How to Solve an Algebraic Equation Involving an Integral in MOOSE

[bdliangxy]

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Question

I would like to solve the following algebraic equation in MOOSE:

$$ \int_{\text{boundary}} j_0(c) \exp\big(f(\phi, c)\big) dL = C $$

​ where there are two variables: $ c $ and $ \phi $ . The variable $ c $ is solved by other modules, while $ \phi $ is the unknown to be determined from this equation. Here, $ j_0(c) $ is a known function of $ c $ , $ f(\phi, c) $ is a given function of both $ \phi $ and $ c $ , and $ C $ is a constant.

In COMSOL, such an equation can be handled using the ODE module. I am trying to use ParsedODEKernel in MOOSE for this purpose. However, the equation involves a boundary integral, and as far as I know, MOOSE's integral operations are only available in postprocessors. This poses a problem. Because postprocessors require all quantities inside the integral to be fully known at the time of evaluation, whereas I need to solve for $ \phi $ that's unknown and appears inside the integrand . Does anyone have suggestions on how to implement this using built-in MOOSE modules ? If writing a custom ODEKernel is unavoidable, what would be a good strategy or reference to follow?

[GiudGiud]

Hello

ODE kernel in MOOSE is meant for equations that live on a single node, possibly coupled to a term at varies (or is being solved for) across multiple nodes.

This equation is living on all the nodes of a boundary at once. I would actually recommend creating an integrated boundary condition to represent this equation.
For the DoFs that do not live on the boundary of interest, use a NullKernel to avoid having rows of 0 in the Jacobian.

[bdliangxy]

Thank you for your suggestion. Your proposal precisely matches my needs—indeed, this integral is intended for use as a boundary condition. However, I have been performing the integration and applying the boundary condition separately, and I hadn’t realized there was an "integrated boundary condition" option available. I’ll follow your advice and give it a try; it should resolve my issue.

[bdliangxy]

@GiudGiud More specifically, ϕ is a scalar variable, c is a field variable . I need to impose the value distribution of the integrand (evaluated at the boundary) as a Neumann boundary condition, while simultaneously ensuring that the total integral over the boundary satisfies the aforementioned equation. Perhaps using an integrated boundary condition combined with a Lagrange multiplier constraint would be a better approach—I have already verified its feasibility in COMSOL. Are there any comparable examples or reference cases in MOOSE that I could draw upon?

[GiudGiud]

if you have two equations for the same variable on the boundary, then yes I think a Lagrange multiplier approach makes sense.

Can you try using a scalar variable for the lagrange multiplier?
With this kernel
https://mooseframework.inl.gov/source/scalarkernels/AverageValueConstraint.html

[bdliangxy]

During my verification in COMSOL, I indeed used a scalar variable as the Lagrange multiplier, and this approach proved effective.
I’ve noticed the kernel you provided—it’s quite close to my requirements, but there are some differences:

1. The integration domain of that kernel is over a domain, whereas my requirement involves integration over a boundary .

2. The integrand in that kernel is a nonlinear variable, while in my case, it’s an equation (i.e., a constraint that must be satisfied).

3. My integral includes an unsolved scalar variable (which also serves as a Lagrange multiplier). Since this variable is unknown, post-processing tools cannot evaluate integrals containing such unknowns.

It seems I cannot directly use this kernel as-is. Do you have any special techniques or workarounds for this situation? Or would I need to modify the kernel’s source code to meet my needs? Of course, I’d prefer to leverage official MOOSE kernels as much as possible

[bdliangxy]

@GiudGiud

Hello,

Due to the time elapsed since my last comment, I would like to reiterate and provide some supplementary information regarding my problem.

Problem Description:

• $\phi$ is a scalar variable (constant throughout the domain, not spatially varying).
• $c$ is a field variable.
• The integral of the function needs to be applied on the boundary as a c flux constraint.

Attempts and Issues:

1. AverageValueConstraint: I attempted to use this as suggested previously. However, its parent class is ScalarKernel, which does not support coupling with field variables or performing boundary integrals directly in the manner I require. With the help of an AI assistant, I tried writing a custom kernel to handle this, but I was unable to get it to run successfully.
2. IntegratedBoundaryCondition + NullKernel: I also tried combining these. However, this approach seems mismatched for my needs. Since $\phi$ must remain a uniform scalar (not varying spatially), using a NullKernel on the boundary might inadvertently allow $\phi$ to vary spatially along the boundary, which violates the physical definition of my variable.

Governing Equation:

For clarity, I omitted the specific expression in my initial inquiry. The equation I am trying to solve at the boundary is the Butler-Volmer equation, which couples the scalar $\phi$ and the field variable $c$ via a boundary integral.

$$ \int_{\text{boundary}} k_0\sqrt{c(1-c)} \left[ \exp\left( \frac{\alpha F}{RT} (\phi - E_{\text{eq}}(c)) \right) - \exp\left( -\frac{\alpha F}{RT} (\phi - E_{\text{eq}}(c)) \right) \right] dL = I_{\text{in}}$$

where $E_eq(c)$is an interpolation function.

This problem has consumed a significant amount of my time, and I have been unable to find a working solution within the MOOSE framework. Could you please provide some guidance on the correct approach to implement a scalar constraint driven by a boundary integral of a field variable?

Thank you very much for your help.

[GiudGiud]

Hello

I think an integrated boundary condition for the variable c remains the best option.

Since ϕ must remain a uniform scalar (not varying spatially), using a NullKernel on the boundary might inadvertently allow ϕ to vary spatially along the boundary, which violates the physical definition of my variable.

Phi should remain a scalar variable, not become a field variable
An integrated BC for variable c should be able to depend on a scalar variable

[bdliangxy]

It really works! Love you so much! It just takes me a lot time while handling a scalar variable in integrated BC .
Due to the different basis functions for scalar and field variables, I had to manually add the residuals and Jacobian matrices for the global constraint equations. Fortunately, I have resolved this issue.