [11], Fried [12] and Oden[13] have proposed the temporal weak formulation of semidiscrete balance laws. Hulme [14] andArgyris et al. [15] have presented the continuous Galerkin method for the discretization of systemsof first-order differential equations. The accuracy of these methods has been improved by Petersand Izadpanah [16] and Hodges ad Hou [17] by using higher order polynomials analogous to thespatial p-finite element method [18].

Discontinuous Galerkin methods have been introduced as spatial discretization techniques byReed and Hill [19] and Le Saint and Raviart [20]. Later, the idea of the weak formulation ofthe continuity condition of primary variables has been applied by Jamet [21] and Eriksson et al.[22] for the development of discontinuous Galerkin time integration schemes. Cockburn et al.[23] present a review on the development of discontinuous Galerkin methods. Furthermore, thetextbooks by Johnson [9] and Eriksson et al. [8] include a number of applications of discontinuousand continuous Galerkin methods.

In the present paper discontinuous and continuous Galerkin time integration schemes for thesolution of non-linear semidiscrete reaction–diffusion problems are developed within a generalizedframework. This generalized formulation is specialized to the discontinuous Bubnov–Galerkin

Copyright q 2006 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2007; 69:1775–1803DOI: 10.1002/nme

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NUMERICAL ANALYSIS OF DISSOLUTION PROCESSES IN CEMENTITIOUS MATERIALS 1777

method and the continuous Petrov–Galerkin method. For the temporal approximation of the statevariables and the weight function Lagrange shape functions of arbitrary polynomial degree pin terms of the natural time co-ordinate �t ∈ [−1, 1] are used. Furthermore, the Galerkin timeintegration schemes are enriched by error estimates of the h- and p-method in oder to obtaininformation on the accuracy of the investigated methods in the context of the non-linear reaction–diffusion problem of calcium leaching of cementitious materials.

1.3. Outline of the paper

In Section 2 the reaction–diffusion model of calcium leaching of cementitious materials is brieflyreviewed including its finite element discretization in the spatial domain and its linearization withrespect to the state variables. Discontinuous and continuous Galerkin time integrations schemesand appropriate error estimates are presented in Section 3. In Section 4 the properties of theproposed Galerkin time integrations schemes are investigated by means of the simulation ofcalcium leaching of cementitious materials. In particular, the robustness of these methods withrespect to non-smooth Dirichlet boundary conditions and pronounced changes of the reaction termas well as their accuracy are studied.

2. SEMIDISCRETE INITIAL VALUE PROBLEM OF CALCIUM LEACHING

In the present section the vector valued non-linear semidiscrete initial value problem of calciumleaching of cementitious materials based on a reaction–diffusion model is summarized. This modelis used as a representative model for the application of Galerkin methods to reaction–diffusionproblems. The linearization of the semidiscrete form yields the tangent storage matrix and thetangent conduction matrix for the simulation of calcium leaching.

2.1. Reaction–diffusion model problem

As a representative highly non-linear reaction–diffusion model problem the calcium leaching ofcementitious materials is used (see e.g. References [24–29]). In this paper the calcium leachingmodel recently developed by Kuhl et al. [4] and Kuhl and Meschke [30] which is based upon thephenomenological chemical equilibrium model by Gerard [24] and the electrolyte diffusion modelby Onsager and Fuoss [31] will be briefly summarized.

Calcium leaching of cementitious materials is governed by the macroscopic mass balance ofcalcium ions solved in the pore fluid. The mass balance

divq + [� c]˙+ s = 0 in � (1)

is described in terms of the concentration of calcium ions Ca2+ within the pore fluid c, the calciumion mass production s from dissociation and the total porosity of the material �. The total porosity� =�0+�c is additively decomposed in the initial porosity �0 and the chemically induced porosity�c. The macroscopic calcium ion mass flux q is calculated based on the microscopic mass fluxD · �, where � denotes the driving force of diffusion and D is the second order electrolyte diffusioncoefficient tensor.

q=�D · �, � = − ∇c, D= D1 (2)


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Initial values, Neumann- and Dirichlet boundary conditions

c(t0) = c0 in �, q · n= q� on �q , c= c� on �c (3)

complete the non-linear initial boundary value problem of calcium leaching. The dissolution processis controlled by the internal variable �c.

�c = �c − c�0 (4)

The associated Kuhn–Tucker conditions �c�0, �c�0 and �c�c = 0 distinguish between a non-linear diffusion problem (�c�0) and a reaction–diffusion problem (�c<0). The chemically inducedporosity �c is calculated by the product of the average molar volume M/� of the cementitiousmaterials constituents and the dissolved amount of calcium s0 − s(�c) of the skeleton

�c(s(�c)) = M

�[s0 − s(�c)] (5)

The current calcium concentration of the skeleton s(�c) is defined by a chemical equilibrium statebetween chemically bounded and dissolved calcium [24, 32]

s(�c) = s0 − [1 − �c]sh[1 − �c

10c+ �2c

400c

]− s0 − sh

1 + [�c/cp]nc − �csh1 + [�c/ccsh]mc

(6)

for 0��c�c0. �c, nc, mc, cp, ccsh and sh are model parameters, c0 and s0 represent the equilibriumconcentrations of the sound material and c= 1mol/m3 represents the physical unit of �c. Basedon Equations (4) and (6), the calcium dissolution rate s is calculated as function of the internalvariable �c and the concentration rate c:

�s�t

= s = �s��c

��c�c

c (7)

Finally, the concentration dependent electrolyte diffusion model by Onsager and Fuoss [31] is usedin the five parameter formulation (see Reference [30] for a more elaborate description):

D = DN + A1�

1 + �a+ DN A3

�

[1 + �a]2 + A1A3�2

[1 + �a]3 (8)

DN is the Nernst diffusion coefficient for infinitely diluted solutions [33], A1, A2, A3 are modelconstants, � is the inverse Debye–Huckel length [34]

�= A2√c (9)

and a is the mean ionic diameter of the electrolyte solution.

2.2. Numerical methods

The initial boundary value problem of calcium leaching described by Equations (1)–(8) is solvednumerically by the weak formulation, the spatial finite element discretization, the linearizationand the time integration. The Newton–Raphson iteration based on the consistent linearization isused to solve the resulting non-linear system of equations. The present paper is focused on thetime integration using Galerkin methods including the linearization of the semidiscrete weak form.First, the remaining parts of spatial discretization and linearization are briefly summarized in thefollowing.


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2.3. Spatial Bubnov–Galerkin discretization

Based on the weak formulation of the mass balance (1) and the Neumann boundary conditions (3)the spatial finite element discretization leads to the following non-linear semidiscrete initial valueproblem:

ri (u, u)= r, u(t0) =u0, u(t0) = u0 (10)

In Equation (10) u and u represent the nodal values of the concentration c and the rate of theconcentration c on the system level, respectively. With the spatial approximations within finiteelement e,

c≈NE∑i=1

Nicei , c≈NE∑i=1

Ni cei (11)

by means of the nodal shape functions Ni and the nodal values cei and cei the generalized internalflux reii and external flux rei associated with the element node i are calculated

reii (c, c) = −∫

�∇Ni · q dV +

∫�Ni [[�c]˙+ s] dV

rei =∫

�q

N iq� dA(12)

Finally, the constituents of Equation (10) are obtained by the assembly of all element quantities

u=NENN⋃i=1e=1

cei , u=NENN⋃i=1e=1

cei , ri =NENN⋃i=1e=1

reii , r=NENN⋃i=1e=1

rei (13)

2.4. Linearization

Linearization of Equation (10) with respect to the state variables u and u:

D(u, u)�u + K(u, u)�u= r − ri (u, u) (14)

defines the tangent storage matrix D and the tangent conduction matrix K:

D(u, u) = �ri (u, u)

�u, K(u, u) = �ri (u, u)

�u(15)

On the element level these tangents are calculated for every nodal pair i, j :

Kei j (c, c) =∫

�Ni �[[�c]˙+ s]

�cN j dV −

∫�

∇Ni · �[�D]�c

· �N j dV

+∫

�∇Ni · �D · ∇N j dV

Dei j (c, c) =∫

�Ni �[[�c]˙+ s]

�cN j dV

(16)


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Note, that �[[�c]˙+ s]/�c depends only on c. Nevertheless, for the reason of a general algorithmicformulation we assume, that Dei j is described in terms of c and c. For details about the derivativesused in Equation (16) see References [4, 30].

3. DISCONTINUOUS AND CONTINUOUS GALERKIN TIMEINTEGRATION SCHEMES

For the time integration of the non-linear semidiscrete initial value problem (10) temporallycontinuous and discontinuous Galerkin methods have been developed (see e.g. References[8, 9, 14, 15, 21, 22]). Since the discontinuous version of Galerkin integration schemes includes thecontinuous Galerkin method as a special case, the development of both methods will be describedby means of the discontinuous Galerkin method of arbitrary polynomial degree p. Subsequently,the generalized method will be specialized to the continuous Galerkin method.

3.1. Time discretization

As basis of the time integration the time interval of interest t ∈ [t0, T ] is subdivided in constant oradaptively controlled time intervals [tn, tn+1] (compare Figure 1)

[t0, T ] =NT−1⋃n=0

[tn, tn+1], �t = tn+1 − tn (17)

The state variables at the time instant tn are known and the state variables at the time instant tn+1are to be computed by means of a time integration scheme.

Figure 1. Galerkin time integration schemes: approximation of the primary variable u, definitionof the jump �un�, illustration of the physical time t , definition of the natural time co-ordinate �t

and position of Gauss points (GP).


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3.2. Continuity condition

The continuity of the primary variable u at the boundaries of the individual time intervals [tn, tn+1]can be enforced by the continuity or the jump condition

�un� =u+n − u−

n =u1 − u0 = 0 (18)

In Equation (18) the primary variables at the end of the previous time step and the beginning ofthe current time step are defined by u−

n = u0 = lim�→0 u(tn − �) and u+n =u1 = lim�→0 u(tn + �),

respectively. u0 and u1 are introduced according to the temporal finite element approximation ofthe primary variable in Section 3.5.

3.3. Temporal weak form

The semidiscrete equilibrium equation (10) is transformed to the temporal weak form by multi-plication with an arbitrary weight function w(t) and integration over the time interval [tn, tn+1].Furthermore, the continuity condition (18) is only weakly enforced using the weight w1 =w(tn).The sum of these integrals defines the weak form of discontinuous Galerkin methods∫ tn+1

tnw · ri (u,u) dt + w1 · A�un(u1)� −

∫ tn+1

tnw · r dt = 0 (19)

A is introduced to balance the summarized weak forms and to adapt their physical units.

3.4. Linearization

The linearization of the weak form (19) with respect to the unknowns u(t), u(t) and u1 leads tothe linearized weak form of discontinuous Galerkin methods∫ tn+1

tnw · [D(u,u)�u + K(u,u)�u] dt + w1 · A�u1

=∫ tn+1

tnw · [r − ri (u, u)] dt − w1 · A�un(u1)� (20)

whereby the increments �u, �u and �u1 of the unknowns and the tangent matrices D and K areidentified according to Equation (14).

3.5. Temporal Galerkin approximation

The state variables, the weight function and the increments are temporally approximated by shapefunctions Ni according to temporal nodes i and associated nodal values (e.g. ui ). Standard Lagrangepolynomials of the polynomial degree p (see e.g. Reference [35]) as function of the naturalco-ordinate �t ∈ [−1, 1] (compare Figure 1) are used as shape functions

Ni (�t ) =p+1∏k=1k �=i

�kt − �t�kt − �it

, �it =2[i − 1]

p− 1 (21)

In accordance with the isoparametric concept, the physical time t and the displacement vectorare approximated by using the shape functions Ni of the polynomial degree p. Independent


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Figure 2. Illustration of the approximation of state variables u, u and the weight function w for discon-tinuous and continuous Galerkin time integration schemes with p= 3 (dG(3) and cG(3)).

approximations of the weight function w by shape functions N i with the polynomial degree p areassumed (compare Figures 1 and 2):

t (�t ) ≈p+1∑j=1

N j (�t )tj , u(�t ) ≈

p+1∑j=1

N j (�t )uj

w(�t ) ≈p+1∑i=1

N i (�t )wi , �u(�t ) ≈

p+1∑j=1

N j (�t )�uj

(22)

The vector of concentration rates u and its increment are obtained by taking the time derivativeof Equation (22).

u(�t ) ≈p+1∑j=1

N j,t (�t )u

j , �u(�t ) ≈p+1∑j=1

N j,t (�t )�u

j (23)

Herein N j,t (�t (t)) = �N j (�t (t))/�t represents the time derivative of the shape function N j with

respect to the physical time t . For the computation of these time derivatives

N j,t (�t ) = �N j (�t )

�t= �N j (�t )

��tJ−1t (�t ) = N j

;t (�t )J−1t (�t ) = N j

;t (�t )Jt (�t )

(24)

and of the time integrals ∫ tn+1

tnf (t) dt =

∫ 1

−1f (�t )|Jt (�t )| d�t (25)

the Jacobian Jt is required.

Jt (�t ) = �t (�t )��t

≈p+1∑j=1

�N j (�t )

��tt j =

p+1∑j=1

N j;t (�t )t

j (26)


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For equidistant nodal times t j − t j−1 =�t/p with j ∈ [2, p+1] the constant Jacobi transformationterm Jt =�t/2 is obtained. The derivatives of the shape functions with respect to the natural timeco-ordinate �t are obtained by taking the derivative of Equation (21).

�Ni (�t )

��t= Ni

;t (�t ) =p+1∑l=1l �=i

−1

�lt − �it

p+1∏k=1k �=ik �=l

�kt − �t�kt − �it

(27)

Substitution of approximations (22) and (23) in the linearized weak form (20) yields the discretizedlinearized temporal weak form of discontinuous Galerkin methods.

p+1∑i=1

p+1∑j=1

wi · [Di jt + Ki j

t ]�u j + w1 · A�u1 =p+1∑i=1

wi · [rit − riti ] − w1 · A�un� (28)

In Equation (28) the time integrals

Di jt =

∫ 1

−1N i N j

,tD|Jt | d�t , riti =∫ 1

−1N iri |Jt | d�t

Ki jt =

∫ 1

−1N i N jK|Jt | d�t , rit =

∫ 1

−1N ir|Jt | d�t

(29)

are used. These integrals are computed numerically by the Gauss–Legendre integration schemeusing standard Gauss points �lt with l ∈ [1,NGT] (compare Figure 1) and weights �l (see e.g.Reference [35]).

∫ 1

−1f (�t ) d�t ≈

NGT∑l=1

�l f (�lt ) (30)

3.6. Discontinuous Bubnov–Galerkin schemes dG(p)

For arbitrary weight functionswi Equation (28) can be transformed into a linear system of equations(i ∈ [2, p + 1] and j ∈ [2, p + 1])⎡

⎣D11t + K11

t + A D1 jt + K1 j

t

Di1t + Ki1

t Di jt + Ki j

t

⎤⎦ [

�u1

�u j

]=

[r1t − r1ti − A�un�

rit − riti

](31)

which can be formally written as an effective linearized system of equations

KdG(ukdG)�udG = rdG(ukdG) (32)

and solved for the increments of the primary variables �udG. For p= p and, consequently, forN i = Ni Equation (31) leads to unique non-singular solutions. Accordingly, discontinuous Galerkintime integrations schemes (dG(p)) of the polynomial degree p are Bubnov–Galerkin methods.


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3.7. Continuous Petrov–Galerkin schemes cG(p)

For continuous Galerkin time integration schemes the continuity condition (18) is strongly ful-filled and the primary variable u1 =u0 =u−

n = u+n is known. Consequently, the derivations of

Sections 3.3–3.6 are adapted to continuous Galerkin methods by setting:

A= 0, �u1 = 0 (33)

Hence, the effective linearized system of equations

[Di jt + Ki j

t ][�u j ] = [rit − riti ] (34)

with i ∈ [1, p + 1] and j ∈ [2, p + 1] can be briefly written as

KcG(ukcG)�ucG = rcG(ukcG) (35)

For the solution of Equation (34) p= p−1 and, consequently, N i �= Ni are required. Accordingly,continuous Galerkin time integrations schemes (cG(p)) of the polynomial degree p are Petrov–Galerkin methods.

3.8. Newton–Raphson iteration

Based on the solution of Equations (31) and (34), respectively, the improved iterative solution iscalculated by using the Newton correction ( j ∈ [1, p + 1] for dG-methods and j ∈ [2, p + 1] forcG-methods)

u jk+1 =u jk + �u j (36)

where the initial values of each time step are given by u j0 =u0 =u−n . The check of convergence

is restricted to the primary variable up+1 =u(tn+1):

‖�up+1‖‖up+1 k − u0‖��u (37)

3.9. Error estimates

In order to investigate the local time integration error, error estimates based on the comparison ofthe primary results with simultaneously performed time integrations of higher (or lower) accuracy(see Figure 4) are used. These error estimates are either based on the h-method

e�t/m =u�tn+1 − u�t/m

n+1 , em�t =u�tn+1 − um�t

n+1 (38)

or, alternatively, on the p-method.

ep/p−m =upn+1 − up−m

n+1 , ep/p+m =upn+1 − up+m

n+1 (39)

A scalar valued relative error measure is given by

e= ‖e‖uref

, uref = ‖un+1 − un‖ (40)


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Figure 3. Algorithmic set-up of discontinuous and continuous (grey and subscript dG → cG) Galerkintime integration schemes for first-order non-linear systems.


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Figure 4. Illustration of error estimates based on the h- and p-method in the context ofGalerkin time integration schemes.

3.10. Algorithmic set-up

Figure 3 shows the algorithmic set-up of continuous and discontinuous Galerkin schemes for thetime integration of non-linear first-order semidiscrete initial value problems of type (10). With thisalgorithmic structure discontinuous and continuous versions of Galerkin time integration schemeswith arbitrary polynomial degree p can be realized within the same programme. Just the additionalclass of shape functions N i with p= p − 1 and the modified dimension j ∈ [2, p + 1] of thelinear system of equations should be considered in the case of continuous Galerkin schemes. Thesmall boxes on the right-hand side of Figure 3 illustrate links of the algorithmic level with theelement and material levels of finite element programs for the computation of r, uu , D, K, riand r as well as the controlling of internal variables �.

4. APPLICATION TO ANALYSES OF REACTION–DIFFUSION PROCESSES

The properties of the present Galerkin time integration schemes dG(p) and cG(p) used for thesolution of semidiscrete reaction–diffusion problems are investigated by means of analyses ofthe prototype examples of calcium leaching of structures made of cementitious materials. For thenumerical assessment benchmark problems using one- and two-dimensional spatial discretizationsof cementitious specimens as shown in Figure 5 are used. In the initial state the cementitiousspecimens are in chemical equilibrium with the environment. The dissolution process and, conse-quently, the degradation of the specimens is initiated by reducing the calcium concentration on theboundary according to the Dirichlet boundary conditions (Figure 5). As the level of the calciumion concentration c in the pore fluid falls below the equilibrium concentration, calcium is dissolved(rate s) from the cementitious skeleton. Consequently, the pore space in the cement paste increases.This results in a weakening of the material. In the framework of the theory of porous media thedissolution process is characterized by the mass balance of calcium ions in terms of the calciumconcentration c in the pore fluid, the remaining calcium concentration s of the skeleton and thecalcium ion production s by dissociation. A detailed description of the calcium leaching model


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Figure 5. Calcium leaching of a cementitious bar and a cementitious beam: geometry, finite elementdiscretization and chemical loading history.

Table I. Dissolution diffusion model for calcium leaching of cementitious materials:standard parameterset [4, 24, 30].

DN = 1.83× 10−9 m2/s c0 = 20.7378mol/m3 �c = 0.565 L = 0.16mA1 = − 2.10× 10−19 m3/s cp = 19mol/m3 n = 85 H = 0.08mA2 = 1.80× 108

√m/mol ccsh = 1.5mol/m3 m = 5 Tc� = 109 s

A3 = − 3.57× 10−10 m s0 = 15 kmol/m3 �0 = 0.2 �t = 107 sa = 4.25× 10−10 m sh = 9 kmol/m3 M

� = 3.5× 10−5 m3/mol A= I

and the evaluation on the basis of benchmark examples is given in detail in References [4, 30, 36].All model parameters, the dimensions of the specimens and algorithmic data are summarized inTable I. Initial conditions are given by the stationary state (c0 = 0) characterized by the equilibriumconcentration between the pore fluid and the skeleton of the virgin material c0. With the exceptionof the analyses in Sections 4.1.2, 4.1.3 and 4.2.2 the chemical loading time is chosen as Tc� = 109 s(see Figure 5). For the present studies of discontinous Galerkin schemes the unity matrix is used asscaling matrix A= I. For the spatial discretization quadratic one- and two-dimensional Lagrangeelements are used.

4.1. Calcium leaching of a cementitious bar

Calcium leaching of the cementitious bar in Figure 5 is simulated for a total time periodT = 1.2× 1010 s. For the simulations both Newmark and Galerkin time integration schemes areapplied. The purpose of this study is to investigate the robustness of the integration schemes forproblems characterized by non-smooth Dirichlet boundary conditions c� and pronounced changesof the reaction rate s as well as the accuracy of time stepping schemes.

4.1.1. Discussion of the numerical results. Figure 6 shows the contour plots of the calcium con-centration of the pore fluid c and the production s obtained from the continuous Galerkin schemewith linear approximations in time as function of the normalized time t/T and the normalizedposition X1/L . The variables c and s are normalized by the initial calcium concentration c0 and


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Figure 6. Calcium leaching of a cementitious bar: numerical results obtained from the cG(1) methodin terms of the calcium concentration of the pore solution c/c0 and the calcium ion production log sn

resulting from dissociation.

the maximum production smax, respectively.

log sn = log[−s109 m3 s/mol] − 1

log[smax109 m3 s/mol] − 1, smax = 1.2019× 10−4 mol/sm3 (41)

The dissolution rate s (Figure 6) shows two pronounced reaction zones which can be distinguishedby their propagation velocity and the value of the reaction rate. A strong and fast dissolutionfront propagating through the complete length of the bar corresponds to the dissolution of calciumhydroxide. In addition a significantly slower propagating dissolution zone is observed, whichpropagates only through approximately 10% of the total length, at an approximately three ordersof magnitude smaller rate s (compare Reference [4]). This second dissolution zone represents thedecalcification of calcium silicate hydrates (CSH). After the calcium hydroxide dissolution zonehas reached the right boundary of the bar the reaction term s within the domain � is significantlyreduced and the propagation of the CSH-dissolution front is accelerated. After the completedissolution of calcium hydroxide within the bar the concentration c decreases at a larger ratecompared with the initial phase. Furthermore, the horizontally aligned kinks in the s-contours att/T ≈ 0.1 and t/T ≈ 0.6 demonstrate, that non-smooth changes of the Dirichlet boundary conditionand the reaction term result in pronounced changes of the system characteristics. The c-plot inFigure 6 allows for the differentiation between the chemical loading range t/T�Tc�/T = 1/12,the calcium hydroxide dissolution phase t/T�0.6 and the decalcification of CSH-phases withinthe remaining time interval of the simulation.

It should be noted, that the results in Figure 6 can be obtained by all investigated timeintegration schemes using constant time steps �t = 107 s. In particular, the Newmark [5] methodwith 2= � = 0.5 and u= 0, dG(p) and cG(p) time integration schemes are applied without anynumerical problems. Even non-smooth Dirichlet boundary conditions and significant changes ofthe reaction term can be computed without numerical problems. However, if the chemical loadingtime Tc� is chosen smaller than Tc� = 109 s, Newmark time integration schemes fail during the


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Newton–Raphson iteration. Therefore, an adaptive time stepping algorithm together with resultsfor cG- and dG-schemes are investigated in the next subsections.

4.1.2. Adaptive Newmark solution. For the solution of calcium leaching using realistic chemicalloading times Tc� the Newmark time integration is enriched by an error-based adaptive time stepcontrol algorithm (for details see Reference [37]). Therefore, the error estimate e�t/2 (compareSection 3.9) is applied for the check of the admissible error range 0.8��e�1.2� with � = 10−6

and for the adaption of the time step �tnew =�told[�/e]1/2. For this study the total time period Tcaptured by the numerical analysis is reduced to 2× 109 s. With the described adaptive Newmarkscheme the dissolution process can be analysed for chemical loading times Tc� set to 1.0× 109,5.0× 108, 2.5× 108 and 1.25× 108s by using 328, 481, 598 and 717 adaptively controlled timesteps. The results of these analyses are given in Figure 7 (t[108 s], �t[s], X1[mm], e[1011]). Thediagrams show the profiles of the concentration c along the bar for selected time instants t , the timehistories of the concentration c for selected positions X1 as well as the time step �t prescribed bythe time step control algorithm and the respective error estimate e�t/2�1.2�.

4.1.3. Robustness of Galerkin solutions. Using Galerkin time integration schemes of the polyno-mial degree p with a constant time step �t = 107 s allows for the simulation of calcium leachingeven for relatively short chemical loading times Tc� . Figures 8 and 9 show the results of discon-tinuous and continuous Galerkin integration schemes with polynomial degrees up to p= 3. Onlythe cubic discontinuous and continuous Galerkin schemes failed to converge during the Newton–Raphson iteration. This failure is caused by oscillations of the primary variable within the timefinite element resulting from the higher order polynomial shape functions (compare Reference[36]). The results obtained from the discontinuous and continuous Galerkin integration are moreor less identical.

4.1.4. Error estimates for Newmark solutions. For comparison reasons the proposed error estimatee�t/m is applied for the analysis based on the Newmark integration scheme. Figure 10 shows thenormalized temporal and spatial local error estimate

e�t/10l = |c�t/10

n+1 − c�tn+1|

c0, log en = log[e�t/10

l 1012] − 1

log[emax1012] − 1(42)

with emax = 2.38× 10−3 for constant time steps �t = 107 s and the temporal local and spatialglobal error estimate e�t/5 according to Equation (38) for different constant time steps �t . Fromthe contour plot on the left-hand side of Figure 10 space–time regions, where large values of timeintegration errors are observed, can be identified: zone a in which an initial error is identified,region b where non-smooth boundary conditions are applied, the dissolution zone c of calciumhydroxide and the zone d characterized by a significant change of the reaction term s. Thesemaxima of the time integration error e�t/10

l are also observed on the right-hand side of Figure 10showing the spatial global error estimate e�t/5. In the first few time steps an extremely largeerror followed by a relatively high level of the error during the chemical loading phase can beobserved. At the end of the chemical loading t = Tc� a peak of the e�t/5-curve is observed, whichcorresponds to the C1-discontinuity of the Dirichlet boundary condition. During the free reaction–diffusion phase the error is reduced until the dissolution of calcium hydroxide stops. This phaseis followed by a significant change of the reaction rate s which causes a large increase of the


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Figure 7. Calcium leaching of a cementitious bar: numerical results and time integration error obtainedfrom adaptive Newmark integration (t[108 s], X1[mm]).


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Figure 8. Calcium leaching of a cementitious bar: numerical results obtained fromdG(p)-integration (t[108 s], X1[mm]).


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Figure 9. Calcium leaching of a cementitious bar: numerical results obtained fromcG(p)-integration (t[108 s], X1[mm]).


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Figure 10. Calcium leaching of a cementitious bar: spatial local error estimate e�t/10l (42) and spatial

global error estimate e�t/5 (38) for Newmark time integrations (t[109 s]). Sources for large errorsand characteristics of the numerical solution are identified in the left-hand figure: zone a: initial er-ror; zone b: non-smooth Dirichlet boundary condition; zone c: dissolution front of calcium hydroxide,

zone d: change of the reaction term s.

error by more than three decades. After this peak the error decreases to a level comparable to theprevious free reaction–diffusion phase. As shown in Figure 10, the error measure e�t/5 increaseswith increasing time steps. The differences in the levels of the error appears to remain constantduring the integration time.

4.1.5. Error estimates for Galerkin solutions. In Figure 11 the error measure e�t/5 is plotted fordiscontinuous Galerkin time integration schemes with polynomial degrees one, two and three.Similar to the analyses based on the Newmark integration scheme the error estimates applied todG-methods reflect the different phases of the dissolution process. The bandwidth of the estimatederrors increases considerably with increasing polynomial degrees. Furthermore, the differencesbetween the error curves obtained for different time steps increase with increasing p. Compared tothe Newmark integration scheme the error obtained for dG(1) is larger, for dG(2) slightly smallerand for dG(3) significantly smaller. The direct comparison of dG(1) and dG(3) within the timeinterval [0, 2.5× 108 s] in Figure 11 illustrates the error of the time integrations at the beginningof the chemical loading process.

Figure 12 contains the plots of the error ep/p+1 for dG-methods of polynomial degrees p ∈ [1, 3].For this study the number of temporal Gauss points is chosen according to the polynomial degreep + 1 of the integration scheme used as a reference measure. The estimated errors e�t/5 (Figure11) and ep/p+1 (Figure 12) are almost identical. This comparison verifies the error estimates of theh- and p-method. The comparison of error estimates ep/p+1 in Figure 12 illustrates the considerableimprovement of the quality of the numerical results with an increasing polynomial degree p.

Figure 13 shows the results of an analogous study for continuous Galerkin schemes.Qualitatively, the same observations as for the dG-methods are made. However, the errorobtained from the cG-methods is significantly smaller. In other words, for the investigated class


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Figure 11. Calcium leaching of a cementitious bar: error estimates e�t/5 for dG-methodswith different time steps �t (t[109 s]).

of problems characterized by dissociation–diffusion problems cG-methods lead to more accu-rate solutions with minor numerical expense compared to dG-methods (compare Equations (31)and (34)).

4.1.6. Order of accuracy of Galerkin schemes. In order to determine the order of accuracy ofNewmark and Galerkin time integration schemes the average errors within the time intervalsI1 =[575, 625]× 107 s and I2 = [50, 100]× 107 s are summarized in Table II. The time intervalsI1 and I2 are chosen as representative intervals for the free reaction–diffusion phase and forthe chemical loading phase of the analysed time interval of the dissolution process. Table IIconfirms again the equivalence of h- and p-error estimates. The errors e�t/5 of the Newmarktime integration scheme and the errors ep/p+1 of the Galerkin integration schemes are plotted inFigure 14 as function of the time step log�t . The slope of the error curves log e= a + b log�tdefines the proportionality of the error e∼�tb and the order of accuracy O(b). For Newmark,


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Figure 12. Calcium leaching of a cementitious bar: error estimates ep/p+1 for dG-methodswith different time steps �t (t[109 s]).

discontinuous Galerkin and continuous Galerkin methods the following proportionality holds:

eN ∼�t2, edG ∼�t p, ecG ∼�t p+1 (43)

As expected, the order of accuracy of Newmark schemes is two (see e.g. Reference [38]). In contrast,Galerkin time integration schemes allow for arbitrary order of accuracy which is controlled by thetemporal polynomial degree p. From Figure 14 follows for discontinuous Galerkin methods theorder of accuracy p and for continuous Galerkin methods the order of accuracy p + 1.

4.2. Calcium leaching of a cementitious beam

In this section the properties of the present time integration schemes are further investigated bymeans a two-dimensional analysis of a cementitious beam subjected to deionized water along


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Figure 13. Calcium leaching of a cementitious bar: error estimates ep/p+1 and e�t/5 for cG-methodswith different time steps �t (t[109 s]).

a part of the bottom surface (Figure 5). The study covers a total time period of T = 5× 109 s.The initial conditions are characterized by the stationary chemical equilibrium state of the virginmaterial.

4.2.1. Analysis of the numerical results. Figure 15 shows the evolution of the calcium concentrationof the pore fluid c, the calcium concentration of the skeleton s and the reaction rate s. Accordingto Equation (41), the reaction rate is normalized with smax = 1.8324× 10−4 mol/m3 s. The peaksin the reaction rate s and the pronounced steps in the concentration field s are associated withthe propagation of the calcium hydroxide dissolution front and the CSH decalcification zone.As discussed in Reference [4] the reaction rate s for calcium hydroxide dissolution is orders ofmagnitude larger than for the dissolution of CSH-phases. Also, the calcium hydroxide dissolutionzone propagates much faster than the zone of CSH-dissolution.


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Table II. Calcium leaching of a cementitious bar.

�t [107 s] 0.2 0.5 1.0 2.0 4.0 e Fig.

N I1 4.244× 10−7 2.628× 10−6 1.040× 10−5 4.169× 10−5 No e�t/5 10I2 8.072× 10−5 4.248× 10−4 1.461× 10−3 2.945× 10−3 Convergence

dG(1) I1 1.262× 10−4 3.113× 10−4 6.150× 10−4 1.238× 10−3 2.714× 10−3 e�t/5 11I2 1.014× 10−3 2.895× 10−3 6.888× 10−3 1.648× 10−2 3.872× 10−2

dG(2) I1 8.405× 10−8 5.184× 10−7 2.096× 10−6 8.752× 10−6 3.558× 10−5

I2 1.368× 10−5 6.626× 10−5 2.203× 10−4 6.703× 10−4 1.643× 10−3

dG(3) I1 8.034× 10−11 1.165× 10−9 8.596× 10−9 5.823× 10−8 3.648× 10−7

I2 4.010× 10−7 4.954× 10−6 3.176× 10−5 1.621× 10−4 5.968× 10−4

dG(1) I1 1.574× 10−4 3.886× 10−4 7.655× 10−4 1.523× 10−3 3.200× 10−3 ep/p+1 12I2 1.262× 10−3 3.553× 10−3 8.445× 10−3 2.058× 10−2 4.713× 10−2

dG(2) I1 8.759× 10−8 5.400× 10−7 2.181× 10−6 9.086× 10−6 3.700× 10−5

I2 1.435× 10−5 7.028× 10−5 2.342× 10−4 7.400× 10−4 1.954× 10−3

dG(3) I1 8.133× 10−11 1.175× 10−9 8.869× 10−9 5.877× 10−8 3.690× 10−7

I2 4.056× 10−7 5.009× 10−6 3.348× 10−5 1.732× 10−4 6.970× 10−4

cG(1) I1 3.592× 10−6 2.159× 10−5 8.156× 10−5 2.949× 10−4 1.011× 10−3 ep/p+1 13I2 2.323× 10−4 1.038× 10−3 2.928× 10−3 7.504× 10−3 1.630× 10−2

cG(2) I1 3.777× 10−10 9.169× 10−9 9.811× 10−8 9.873× 10−7 9.076× 10−6

I2 1.191× 10−6 1.844× 10−5 1.140× 10−4 7.336× 10−4 3.133× 10−3

cG(3) I1 2.623× 10−12 8.644× 10−12 2.257× 10−10 5.425× 10−9 1.111× 10−7

I2 1.077× 10−8 4.782× 10−7 8.843× 10−6 1.218× 10−4 1.180× 10−3

Average relative errors of the Newmark method (N), discontinuous Galerkin methods (dG) and continu-ous Galerkin methods (cG) within the time intervals I1 = [575, 625]× 107 s (reaction–diffusion phase) andI2 = [50, 100]× 107 s (chemical loading phase).

At t = 3× 109 s oscillations of the reaction rate s are observed in Figure 15. The reason for theseoscillations and the effect of linear and quadratic continuous Galerkin time integration schemeson this oscillatory behaviour are investigated in Figure 16. The left column shows the spatial localerror e�t/2

l (compare Equation (42)) of the cG(1)-solution. In the middle and right columns cG(1)-and cG(2)-solutions for the reaction rate s are compared. The s-plots show, that oscillations startafter the calcium hydroxide dissolution front has arrived at the top face of the beam (X2 = H ).Since the dissolution front approaches the upper surface of the beam parallel to the boundary, thespeed of the dissolution front in horizontal direction grows to infinity. Consequently, a singularityof the rate s (or c, compare Equation (7)) causes the oscillations. The spatial position of theresulting oscillations is fixed (compare also Figure 15). Only a slow decay of the oscillations inthe time domain can be observed. In conclusion, the error plots in Figures 16 and 17 demonstratetwo characteristic properties of the numerical solution: Firstly, a local error maximum followsthe movement of the dissolution front of calcium hydroxide through the structure. Secondly, theaforementioned singularity of s at the upper surface of the beam causes a large time integrationerror. The location of this error remains unchanged and its amplitude is only slowly reducedin time.


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Figure 14. Calcium leaching of a cementitious bar: average relative errors of the Newmark method(N), discontinuous and continuous Galerkin methods (dG und cG, ep/p+1) within the time intervalsI1 = [575, 625] × 107 s (reaction–diffusion phase) and I2 =[50, 100] × 107 s (chemical loading phase).

4.2.2. Robustness of continuous Galerkin solutions. In Figure 17 the robustness of continuousGalerkin schemes is investigated for analyses, in which realistically small chemical loading timesTc� are used. The middle column shows again the spatial distribution of the calcium concentration sof the skeleton obtained from using the standard chemical loading time Tc� = 109 s. This solutionrepresents the minimal chemical loading time which can be employed if non-adaptive Newmarktime integration schemes are used (compare Reference [36]). As illustrated in the right column ofFigure 17, the cG(1)-integration allows a robust numerical simulation of this initial boundary valueproblem, even for a very small value for the chemical loading time taken as Tc� = 6.25× 107 s thefull chemical loading can be applied within seven time steps.

5. CONCLUSIONS

In the present paper discontinuous and continuous Galerkin time integration schemes with Lagrangeshape functions of arbitrary polynomial degree p have been investigated in the context of non-linear reaction–diffusion problems. As a representative initial boundary value problem analysesof calcium leaching by means of a dissociation–diffusion model for cementitious materials hasbeen adopted. The model problem has been spatially discretized by finite elements. Discontinuousand continuous Galerkin time integration schemes have been developed for the solution of thecorresponding semidiscrete initial value problem. It has been shown, that the discontinuous versionsare Bubnov–Galerkin methods and the continuous versions belong to the class of Petrov–Galerkinmethods. The time integration schemes have been enriched by error estimates of the h- and thep-method. The Galerkin time integration schemes have been applied to the simulation of calciumleaching of cementitious materials and compared with classical and adaptively controlled Newmarktime integration schemes. It has been shown, that Galerkin time integration schemes allow for therobust solution of highly non-linear reaction–diffusion problems even for non-smooth Dirichletboundary conditions and strong alterations of the reaction term. Furthermore, the order of accuracy


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Figure 15. Calcium leaching of a cementitious beam: numerical results obtained from cG(1).


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Figure 16. Calcium leaching of a cementitious beam: investigation of the oscillations appearing in thenumerical results obtained from cG(1)- and cG(2)-solutions.


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Figure 17. Calcium leaching of a cementitious beam: investigation of the robustness of the cG(1)-solutionfor small values of the chemical loading time Tc� .


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of Galerkin methods has been analysed by means of numerical studies. It has been shown, thatdiscontinuous Galerkin methods are characterized by an order of accuracy O(p) of the polynomialdegree p and continuous Galerkin methods by the order O(p + 1).

ACKNOWLEDGEMENTS

Financial support was provided by the German National Science Foundation (DFG) in the framework ofproject A9 of the collaborative research center (SFB) 398. This support is gratefully acknowledged.

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