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Keywords:

  • thermodynamics;
  • domain decomposition;
  • mortar method;
  • energy momentum;
  • coupled problems;
  • time integration

Abstract

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. FINITE STRAIN THERMOELASTODYNAMICS
  5. 3. DISCRETIZATION IN TIME
  6. 4. DISCRETIZATION IN SPACE
  7. 5. FOUR-DIMENSIONAL MORTAR METHOD
  8. 6. EXAMPLES
  9. 7. CONCLUSIONS
  10. Acknowledgements
  11. APPENDIX A: CONSTITUTIVE EQUATIONS
  12. APPENDIX B:ENHANCED ASSUMED STRAIN METHOD
  13. REFERENCES

An energy-momentum consistent integrator for non-linear thermoelastodynamics is newly developed and extended to domain decomposition problems. The energy-momentum scheme is based on the first law of thermodynamics for strongly coupled, non-linear thermoelastic problems. In contrast to staggered algorithms, a monolithic approach, which solves the mechanical as well as the thermal part simultaneously, is introduced. The approach is thermodynamically consistent in the sense that the first law of thermodynamics is fulfilled. Furthermore, a domain decomposition method for the thermoelastic system is developed based on previous developments in the context of the mortar method. The excellent performance of the new approach is illustrated in representative numerical examples. Copyright © 2011 John Wiley & Sons, Ltd.


1. INTRODUCTION

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. FINITE STRAIN THERMOELASTODYNAMICS
  5. 3. DISCRETIZATION IN TIME
  6. 4. DISCRETIZATION IN SPACE
  7. 5. FOUR-DIMENSIONAL MORTAR METHOD
  8. 6. EXAMPLES
  9. 7. CONCLUSIONS
  10. Acknowledgements
  11. APPENDIX A: CONSTITUTIVE EQUATIONS
  12. APPENDIX B:ENHANCED ASSUMED STRAIN METHOD
  13. REFERENCES

General thermoelastic material models have been a major topic in research for the past decades (see e.g. Reese and Govindjee 1 and Miehe 2 among many others). Especially the non-linear stability of the time-discrete systems has been addressed in several papers (see Simo 3). The present paper is based on an approach by Holzapfel and Simo 4, in which rubber elasticity has been extended to a class of entropic elastic materials, written entirely in the material configuration.

Energy-momentum schemes are well known in the context of non-linear elastodynamics (see, for example, Gonzalez 5 and Betsch and Steinmann 6, 7 and the references therein) and have been applied to a wide range of applications (see e.g. for contact problems Hauret and Le Tallec 8 and Hesch and Betsch 9, 10). They are able to conserve qualitative features of the systems and, more importantly, they exhibit an excellent performance in long-term simulations and are numerically stable. In a nutshell, in the present work we merge the concept of structure preserving integrators with the concept of entropic thermoelastic materials. In contrast to actual developments (see Romero 11, 12 and Gross 13) we apply only a minor modification to the concept of the discrete gradient (see Gonzalez 14), based on the first law of thermodynamics to achieve our goal.

Additionally, we introduce a structure preserving, variationally consistent mortar domain decomposition method for thermoelastic systems, based on previous developments in Hesch and Betsch 15 confined to the isothermal case. The present domain decomposition method provides a flexible approach for the coupling of different discretization schemes or for nonmatching triangularizations. The development of variationally consistent domain decomposition constraints started two decades ago with the work of Bernadi et al. 16, 17. Further advances can be found in Krause and Wohlmuth 18 and in Dohrmann et al. 19. An extension to non-linear solid mechanics is given in Puso 20. Thermomechanical contact problems using the mortar method are addressed in Hüeber and Wohlmuth 21.

An outline of the present work is as follows. The fundamental equations in the context of the first law of thermodynamics are outlined in Section 2. In particular, the weak form of the balance equations is derived within this section. The equations of motion of the thermoelastic system under consideration along with the energy-momentum consistent discretization in time will be dealt with in Section 3. In Section 4 we apply a spatial discretization based on finite elements. In this connection, the use of nonconforming meshes is facilitated. Therefore, we introduce in Section 5 the mortar method for the domain decomposition of thermoelastic systems along with an energy-momentum consistent time-stepping scheme. Representative numerical examples are presented in Section 6. Eventually, conclusions are drawn in Section 7.

2. FINITE STRAIN THERMOELASTODYNAMICS

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. FINITE STRAIN THERMOELASTODYNAMICS
  5. 3. DISCRETIZATION IN TIME
  6. 4. DISCRETIZATION IN SPACE
  7. 5. FOUR-DIMENSIONAL MORTAR METHOD
  8. 6. EXAMPLES
  9. 7. CONCLUSIONS
  10. Acknowledgements
  11. APPENDIX A: CONSTITUTIVE EQUATIONS
  12. APPENDIX B:ENHANCED ASSUMED STRAIN METHOD
  13. REFERENCES

We start with a short summary of non-linear thermoelasticity. More details on the continuum description of thermoelastic solids can be found in textbooks, such as Holzapfel 22 and Gonzalez and Stuart 23. Consider a continuum body with reference configuration ℬ0⊂ℝ3 undergoing a motion characterized by a time-dependent deformation mapping φ: ℬ0 × [0, T][RIGHTWARDS ARROW]3, where [0, T] is the time interval elapsed during the motion. The current configuration is denoted by ℬt = φt(ℬ0). Material points are labeled by X∈ℬ0, the material velocity is given by v: ℬ0 × [0, T][RIGHTWARDS ARROW]3, v = ∂φ/∂t, and the deformation gradient is denoted by F: ℬ0 × [0, T][RIGHTWARDS ARROW]3 × 3, F = Dφ. The linear momentum is given by π = ϱ0v, where ϱ0 stands for the density in the reference configuration.

The absolute temperature θ: ℬ0 × [0, T][RIGHTWARDS ARROW]ℝ is assumed to be a smooth function of (X, t)∈ℬ0 × [0, T]. We further assume that the material behavior is governed by the free energy function Ψ: ℬ0 × [0, T][RIGHTWARDS ARROW]ℝ, equation image, where C: ℬ0 × [0, T][RIGHTWARDS ARROW]3 × 3, C = FTF is the right Cauchy-Green deformation tensor. Accordingly, the nominal (or first Piola-Kirchhoff) stress tensor P: ℬ0 × [0, T][RIGHTWARDS ARROW]3 × 3 and the entropy η: ℬ0 × [0, T][RIGHTWARDS ARROW]ℝ are defined by

  • equation image(1)

Moreover, the nominal heat flux vector Q is defined by

  • equation image(2)

Here, equation image is a thermal conductivity tensor which must be positive semi-definite. Note that the constitutive laws (1) and (2) are thermodynamically consistent in the sense that they satisfy the restrictions imposed by the second law of thermodynamics (in the form of the Clausius–Duhem inequality). The Lagrangian form of the local balance of linear momentum and energy for a thermoelastic body can be written as

  • equation image(3)

where equation image and equation image denote the material descriptions of prescribed body force and heat supply per unit volume. The above equations have to be satisfied for all X∈ℬ0 and t⩾0. To complete the initial-boundary value problem for the thermoelastic body under consideration, the equations in (3) have to be supplemented by appropriate initial and boundary conditions. Accordingly, initial conditions in ℬ0 and at time t = 0 are specified by

  • equation image(4)

where v0 and θ0 are prescribed fields. Moreover, boundary conditions on ∂ℬ0 at times t⩾0 are specified by

  • equation image(5)

where ∂ℬmath image and ∂ℬmath image are subsets of ∂ℬ0 with the properties ∂ℬmath image∪∂ℬmath image = ∂ℬ0 and ∂ℬmath image∩∂ℬmath image = ∅. Similarly, ∂ℬmath image and ∂ℬmath image are subsets of ∂ℬ0 with the properties ∂ℬmath image∪∂ℬmath image = ∂ℬ0 and ∂ℬmath image∩∂ℬmath image = ∅. Furthermore, N denotes the unit outward normal field on ∂ℬ0, and equation image, equation image, equation image, and equation image are prescribed fields.

2.1. Weak formulation

To perform a finite element discretization in space we next recast the coupled thermoelastic problem in weak form. To this end we introduce the space of test functions ��φ defined as

  • equation image(6)

along with

  • equation image(7)

Scalar multiplication of (3)2 by the test function δφ∈��φ and subsequent integration yields

  • equation image(8)

Similarly, (3)3 leads to

  • equation image(9)

Applying integration by parts along with the divergence theorem, (8) and (9) can be written as

  • equation image(10)

These equations have to hold for all δφ∈��φ and δθ∈��θ. In (10)1, the second Piola-Kirchhoff stress field Σ = Dφ−1P has been introduced. While the balance of linear momentum and the balance of energy are stated in weak form, we retain the kinematic relationship (3)1 in local form.

2.2. Balance laws in global form

We next summarize fundamental balance laws in global form which should be preserved under discretization in space and time. This viewpoint leads to the notion of energy-momentum consistent integrators. The design of a specific energy-momentum consistent integrator is one of the main goals of the present work. For simplicity of exposition in what follows we restrict our attention on the thermoelastic problem with pure Neumann data, i.e. ∂ℬmath image = ∂ℬmath image = ∅. While Dirichlet boundary conditions on the mechanical part affect the momentum balance equations, Dirichlet boundary conditions on the thermal part affect the energy balance.

Setting δφ = µ, where µ∈ℝ3 is arbitrary and constant, showing that (10)1 yields the balance law for linear momentum in integral form in a straightforward procedure:

  • equation image(11)

Here, the total linear momentum is given by L = ∫math imageπdV, and the right-hand side of (11) characterizes the resultant external force applied to the continuum body.

Similarly, substituting δφ = µ × φ into (10)1, the integral form of the balance law for angular momentum is recovered:

  • equation image(12)

In this connection, J = ∫math imageφ × πdV is the total angular momentum of the continuum body with respect to the origin of the inertial frame of reference. The right-hand side of (12) equals the resultant external torque about the origin.

2.2.1. Balance of energy.

We next consider the integral form of the balance law for energy. As before we focus on the thermoelastic problem with pure Neumann data. Substituting equation image into (10)1 yields

  • equation image(13)

Here, T denotes the total kinetic energy of the continuum body, W is the net working, and Pext is the power of external forces. The relationship between the free energy Ψ and the specific internal energy e is given by

  • equation image(14)

Differentiation with respect to time yields

  • equation image(15)

On the other hand the constitutive Equations (1) are based on the free energy function equation image, and thus imply

  • equation image(16)

Accordingly, taking into account (15) and (16), the stress power can be written as

  • equation image(17)

Introducing the total internal energy of the continuum body E = ∫math imageedV, the net working can now be written in the form

  • equation image(18)

Testing (10)2 on the constant δθ = µ, µ∈ℝ, gives

  • equation image(19)

where Q is the total net heating of the continuum body. Substituting (19) into (18), we recover the first law of thermodynamics in the form d E/dt = W + Q. Moreover, in view of (13), the global form of the energy balance law can be written as

  • equation image(20)

In case the external forces are associated with a potential energy Vext, i.e. Pext = − dVext/dt, the energy balance law reads as

  • equation image(21)

Concerning the discretization in space and time of the coupled thermoelastic problem under consideration we aim at numerical methods that correctly reproduce the above balance laws for any time step. For example, if the external forces vanish the linear momentum as well as the angular momentum of the system should be conserved exactly. Moreover, if (21) applies and the system is insulated (i.e. Q = 0), the total energy should be exactly conserved in the discrete setting thus correctly reproducing the continuous law (21).

3. DISCRETIZATION IN TIME

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. FINITE STRAIN THERMOELASTODYNAMICS
  5. 3. DISCRETIZATION IN TIME
  6. 4. DISCRETIZATION IN SPACE
  7. 5. FOUR-DIMENSIONAL MORTAR METHOD
  8. 6. EXAMPLES
  9. 7. CONCLUSIONS
  10. Acknowledgements
  11. APPENDIX A: CONSTITUTIVE EQUATIONS
  12. APPENDIX B:ENHANCED ASSUMED STRAIN METHOD
  13. REFERENCES

We next perform the discretization in time of the coupled thermoelastic problem under consideration. In particular, we present a new energy-momentum consistent integrator for thermoelastodynamics.

Consider a sequence of times t0, …, tn, tn + 1, … and assume that the state at tn, denoted by (φn, θn), is known. Then the goal is to approximate the state (φn + 1, θn + 1) at tn + 1, where the time-step size Δt = tn + 1tn is prescribed. Consider the algorithmic approximation to the weak form (10) defined by

  • equation image(22)

The above two equations are supplemented by the mid-point-type approximation to the kinematic relationship (3)1 given by

  • equation image(23)

together with

  • equation image(24)

In the above formulas, (·)n + 1/2 denotes the standard mid-point approximation, e.g. equation image. Moreover, in (22) 1, Σn, n + 1 denotes a consistent algorithmic version of the second Piola-Kirchhoff stress tensor defined by

  • equation image(25)

where

  • equation image(26)

In this connection, (14) gives rise to

  • equation image(27)

for k∈{n, n + 1}. Accordingly, the algorithmic stress Σn, n + 1 just depends on the state variables (φn, θn) and (φn + 1, θn + 1). We further remark that in the isothermal limit formula (25) boils down to the discrete gradient in the sense of Gonzalez 14 corresponding to De(C), see also Gonzalez 5 and Betsch and Steinmann 6.

In (22)2, Qn, n + 1 denotes a consistent algorithmic version of the nominal heat flux vector defined by

  • equation image(28)

3.1. Algorithmic versions of the global balance laws

We next verify that the newly proposed energy-momentum consistent integrator does indeed satisfy the global balance laws summarized in Section 2.2. Again we focus on pure Neumann data. Inserting δφ = µ into (22)1 yields

  • equation image(29)

Furthermore, inserting δφ = µ × φn + 1/2 into (22) 1 yields

  • equation image(30)

The last two results can be viewed as time-discrete counterparts of the global balance laws for linear and angular momentum. To show algorithmic satisfaction of the global balance law for energy, substitute vn + 1/2 for δφ in (22)1 to obtain

  • equation image(31)

where use has been made of (23). On the right-hand side of the last equation Pmath image stands for the algorithmic version of the power of external forces given by

  • equation image(32)

Taking into account the symmetry of Σn, n + 1 along with the definition of the kinetic energy, (31) can be recast in the form

  • equation image(33)

In view of the definition of the algorithmic stress tensor (25), the last equation can be written as

  • equation image(34)

Now, setting δθ = µ in (22) 2 yields

  • equation image(35)

Here, Qn, n + 1 denotes the discrete version of the total net heating. Combining (34) and (35) we arrive at the result

  • equation image(36)

The last equation confirms algorithmic energy consistency of the integrator under consideration. That is, the balance law for energy (20) is correctly reproduced in the discrete setting for any time-step size Δt.

4. DISCRETIZATION IN SPACE

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. FINITE STRAIN THERMOELASTODYNAMICS
  5. 3. DISCRETIZATION IN TIME
  6. 4. DISCRETIZATION IN SPACE
  7. 5. FOUR-DIMENSIONAL MORTAR METHOD
  8. 6. EXAMPLES
  9. 7. CONCLUSIONS
  10. Acknowledgements
  11. APPENDIX A: CONSTITUTIVE EQUATIONS
  12. APPENDIX B:ENHANCED ASSUMED STRAIN METHOD
  13. REFERENCES

To achieve a numerical solution for the semi-discrete, coupled thermoelastic problem, we apply a finite element framework to both, the mechanical as well as the thermal field. In particular, we consider a standard displacement-based finite element approach, where we introduce finite dimensional approximations of φ and δφ so that

  • equation image(37)

Here, A∈ω = {1, …, nnode} such that qA∈ℝ3 denotes the position vector of node A and NA(X): ℬ0[RIGHTWARDS ARROW]ℝ are global shape functions. In the present work, we make use of standard trilinear shape functions. In a second step, we discretize the temperature field using the same shape functions as before

  • equation image(38)

where ΘA∈ℝ denotes the temperature at node A∈ω = {1, …, nnode}. For later use we introduce the system vectors q = [q1, …, qmath image], Θ = [Θ1, …, Θmath image] and the variations thereof as δq = [δq1, …, δqmath image] and δΘ = [δΘ1, …, δΘmath image].

Next we develop the fully discretized weak form based on the semi-discrete formulation (22), starting with the first term in (22)1

  • equation image(39)

where (23) and (24) have been taken into account. For simplicity of exposition we make use of the summation convention for repeated indices. Similarly, we obtain for the first term in (22)2

  • equation image(40)

where, in view of (1)2, the discrete entropy reads as

  • equation image(41)

and the discrete right Cauchy-Green tensor is given by

  • equation image(42)

Next we approximate the second term in (22)1 as follows:

  • equation image(43)

Analogue to (25), Σmath image denotes a consistent algorithmic version of the discrete second Piola-Kirchhoff stress tensor, defined by

  • equation image(44)

As before, we formulate the inner energy in terms of the free Helmholtz energy

  • equation image(45)

Similarly, the second term in (22)2 yields

  • equation image(46)

At last, the terms on the right side of (22)1 and (22)2 can be written as

  • equation image(47)

which completes the spatial discretization process. In summary, we receive the following equations:

  • equation image(48)

Next, we verify that the spatial discretization of the coupled thermoelastic system inherits the fundamental conservation properties of the underlying semi-discrete system.

4.1. Conservation properties

4.1.1. Linear momentum.

To verify conservation of linear momentum, we focus again on pure Neumann data. Insertion of δqA = µ into (48)1 and summation yields

  • equation image(49)

With regard to (43) we can state

  • equation image(50)

and obtain for the discrete counterpart of (29)

  • equation image(51)
4.1.2. Angular momentum.

Next, we substitute δqA = µ × qA, n + 1/2 into (48)1 and obtain

  • equation image(52)

Due to the skew-symmetry of qA, n + 1/2 × qB, n + 1/2 and the symmetry of Smath image we obtain the discrete counterpart of (30)

  • equation image(53)
4.1.3. Total energy.

Eventually, we verify algorithmic conservation of energy. Replacing δqA in (48)1 with vA, n + 1/2 and summation over all nodes yields

  • equation image(54)

With regard to (43) one can rewrite the last equation as

  • equation image(55)

Taking (44) into account, we obtain

  • equation image(56)

In a second step we set δθA = µ and obtain from (48)2

  • equation image(57)

Thus, we arrive at

  • equation image(58)

which proves algorithmic energy consistency.

5. FOUR-DIMENSIONAL MORTAR METHOD

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. FINITE STRAIN THERMOELASTODYNAMICS
  5. 3. DISCRETIZATION IN TIME
  6. 4. DISCRETIZATION IN SPACE
  7. 5. FOUR-DIMENSIONAL MORTAR METHOD
  8. 6. EXAMPLES
  9. 7. CONCLUSIONS
  10. Acknowledgements
  11. APPENDIX A: CONSTITUTIVE EQUATIONS
  12. APPENDIX B:ENHANCED ASSUMED STRAIN METHOD
  13. REFERENCES

The goal of this section is to extend our previous developments to domain decomposition problems in dynamic thermoelasticity. In what follows, we focus on specific coupling terms in the weak form arising from the tying of dissimilarly meshed regions using quadrilateral meshes. Note, however, that the presented segmentation process can be easily applied to other meshes as well.

Consider a body subdivided into two parts as depicted in Figure 1. The two parts are tied together at a common interface Γd. While the weak form corresponding to each subdomain ℬmath image, i∈{1, 2}, is again given by (10), the coupling of both subdomains gives rise to the additional interface contributions

  • equation image(59)

The quantity t(1)∈ℝ4 will later on denote the Lagrange multipliers, which can be interpreted as Piola tractions for the mechanical part and have the dimension of the entropy for the thermal part. Applying the mortar method, we discretize t(1) using the shape functions of the underlying geometry, i.e.

  • equation image(60)

where equation image denotes the set of nodes on the internal interfaces Γmath image. Insertion of the approximations (37), (38) and (60) in (59) yields

  • equation image(61)

where nAB and nAC denote the so-called mortar integrals given by

  • equation image(62)

The nodal Lagrange multipliers λA∈ℝ4 characterize the generalized forces of constraint for enforcing the mortar mesh-tying constraint, given by

  • equation image(63)

To evaluate the mortar integrals, we have to divide both sides of the discrete interface into segments. We will summarize this process in the following, for further details see Puso 20 and Hesch and Betsch 15.

thumbnail image

Figure 1. Decomposition of a body into two domains with the internal interface Γd.

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5.1. Segmentation

We consider a typical element equation image, where equation image denotes the set of elements on the discrete surface Γmath image of an arbitrarily chosen side (2), referred to as mortar side, with the nodes qmath image, qmath image, qmath image, and qmath image of the element e2. These nodes are projected orthogonally to the opposing non-mortar side Γmath image. Each orthogonally projected node I can be written in terms of convective coordinates equation image (see Figure 2(b)) of the opposing elements, which are collected in a vector

  • equation image(64)
thumbnail image

Figure 2. Segmentation process: (a) Representative element e on the non-mortar side Γmath image and one opposing element e2 on the mortar side; (b) Projection of the nodal points of element e2 onto the non-mortar side and determination of the relevant segments; (c) Location of the segments in the ξmath image, ξmath image coordinate system; and (d) Coordinate transformation of each segment to a reference triangle with coordinates η1, η2.

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Next a clipping algorithm is used to determine the segments, corresponding to each pair of elements equation image and e2, where equation image denotes the set of elements on the interface Γmath image. Each specific segment depends on the corresponding nodal coordinates which can be collected in the ordered set

  • equation image(65)

relevant to the segment at hand. For each segment a linear transformation η[RIGHTWARDS ARROW]ξmath image of the form (see Figures 2(c) and (d))

  • equation image(66)

is introduced, where ξmath image denotes the vertices of the segment. In accordance with the results of the clipping algorithm, linear triangular shape functions MK are used. Now we obtain for the discrete values on the interface

  • equation image(67)
  • equation image(68)
  • equation image(69)

Here, κ, β, and ζ denote the nodes corresponding to each particular segment. The mortar integrals for each segment can now be calculated as follows:

  • equation image(70)

Furthermore, the Jacobian Jseg is required

  • equation image(71)

where we have made use of the convective base vectors

  • equation image(72)

in the initial configuration. Eventually, the mortar integrals on each segment can be written as

  • equation image(73)

The segment contributions to the mortar mesh-tying constraints are collected in the vector

  • equation image(74)

where

  • equation image(75)

To perform the assembly of the contributions of all elements equation image on the non-mortar side, the connection between local and global node numbers is stored in the location array LM, so that A = LM(κ, e), for equation image, κ∈{1, …, 4} and equation image. Accordingly, the mortar constraints follow from

  • equation image(76)

Or equivalently

  • equation image(77)

5.2. Augmentation

The above mesh-tying constraints are only frame indifferent, if they are fulfilled in the reference configuration (see Puso 20), which would not be the case in general. To remedy this drawback, we reformulate the domain decomposition constraints in terms of invariants being at most quadratic. Based on the developments in Hesch and Betsch 15 we introduce augmented coordinates dA, which equal the nodal normal vector of the discrete interface Γmath image. To determine the value of the augmented coordinates, we have to apply three additional constraints for each vector dA

  • equation image(78)

where ai are the tangential vectors, evaluated at the placement of the vector dA on the interface . Similar to the above introduction of system vectors q and Θ we collect the augmented coordinates into a system vector d = [d1, …, dmath image]. In contrast to the use of nodal normal vectors, we have applied in our previous developments 15 a single vector dseg for each segment. Thus, there are overall six additional unknowns (three additional coordinates and three Lagrange multipliers) for each segment. In the present work, a dramatic decrease of unknowns is achieved by using a single vector dκ at each node κ on the mortar side of the interface. Interpolating this vector using standard shape functions of the underlying geometry (cf. Betsch and Sänger 24, Section 3) yields

  • equation image(79)

To simplify the numerical quadrature of the mortar integrals, we keep dseg constant within each segment. In particular, we evaluate dseg at the fourth Gauss point equation image (see Figure 2(d)) in the segment, where we also evaluate the tangential vectors

  • equation image(80)

At last, we obtain the following modified constraint functions:

  • equation image(81)

Collecting the above segment contributions into a system vector yields

  • equation image(82)

Similarly, the Lagrange multipliers corresponding to the coordinate augmentation and the mortar constraints are collected in the system vector λ. Overall we obtain seven constraints per node on the mortar side.

5.3. Time discretization

The last step in our development is to discretize the constrained, finite dimensional system in time and to verify the algorithmic conservation properties. Basically we have to discretize a thermomechanical system subject to holonomic constraints. We refer to previous developments (see Hesch and Betsch 15) concerning purely mechanical systems in the context of domain decomposition problems and to Betsch & Steinmann 7 and Gonzalez 25 for details concerning the time discretization of holonomic constraints.

As before, we evaluate the primary variables (position, temperature, and augmented coordinates) using a mid-point-type approximation, whereas the Lagrange multipliers λ[RIGHTWARDS ARROW]λn, n + 1 remain constant within each time step

  • equation image(83)

As we do not change the energy-momentum consistent integrator used for the thermoelastic system, we will focus on the additional terms due to the constraints to verify the conservation properties.

5.3.1. Linear momentum.

Proceeding along the lines of Section 4.1.1, we now obtain

  • equation image(84)

Frame-indifference of the vector of constraints Φ has already been shown in detail in Hesch and Betsch 15 for the purely mechanical case. It can be easily verified that

  • equation image(85)

Here, qmath image = qmath image + εµ, ∀A∈ω and ε∈ℝ is arbitrary. Equation (85) implies

  • equation image(86)

Insertion in (84) yields

  • equation image(87)

which confirms that the constraints do not affect linear momentum conservation.

5.3.2. Angular momentum.

Proceeding along the lines of Section 4.1.2, (53) is replaced by

  • equation image(88)

Based on the results in Hesch and Betsch 15 we can postulate for the domain decomposition constraints

  • equation image(89)

where equation image, ∀A∈ω and equation image, equation image. Accordingly, we can write in analogy to (86)

  • equation image(90)

Due to (83)3 the last term in the square brackets of (90)2 vanishes. Inserting the last equation into (88) yields

  • equation image(91)

which confirms that the constraints do not affect angular momentum conservation as well.

5.3.3. Total energy.

Eventually, we verify algorithmic energy consistency. Proceeding along the lines of Section 4.1.3, we can restrict ourselves to the contributions of the constraints. In particular, we have to show that these contributions are workless. For the mechanical part, conservation of energy can be proven following the arguments in Hesch and Betsch 15, Section 6.1. Similar to result (58), in the present case we obtain

  • equation image(92)

With regard to (63) and (62) we can rewrite the last term in the square brackets (92)

  • equation image(93)

where we have made use of the property equation image. Taking into account the last result, (92) yields

  • equation image(94)

which confirm that the constraints are workless.

Remark

The last statement depends crucially on the accuracy of the numerical evaluation of the mortar integrals for each segment. Remarkably, the proof for linear momentum conservation in Puso 20 leads to the same conclusion. Our numerical experiments have shown that a four point Gauss integration is sufficient.

6. EXAMPLES

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. FINITE STRAIN THERMOELASTODYNAMICS
  5. 3. DISCRETIZATION IN TIME
  6. 4. DISCRETIZATION IN SPACE
  7. 5. FOUR-DIMENSIONAL MORTAR METHOD
  8. 6. EXAMPLES
  9. 7. CONCLUSIONS
  10. Acknowledgements
  11. APPENDIX A: CONSTITUTIVE EQUATIONS
  12. APPENDIX B:ENHANCED ASSUMED STRAIN METHOD
  13. REFERENCES

In this section we evaluate the accuracy and performance of the newly proposed method. An incremental iterative Newton–Raphson solution procedure has been implemented in MATLAB to solve monolithically the non-linear system of equations (48) or (83). The data for the used Ogden model are similar to Holzapfel and Simo 4 and are summarized in Table I (see Appendix A for a summary of the constitutive equations). Enhanced assumed strain elements (see Appendix B for details) have been implemented as well. For all examples we define a stress free reference state in thermal equilibrium based on a homogeneous temperature field Θ0 = 293.15K.

Table I. Material properties.
Ogden modelµ1 = 6.30 × 105N/m2 α1 = 1.3 µ2 = 0.012 × 105N/m2 α2 = 5.0 µ3 = − 0.10 × 105N/m2 α3 = − 2.0
Heat capacityc0 = 1830Nm/kgK
Densityρ0 = 950kg/m3
Linear expansion coefficientα0 = 22.333 × 10−5K−1
Bulk modulusκ(Θ0) = 2.0 × 108N/m2
Empirical coefficientsβ = 9.0
 γ = 2.50
Thermal conductivityK00) = 0.15N/sK
Softening parameterωK = 0.004

6.1. Three-dimensional cracked rectangular bar

As a first example we consider a quasi-static system by setting the density artificially to zero. Apart from that, the coupled transient system has been calculated. Based on the last example in Holzapfel and Simo 4 we define a rectangular bar of the size 10m × 4.8m × 1m. A crack of 1/3 of the width has been inserted into the middle of the system. The reference mesh is displayed in Figure 3.

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Figure 3. Reference configuration.

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The mesh consists of 6912 elements with 36 740 thermal and mechanical degrees of freedom. Furthermore, 62 208 enhanced strain modes are used. Both sides are clamped, i.e. Dirichlet boundary conditions have been applied to the mechanical as well as the thermal boundary, keeping the temperature constant on the boundary at Θ0 = 293.15K. Within each time step Δt = 1s both sides are moved apart with Δx = 0.02m, until an increase of the length of 65% has been reached. The resulting temperature distribution is displayed in Figure 4. As can be seen, the Dirichlet boundaries on both sides are fulfilled exactly. As expected, we receive the highest changes in temperature at the crack tip.

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Figure 4. Deformed configuration of the cracked rectangular bar. The temperature distribution is color coded.

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6.2. Thermoelastic problem

The next problem under consideration consists of a three-dimensional L-shaped block of the size 2.4m × 3.6m × 1.2m. The L-shape has been discretized with 864 elements leading to overall 4900 mechanical and thermal degrees of freedom. We apply a sinusoidal pressure load pmax = 25000N/m2during the time interval t∈[0, 1] to the outer surfaces, as shown in Figure 5. No thermal boundary conditions have been set, hence the system is adiabatically isolated after the load phase. Thus, the body moves freely in space for t∈[1, 5]. Due to the initial loading conditions, the body rotates about three times around the y-axis within 5 s.

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Figure 5. Pressure load on boundaries.

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Three different simulations have been performed:

  • Integrator 1: Newly developed energy-momentum consistent scheme (48) or (83).

  • Integrator 2: Standard mid-point rule, however the right Cauchy-Green tensor has been evaluated according to (26).

  • Integrator 3: Standard mid-point rule.

All three integrators conserve linear and angular momentum. For the proposed energy-momentum scheme the fulfillment of the discrete balance of angular momentum (cf. (53) and (91)) is confirmed in Figure 7. A time step size of Δt = 0.01s has been used during the load phase. After the load phase, the time step size has been doubled to Δt = 0.02s for integrators 1 and 2, whereas for the integrator 3 the time step size has been changed to Δt = 0.011s (larger time steps for the integrator 3 are not possible).

In Figure 6 the absolute values of change in total energy (i.e. the total mechanical and the thermal energy) in each time step after the end of the load phase are plotted in a semilogarithmic scale.

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Figure 6. Change in total energy within each time step.

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Figure 7. Change in angular momentum within each time step.

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As expected, the change in total energy for the energy-momentum scheme is below the stop criterion of the Newton iterations for which the value 10−5 for the norm of the residual vector has been prescribed. Without the concept of the discrete gradient, total energy will not be conserved, but as the results clearly show the simulation remains stable using the integrator 2. The third integrator diverges 19 time steps after the end of the load phase.

6.3. Convergence analysis

To evaluate the accuracy of the proposed algorithm, we simplify the thermoelastic problem in Section 6.2. Four elements are used to discretize the L-Shape, furthermore the maximum pressure has been changed to pmax = 65000N/m2and applied for the first 0.5 s as shown in Figure 8.

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Figure 8. Load curve for the convergence analysis.

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Additionally, we changed the Bulk modulus to κ(Θ0) = 2.0 × 105N/m2, such that the body can change its volume. The maximum temperature reached Θmax≈296K and the minimum temperature Θmin≈291K. After the load phase, the calculation was terminated at t = 1s. We calculate a reference solution qRef using a time step size of Δt = 10−4s. Note that we set the tolerance of the Newton iteration to 10−6 with regard to the L2-norm of the residual vector for all calculations. To run the analysis, we have calculated the system using Δt∈{10−2s, 5 × 10−3s, 2.5 × 10−3s, 10−3s, 5 × 10−4s, 2.5 × 10−4s}. The relative error for the discrete system has been determined as follows:

  • equation image(95)

The values for the velocity as well as for the temperature have been generated analogously. As can be observed from Figure 9, the proposed algorithm is second-order accurate.

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Figure 9. Convergence results.

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6.4. Thermal domain decomposition problem

This example utilizes the same geometry of a three-dimensional L-shape as before. However, two subdomains of the L-shaped block have been meshed independently (see Figure 10). The larger subdomain consists of 840 elements, whereas the smaller subdomain consists of 675 elements. 1584 segments have to be computed for the domain decomposition interface. Additional 77 normal vectors with altogether 231 augmented coordinates as unknowns are used. In contrast to the present node-based augmentation, our original segment-based augmentation technique (see Hesch and Betsch 15) leads to 1584 normal vectors with altogether 4785 additional unknowns.

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Figure 10. Reference configuration.

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Thus, the system consists of overall 9110 thermal, mechanical, and augmented degrees of freedom in conjunction with 539 constraints (231 constraints for the augmented coordinates and 308 for the mortar constraints). First we examine the influence of the mortar method on the pure heat conduction problem, i.e. we exclude all mechanical equations from the system and apply a linear distribution of temperature to the adiabatically isolated L-shape (cf. Figure 11, left). Figure 11 (central) shows the temperature distribution after 2 h 45 min, whereas Figure 11 (right) shows the temperature distribution after 5 h 30 min. At this time, thermodynamic equilibrium has nearly been reached. Figure 12 shows the total energy of the system over time.

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Figure 11. Temperature distribution at different times.

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Figure 12. Energy over time (in s).

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As can be seen, energy is conserved, which reflects the first law of thermodynamics. The second law states that the entropy production remains equal or greater zero, which can be observed in this particular example from Figure 13.

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Figure 13. Entropy production over time (in s).

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6.5. Thermoelastic domain decomposition problem

We next deal with the completely coupled, transient thermoelastic problem. The same mechanical configuration as in Example 6.4 has been used combined with the load phase described in Example 6.2. Again, after the load phase the adiabatically isolated system moves freely in space and the time step size has been set to Δt = 0.02s. The proposed energy-momentum consistent scheme obeys the first law of thermodynamics, as shown in Figure 14. For the problem at hand, the entropy production is displayed in Figure 15.

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Figure 14. Energy over time (in s).

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Figure 15. Entropy production over time (in s).

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7. CONCLUSIONS

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. FINITE STRAIN THERMOELASTODYNAMICS
  5. 3. DISCRETIZATION IN TIME
  6. 4. DISCRETIZATION IN SPACE
  7. 5. FOUR-DIMENSIONAL MORTAR METHOD
  8. 6. EXAMPLES
  9. 7. CONCLUSIONS
  10. Acknowledgements
  11. APPENDIX A: CONSTITUTIVE EQUATIONS
  12. APPENDIX B:ENHANCED ASSUMED STRAIN METHOD
  13. REFERENCES

A novel energy-momentum consistent scheme for thermoelastodynamics along with its extension to domain decomposition problems has been presented. The present approach can be viewed as straightforward extension of energy-momentum integrators to the realm of thermoelasticity. In essence, the present method relies on an enhancement of the notion of a discrete gradient to the coupled problem at hand. Arbitrary constitutive laws for thermoelasticity can be directly used. This is in contrast to the design of thermodynamically consistent integrators proposed by Romero 11, 12, which requires a reformulation within the GENERIC framework.

The extension of the present energy-momentum consistent approach to domain decomposition problems is based on a four-dimensional mortar method for thermoelastic systems. Our new developments imply a significant modification of our original approach which has been confined to isothermal problems. In particular the present work reduces the number of augmented unknowns as well as the number of constraints drastically, improving the performance of the mortar method far beyond the performance of previous approaches.

The present work will provide the foundation for the development of new energy-momentum consistent integrators for thermomechanically coupled contact problems. Mortar-based algorithms for large deformation contact will be the subject of a follow-up paper.

APPENDIX A: CONSTITUTIVE EQUATIONS

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. FINITE STRAIN THERMOELASTODYNAMICS
  5. 3. DISCRETIZATION IN TIME
  6. 4. DISCRETIZATION IN SPACE
  7. 5. FOUR-DIMENSIONAL MORTAR METHOD
  8. 6. EXAMPLES
  9. 7. CONCLUSIONS
  10. Acknowledgements
  11. APPENDIX A: CONSTITUTIVE EQUATIONS
  12. APPENDIX B:ENHANCED ASSUMED STRAIN METHOD
  13. REFERENCES

The constitutive equations rely on the developments of Holzapfel and Simo 4, where a detailed derivation of the equations can be found. Since the development of constitutive equations is not the main focus of this paper, we restrict ourselves to the well-known Ogden material given in the aforementioned paper, written in terms of the eigenvalues λmath image, A∈[1, 2, 3] of the right Cauchy-Green tensor.

A.1. Free Helmholtz energy

Since materials like rubber behave differently in bulk and shear, we split the free Helmholtz energy function additive into a volumetric and deviatoric part

  • equation image(A1)

with J = λ1λ2λ3 and equation image. The volumetric part reads as

  • equation image(A2)

with

  • equation image(A3)

On the other hand, the deviatoric part reads as

  • equation image(A4)
A.2. Derivatives of the free Helmholtz energy

With regard to the additive split of the free Helmholtz energy (A2), we can split the second Piola-Kirchhoff stress tensor as

  • equation image(A5)

The volumetric part of the stress tensor reads as

  • equation image(A6)

with

  • equation image(A7)

For the derivatives of equation image we recall the relation

  • equation image(A8)

The isochoric contribution to the stress tensor reads as

  • equation image(A9)

with

  • equation image(A10)

and

  • equation image(A11)

Thus, we obtain

  • equation image(A12)

For the sake of completeness we deduce the derivative of the free Helmholtz energy function with respect to the temperature

  • equation image(A13)
A.3. Duhamel's law

Concerning the thermal conductivity tensor in (2) we set

  • equation image(A14)

with

  • equation image(A15)

Note that the above constitutive laws are in accordance with the second law of thermodynamics.

APPENDIX B:ENHANCED ASSUMED STRAIN METHOD

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. FINITE STRAIN THERMOELASTODYNAMICS
  5. 3. DISCRETIZATION IN TIME
  6. 4. DISCRETIZATION IN SPACE
  7. 5. FOUR-DIMENSIONAL MORTAR METHOD
  8. 6. EXAMPLES
  9. 7. CONCLUSIONS
  10. Acknowledgements
  11. APPENDIX A: CONSTITUTIVE EQUATIONS
  12. APPENDIX B:ENHANCED ASSUMED STRAIN METHOD
  13. REFERENCES

Standard displacement-based elements are subject to volumetric locking effects in the incompressible limit. To enhance the performance of the tri-linear displacement elements we incorporate the enhanced assumed strain method developed by Simo et al. 26, 27 into the present energy-momentum consistent framework.

Based on the developments in Simo et al. 27 the following operator is introduced:

  • equation image(B1)

with

  • equation image(B2)

The enhanced deformation gradient can now be written as

  • equation image(B3)

where

  • equation image(B4)

Here, γJ are gamma-stabilization vectors and ℋJ hourglass functions, defined in Belytschko et al. 28. Additionally we introduce Wilson's incompatible shape functions (see Wilson et al. 29) for tri-linear brick elements

  • equation image(B5)

Now the discrete enhanced version of (10)1 reads as

  • equation image(B6)

where equation image. Concerning the discretization in time the additional α-modes are evaluated in the mid-point

  • equation image(B7)

The enhanced form of the fully discrete system (48) can now be written in the form

  • equation image(B8)

Note that the additional equations (118)2 can be eliminated using standard condensation procedures. We further remark that the enhancement of the space discretization outlined above does not affect the conservation and consistency properties of the present energy-momentum integrator.

  • Note that the tangential vectors are not uniquely defined at the nodes on a C(0) surface. A common approach is to use an average of the tangential vectors of the adjacent elements.

REFERENCES

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. FINITE STRAIN THERMOELASTODYNAMICS
  5. 3. DISCRETIZATION IN TIME
  6. 4. DISCRETIZATION IN SPACE
  7. 5. FOUR-DIMENSIONAL MORTAR METHOD
  8. 6. EXAMPLES
  9. 7. CONCLUSIONS
  10. Acknowledgements
  11. APPENDIX A: CONSTITUTIVE EQUATIONS
  12. APPENDIX B:ENHANCED ASSUMED STRAIN METHOD
  13. REFERENCES
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