src-local/log-conform-elastoviscoplastic-HB-scalar-2D.h

Log-Conformation (Scalar 2D/Axi EVP-HB)

Scalar log-conformation implementation for 2D and axisymmetric elasto-viscoplastic Herschel-Bulkley flows.

Key Features

• Conformation tensor components stored as scalars.
• Supports 2D and axisymmetric configurations.
• Mirrors log-conform-elastoviscoplastic-HB-scalar-3D.h.

Change Log

• 2024-10-18: Initial 2D/axi implementation.
• 2024-11-03: Axisymmetric mirror of 3D scalar version.
• 2024-11-14: Infinite Deborah support.
• 2024-11-23: Documentation updates.
• 2026-02-02: Unify MultiRheoFlow docs across all submodules.
• 2026-02-09: Add HB variant with nHB and HB relaxation factor.

Future Work

• Convert to tensor formulation for consistency and maintainability.
• Track: comphy-lab/Viscoelastic3D#11, #5.

Author

Vatsal Sanjay, Arivazhagan G. Balasubramanian Email: [email protected] Last updated: 2025-06-30

The log-conformation method for EVP-HB constitutive models

Introduction

Elasto-viscoplastic fluids exhibit viscous, elastic and plastic behaviour when subjected to deformation. Therefore these materials are governed by the Navier–Stokes equations enriched with an extra stress \(Tij\) \[ \rho\left[\partial_t\mathbf{u}+\nabla\cdot(\mathbf{u}\otimes\mathbf{u})\right] = - \nabla p + \nabla\cdot(2\mu_s\mathbf{D}) + \nabla\cdot\mathbf{T} + \rho\mathbf{a} \] where \(\mathbf{D}=[\nabla\mathbf{u} + (\nabla\mathbf{u})^T]/2\) is the deformation tensor and \(\mu_s\) is the solvent viscosity of the EVP fluid.

The extra stress \(\mathbf{T}\) includes memory effects due to the polymers and the plasticity effects associated with internal structural changes in the material that constitutes to yielding the material and fluidizes it. Several constitutive rheological models are available in the literature where the extra stress \(\mathbf{T}\) is typically a function \(\mathbf{f_s}(\cdot)\) of the conformation tensor \(\mathbf{A}\) such as \[ \mathbf{T} = G_p \mathbf{f_s}(\mathbf{A}) \] where \(G_p\) is the elastic modulus and \(\mathbf{f_s}(\cdot)\) is the relaxation function.

The conformation tensor \(\mathbf{A}\) is related to the deformation of the elasto-viscoplastic matrix. \(\mathbf{A}\) is governed by the equation \[ D_t \mathbf{A} - \mathbf{A} \cdot \nabla \mathbf{u} - \nabla \mathbf{u}^{T} \cdot \mathbf{A} = -\frac{\mathbf{f_r}(\mathbf{A})}{\lambda} \] where \(D_t\) denotes the material derivative and \(\mathbf{f_r}(\cdot)\) is the relaxation function. Here, \(\lambda\) is the relaxation time.

In the case of elasto-viscoplastic Herschel-Bulkley (HB) model, where the material before yielding is represented by Kelvin-Voigt viscoelastic solid and long after yielding recovers the Oldroyd-B type viscoelastic fluid behaviour, \(\mathbf{f}_s (\mathbf{A}) = \mathbf{A} -\mathbf{I}\), \(\mathbf{f}_r (\mathbf{A}) = K_{HB}(\mathbf{A} -\mathbf{I})\), and the above equations can be combined to avoid the use of \(\mathbf{A}\) \[ (K_{HB}/\lambda) \mathbf{T} + (D_t \mathbf{T} - \mathbf{T} \cdot \nabla \mathbf{u} - \nabla \mathbf{u}^{T} \cdot \mathbf{T}) = 2 G_p\lambda \mathbf{D} \]

Here, \[ K_{HB} = max\left[0, \left(\frac{\|\tau_d\|-\tau_y} {(G_p\lambda)^{1-n_{HB}} \|\tau_d\|}\right)^{1/n_{HB}}\right] \] where \(n_{HB}=1\) recovers the Saramito yield factor. The stress closure is \[ \mathbf{\tau}_p = \frac{Oh_p}{De}(\mathbf{A}-\mathbf{I}), \quad Oh_p = G_p\lambda \] which is implemented as \(\mathbf{T} = G_p(\mathbf{A}-\mathbf{I})\). Comminal et al. (2015) gathered the functions \(\mathbf{f}_s (\mathbf{A})\) and \(\mathbf{f}_r (\mathbf{A})\) for different viscoelastic constitutive models. This work is an extension of such formulation for elasto-viscoplastic materials.

Parameters

The primary parameters are the relaxation time \(\lambda\), elastic modulus \(G_p\), yield stress \(\tau_y\) and HB exponent \(n_{HB}\). The solvent viscosity \(\mu_s\) is defined in the Navier-Stokes solver.

Gp, lambda, tau_y and nHB are defined in two-phaseEVP-HB.h.

The log conformation approach

The numerical resolution of viscoelastic and EVP fluid problems often faces the High-Weissenberg Number Problem. This is a numerical instability appearing when strongly elastic flows create regions of high stress and fine features. This instability poses practical limits to the values of the relaxation time of the viscoelastic and EVP fluid, \(\lambda\). Fattal & Kupferman (2004, 2005) identified the exponential nature of the solution as the origin of the instability. They proposed to use the logarithm of the conformation tensor \(\Psi = \log \, \mathbf{A}\) rather than the extra stress tensor to circumvent the instability.

The constitutive equation for the log of the conformation tensor is \[ D_t \Psi = (\Omega \cdot \Psi -\Psi \cdot \Omega) + 2 \mathbf{B} + \frac{e^{-\Psi} \mathbf{f}_r (e^{\Psi})}{\lambda} \] where \(\Omega\) and \(\mathbf{B}\) are tensors that result from the decomposition of the transpose of the tensor gradient of the velocity \[ (\nabla \mathbf{u})^T = \Omega + \mathbf{B} + N \mathbf{A}^{-1} \]

The antisymmetric tensor \(\Omega\) requires only the memory of a scalar in 2D since, \[ \Omega = \left( \begin{array}{cc} 0 & \Omega_{12} \\ -\Omega_{12} & 0 \end{array} \right) \]

For 3D, \(\Omega\) is a skew-symmetric tensor given by

\[ \Omega = \left( \begin{array}{ccc} 0 & \Omega_{12} & \Omega_{13} \\ -\Omega_{12} & 0 & \Omega_{23} \\ -\Omega_{13} & -\Omega_{23} & 0 \end{array} \right) \]

The log-conformation tensor, \(\Psi\), is related to the polymeric stress tensor \(\mathbf{T}\), by the strain function \(\mathbf{f}_s (\mathbf{A})\) \[ \Psi = \log \, \mathbf{A} \quad \mathrm{and} \quad \mathbf{T} = \frac{G_p}{\lambda} \mathbf{f}_s (\mathbf{A}) \] where \(Tr\) denotes the trace of the tensor and \(L\) is an additional property of the viscoelastic fluid.

We will use the Bell–Collela–Glaz scheme to advect the log-conformation tensor \(\Psi\).

/*
TODO:
- Perhaps, instead of the Bell--Collela--Glaz scheme, we can use the conservative form of the advection equation and transport the log-conformation tensor with the VoF color function, similar to [http://basilisk.fr/src/navier-stokes/conserving.h](http://basilisk.fr/src/navier-stokes/conserving.h)
*/

#include "bcg.h"

(const) scalar Gp = unity; // elastic modulus
(const) scalar lambda = unity; // relaxation time
(const) scalar tau0 = unity; // yield-stress
(const) scalar nHB = unity; // HB power-law exponent
double saramitoHB_yield_eps = 1e-6; // regularization for HB denominator

scalar A11[], A12[], A22[]; // conformation tensor
scalar T11[], T12[], T22[]; // stress tensor
#if AXI
scalar AThTh[], T_ThTh[];
#endif

event defaults (i = 0) {
  if (is_constant (a.x))
    a = new face vector;

  /*
  initialize A and T
  */
  for (scalar s in {A11, A22}) {
    foreach () {
      s[] = 1.;
    }
  }
  for (scalar s in {T11, T12, T22, A12}) {
    foreach(){
      s[] = 0.;
    }
  }
#if AXI
  foreach(){
    T_ThTh[] = 0;
    AThTh[] = 1.;
  }
#endif

  for (scalar s in {T11, T12, T22}) {
    if (s.boundary[left] != periodic_bc) {
        s[left] = neumann(0);
          s[right] = neumann(0);
      }
  }

  for (scalar s in {A11, A12, A22}) {
    if (s.boundary[left] != periodic_bc) {
        s[left] = neumann(0);
          s[right] = neumann(0);
    }
  }

#if AXI
  T12[bottom] = dirichlet (0.);
  A12[bottom] = dirichlet (0.);
#endif
}

Useful functions in 2D

The first step is to implement a routine to calculate the eigenvalues and eigenvectors of the conformation tensor \(\mathbf{A}\).

These structs ressemble Basilisk vectors and tensors but are just arrays not related to the grid.

typedef struct { double x, y;}   pseudo_v;
typedef struct { pseudo_v x, y;} pseudo_t;

// Function to initialize pseudo_v
static inline void init_pseudo_v(pseudo_v *v, double value) {
    v->x = value;
    v->y = value;
}

// Function to initialize pseudo_t
static inline void init_pseudo_t(pseudo_t *t, double value) {
    init_pseudo_v(&t->x, value);
    init_pseudo_v(&t->y, value);
}

static void diagonalization_2D (pseudo_v * Lambda, pseudo_t * R, pseudo_t * A)
{

The eigenvalues are saved in vector \(\Lambda\) computed from the trace and the determinant of the symmetric conformation tensor \(\mathbf{A}\).

  if (sq(A->x.y) < 1e-15) {
    R->x.x = R->y.y = 1.;
    R->y.x = R->x.y = 0.;
    Lambda->x = A->x.x; Lambda->y = A->y.y;
    return;
  }

  double T = A->x.x + A->y.y; // Trace of the tensor
  double D = A->x.x*A->y.y - sq(A->x.y); // Determinant

The eigenvectors, \(\mathbf{v}_i\) are saved by columns in tensor \(\mathbf{R} = (\mathbf{v}_1|\mathbf{v}_2)\).

  R->x.x = R->x.y = A->x.y;
  R->y.x = R->y.y = -A->x.x;
  double s = 1.;
  for (int i = 0; i < dimension; i++) {
    double * ev = (double *) Lambda;
    ev[i] = T/2 + s*sqrt(sq(T)/4. - D);
    s *= -1;
    double * Rx = (double *) &R->x;
    double * Ry = (double *) &R->y;
    Ry[i] += ev[i];
    double mod = sqrt(sq(Rx[i]) + sq(Ry[i]));
    Rx[i] /= mod;
    Ry[i] /= mod;
  }
}

The stress tensor depends on previous instants and has to be integrated in time. In the log-conformation scheme the advection of the stress tensor is circumvented, instead the conformation tensor, \(\mathbf{A}\) (or more precisely the related variable \(\Psi\)) is advanced in time.

In what follows we will adopt a scheme similar to that of Hao & Pan (2007). We use a split scheme, solving successively

1. the upper convective term: \[ \partial_t \Psi = 2 \mathbf{B} + (\Omega \cdot \Psi -\Psi \cdot \Omega) \]
2. the advection term: \[ \partial_t \Psi + \nabla \cdot (\Psi \mathbf{u}) = 0 \]
3. the model term (but set in terms of the conformation tensor \(\mathbf{A}\)). In an Oldroyd-B viscoelastic fluid, the model is \[ \partial_t \mathbf{A} = -\frac{\mathbf{f}_r (\mathbf{A})}{\lambda} \]

event tracer_advection(i++)
{
  scalar Psi11 = A11;
  scalar Psi12 = A12;
  scalar Psi22 = A22;
#if AXI
  scalar Psiqq = AThTh;
#endif

Computation of \(\Psi = \log \mathbf{A}\) and upper convective term

  foreach() {

We assume that the stress tensor \(\mathbf{\tau}_p\) depends on the conformation tensor \(\mathbf{A}\) as follows \[ \mathbf{\tau}_p = G_p (\mathbf{A}) = G_p (\mathbf{A} - I) \]

    pseudo_t A;

    A.x.x = A11[]; A.y.y = A22[];
    A.x.y = A12[];

#if AXI
    double Aqq = AThTh[];
    Psiqq[] = log (Aqq);
#endif

The conformation tensor is diagonalized through the eigenvector tensor \(\mathbf{R}\) and the eigenvalues diagonal tensor, \(\Lambda\).

    pseudo_v Lambda;
    init_pseudo_v(&Lambda, 0.0);
    pseudo_t R;
    init_pseudo_t(&R, 0.0);
    diagonalization_2D (&Lambda, &R, &A);

    /*
    Check for negative eigenvalues -- this should never happen. If it does, print the location and value of the offending eigenvalue.
    Please report this bug by opening an issue on the GitHub repository.
    */
    if (Lambda.x <= 0. || Lambda.y <= 0.) {
      fprintf(ferr, "Negative eigenvalue detected: Lambda.x = %g, Lambda.y = %g\n", Lambda.x, Lambda.y);
      fprintf(ferr, "x = %g, y = %g\n", x, y);
      exit(1);
    }

\(\Psi = \log \mathbf{A}\) is easily obtained after diagonalization, \(\Psi = R \cdot \log(\Lambda) \cdot R^T\).

    Psi12[] = R.x.x*R.y.x*log(Lambda.x) + R.y.y*R.x.y*log(Lambda.y);
    Psi11[] = sq(R.x.x)*log(Lambda.x) + sq(R.x.y)*log(Lambda.y);
    Psi22[] = sq(R.y.y)*log(Lambda.y) + sq(R.y.x)*log(Lambda.x);

We now compute the upper convective term \(2 \mathbf{B} + (\Omega \cdot \Psi -\Psi \cdot \Omega)\).

The diagonalization will be applied to the velocity gradient \((\nabla u)^T\) to obtain the antisymmetric tensor \(\Omega\) and the traceless, symmetric tensor, \(\mathbf{B}\). If the conformation tensor is \(\mathbf{I}\), \(\Omega = 0\) and \(\mathbf{B}= \mathbf{D}\).

Otherwise, compute M = R * (nablaU)^T * R^T, where nablaU is the velocity gradient tensor. Then,

1. Calculate omega using the off-diagonal elements of M and eigenvalues: omega = (Lambda.yM.x.y + Lambda.xM.y.x)/(Lambda.y - Lambda.x) This represents the rotation rate in the eigenvector basis.

2. Transform omega back to physical space to get OM: OM = (R.x.xR.y.y - R.x.yR.y.x)*omega This gives us the rotation tensor Omega in the original coordinate system.

3. Compute B tensor components using M and R: B is related to M and R through:

   In 2D: \[ B_{xx} = R_{xx}^2 M_{xx} + R_{xy}^2 M_{yy} \\ B_{xy} = R_{xx}R_{yx} M_{xx} + R_{xy}R_{yy} M_{yy} \\ B_{yx} = B_{xy} \\ B_{yy} = -B_{xx} \]

   Where:

   • R is the eigenvector matrix of the conformation tensor
   • M is the velocity gradient tensor in the eigenvector basis
   • The construction ensures B is symmetric and traceless

    pseudo_t B;
    init_pseudo_t(&B, 0.0);
    double OM = 0.;
    if (fabs(Lambda.x - Lambda.y) <= 1e-20) {
      B.x.y = (u.y[1,0] - u.y[-1,0] + u.x[0,1] - u.x[0,-1])/(4.*Delta);
      foreach_dimension()
        B.x.x = (u.x[1,0] - u.x[-1,0])/(2.*Delta);
    } else {
      pseudo_t M;
      init_pseudo_t(&M, 0.0);
      foreach_dimension() {
        M.x.x = (sq(R.x.x)*(u.x[1] - u.x[-1]) +
        sq(R.y.x)*(u.y[0,1] - u.y[0,-1]) +
        R.x.x*R.y.x*(u.x[0,1] - u.x[0,-1] +
        u.y[1] - u.y[-1]))/(2.*Delta);

        M.x.y = (R.x.x*R.x.y*(u.x[1] - u.x[-1]) +
        R.x.y*R.y.x*(u.y[1] - u.y[-1]) +
        R.x.x*R.y.y*(u.x[0,1] - u.x[0,-1]) +
        R.y.x*R.y.y*(u.y[0,1] - u.y[0,-1]))/(2.*Delta);
      }
      double omega = (Lambda.y*M.x.y + Lambda.x*M.y.x)/(Lambda.y - Lambda.x);
      OM = (R.x.x*R.y.y - R.x.y*R.y.x)*omega;

      B.x.y = M.x.x*R.x.x*R.y.x + M.y.y*R.y.y*R.x.y;
      foreach_dimension()
        B.x.x = M.x.x*sq(R.x.x)+M.y.y*sq(R.x.y);
    }

We now advance \(\Psi\) in time, adding the upper convective contribution.

    double s = -Psi12[];
    Psi12[] += dt * (2. * B.x.y + OM * (Psi22[] - Psi11[]));
    s *= -1;
    Psi11[] += dt * 2. * (B.x.x + s * OM);
    s *= -1;
    Psi22[] += dt * 2. * (B.y.y + s * OM);

In the axisymmetric case, the governing equation for \(\Psi_{\theta \theta}\) only involves that component, \[ \Psi_{\theta \theta}|_t - 2 L_{\theta \theta} = \frac{\mathbf{f}_r(e^{-\Psi_{\theta \theta}})}{\lambda} \] with \(L_{\theta \theta} = u_y/y\). Therefore step (a) for \(\Psi_{\theta \theta}\) is

#if AXI
    Psiqq[] += dt*2.*u.y[]/max(y, 1e-20);
#endif

}

Advection of \(\Psi\)

We proceed with step (b), the advection of the log of the conformation tensor \(\Psi\).

#if AXI
  advection ({Psi11, Psi12, Psi22, Psiqq}, uf, dt);
#else
  advection ({Psi11, Psi12, Psi22}, uf, dt);
#endif

Convert back to Aij

  foreach() {

It is time to undo the log-conformation, again by diagonalization, to recover the conformation tensor \(\mathbf{A}\) and to perform step (c).

    pseudo_t A = {{Psi11[], Psi12[]}, {Psi12[], Psi22[]}}, R;
    init_pseudo_t(&R, 0.0);
    pseudo_v Lambda;
    init_pseudo_v(&Lambda, 0.0);
    diagonalization_2D (&Lambda, &R, &A);
    Lambda.x = exp(Lambda.x), Lambda.y = exp(Lambda.y);

    A.x.y = R.x.x*R.y.x*Lambda.x + R.y.y*R.x.y*Lambda.y;
    foreach_dimension()
      A.x.x = sq(R.x.x)*Lambda.x + sq(R.x.y)*Lambda.y;
#if AXI
      double Aqq = exp(Psiqq[]);
#endif

We perform now step (c) by integrating \(\mathbf{A}_t = -\mathbf{f}_r (\mathbf{A})/\lambda\) to obtain \(\mathbf{A}^{n+1}\). This step is analytic, \[ \int_{t^n}^{t^{n+1}}\frac{d \mathbf{A}}{\mathbf{I}- \mathbf{A}} = \frac{\Delta t}{\lambda} \]

#if AXI
     double tauD = sqrt((1./6.)*((T11[] - T22[])*(T11[] - T22[])+(T22[] - T_ThTh[])*(T22[] - T_ThTh[])+(T_ThTh[] - T11[])*(T_ThTh[] - T11[])) + T12[]*T12[]);
#else
    double tauD = sqrt(0.25*(T11[] - T22[])*(T11[] - T22[]) + T12[]*T12[]);
#endif
    double nHB_loc = max(nHB[], 1e-12);
    double ohp = max(fabs(Gp[]*lambda[]), saramitoHB_yield_eps);
    double hbArgument = (tauD - tau0[])/
      (pow(ohp, 1. - nHB_loc)*(tauD + saramitoHB_yield_eps));
    double KHB = max(0., pow(max(hbArgument, 0.), 1./nHB_loc)); // $K_{HB}$
    double intFactor = (lambda[] != 0. ?
      (lambda[] == 1e30 ? 1: exp(-dt*KHB/lambda[])): 0.);

#if AXI
      Aqq = (1. - intFactor) + intFactor*exp(Psiqq[]);
#endif

    A.x.y *= intFactor;
    foreach_dimension()
      A.x.x = (1. - intFactor) + A.x.x*intFactor;

Then the Conformation tensor \(\mathcal{A}_p^{n+1}\) is restored from \(\mathbf{A}^{n+1}\).

    A12[] = A.x.y;
    T12[] = Gp[]*A.x.y;
#if AXI
      AThTh[] = Aqq;
      T_ThTh[] = Gp[]*(Aqq - 1.);
#endif

    A11[] = A.x.x;
    T11[] = Gp[]*(A.x.x - 1.);
    A22[] = A.y.y;
    T22[] = Gp[]*(A.y.y - 1.);
  }
}

Divergence of the extra stress tensor

The extra stress tensor \(\mathbf{\tau}_p\) is defined at cell centers while the corresponding force (acceleration) will be defined at cell faces. Two terms contribute to each component of the momentum equation. For example the \(x\)-component in Cartesian coordinates has the following terms: \(\partial_x \mathbf{\tau}_{p_{xx}} + \partial_y \mathbf{\tau}_{p_{xy}}\). The first term is easy to compute since it can be calculated directly from center values of cells sharing the face. The other one is harder. It will be computed from vertex values. The vertex values are obtained by averaging centered values. Note that as a result of the vertex averaging cells [] and [-1,0] are not involved in the computation of shear.

event acceleration (i++)
{
  face vector av = a;

  foreach_face(x){
    if (fm.x[] > 1e-20) {

      double shearX = (T12[0,1]*cm[0,1] + T12[-1,1]*cm[-1,1] -
      T12[0,-1]*cm[0,-1] - T12[-1,-1]*cm[-1,-1])/4.;

      av.x[] += (shearX + cm[]*T11[] - cm[-1]*T11[-1])*
      alpha.x[]/(sq(fm.x[])*Delta);

    }
  }

  foreach_face(y){
    if (fm.y[] > 1e-20) {

      double shearY = (T12[1,0]*cm[1,0] + T12[1,-1]*cm[1,-1] -
      T12[-1,0]*cm[-1,0] - T12[-1,-1]*cm[-1,-1])/4.;

      av.y[] += (shearY + cm[]*T22[] - cm[0,-1]*T22[0,-1])*
      alpha.y[]/(sq(fm.y[])*Delta);

    }
  }

#if AXI
  foreach_face(y)
    if (y > 1e-20)
      av.y[] -= (T_ThTh[] + T_ThTh[0,-1])*alpha.y[]/sq(y)/2.;
#endif
}