# Strain from FEM simulations to strain gauge measurements

I am looking for some intuition in making comparisons of FEM simulations to experimental measurements. In particular, I am interested in comparisons to strain gauge readings, and perhaps even LVDTs.

From an FEM simulation, the strains you compute would typically be at a certain node's location. For an FVM simulation, the strain you compute would typically be at a cell centered location.

If you make a strain gauge measurement at a location where your simulation is outputting a strain computation, could you make a direct comparison with the experimental measurement, or is some manipulation of the simulation results needed prior to making a direct comparison?

Your physical strain gauge gives you a single number, which means that it is represented by a functional $$\varphi(\cdot)$$ applied to the solution and its gradient (strain) everywhere. Let's say that it only depends on the strain; then the Riesz representation theorem tells us that the operation the strain gauge performs must be of the form $$\varphi(\varepsilon(\mathbf u)) = \int_\Omega a(\mathbf x) : \varepsilon(\mathbf u(\mathbf x)) \, \text{d}x$$ for some $$a(\mathbf x)\in L^2(\Omega)^{d\times d}$$. ($$a$$ must be in $$L^2$$ because the strain is in $$L^2$$ because the solution is in $$H^1$$.)
So then you have to model how exactly this looks. Since strain gauges have a finite size, your $$a(\mathbf x)$$ is likely a function that is constant in some small subdomain and zero outside. Then what you will need to evaluate for your numerical solution is the approximate value of the predicted strain: $$\varphi(\varepsilon(\mathbf u_h))$$. If the area where $$a$$ is nonzero is smaller than a cell, then you can approximate the integral by taking the value of the (computed approximation to the) strain at an individual node or face, but in general you will want to actually compute this integral: if your mesh is fine enough, then this will involve integration over more than one cell, and you will want to approximate the integral using quadrature.
• Sorry it took me so long to follow up on this. I am a bit confused on how to determine $a(x)$. Could you explain this more? Thanks! – anonuser01 Dec 29 '18 at 6:45