Combining Machine Learning and the Nonlinear Finite Element Analysis for Reinforced Concrete Membranes – Concrete E-Learning Platform

Link zur deutschen Version: Kombination von Maschinellem Lernen und der Nichtlinearen Finite Elemente Methode für Stahlbetonscheiben

As you might remember from my previous blogposts, my research focuses on developing a pipeline for integrating machine learning within the nonlinear finite element analysis (NLFEA) for reinforced concrete (RC) structures. In this blogpost, I will focus on demonstrating the development of a novel so-called “material surrogate model” for RC membranes loaded purely in plane. Figure 1shows an overview of my research so far, and which blogposts you can revisit to understand previous work.

Figure 1: Overview of my research so far (Blogpost 1: here, Blogpost 2: here); topic of this blogpost highlighted with yellow outlines.

Initially, machine learning techniques for the analysis of RC beams were investigated, and a physics-informed neural network (PINN) was developed for a partially cracked RC beam (Blogpost 1, Figure 1). Many RC structures however comprise thin-walled planar elements as main load-bearing elements, such as ceilings, bridge decks, walls or girder webs, which can be modelled with planar shell elements. Understanding the behaviour of these elements and enabling mechanically meaningful simulations is thus crucial to reliably assess the load-bearing behaviour of existing structures and design efficient new ones. To develop a meaningful ML-FEA pipeline for these more complex elements, a linear elastic 3D shell element was analysed (Blogpost 2, Figure 1). Before moving to the final goal of analysing nonlinear 3D shell elements with a ML-FEA pipeline, a final step is however required: The development of a ML-NLFEA pipeline for nonlinear 2D membrane elements, serving as a last proof of concept.

This blogpost will present the results of this latest development. The goal of this work is to show that such a pipeline is feasible and can be utilised within simple analyses of in-plane loaded structures. The case study that will be presented in this blogpost is visualised in Figure 2: A membrane modelled with 100 finite elements loaded in pure shear, which could e.g. represent a load-bearing wall of a building. In this case we fix L = 6 m, but the framework can generalise for any length. The goal is to find the load-deformation behaviour of this membrane with the ML-NLFEA framework and compare it to the standard NLFEA and its underlying models (i.e., the cracked membrane model).

Figure 2: *Case study of pure shear* (n_xy)

In general, for an element to be analysed as a membrane, a small thickness to length ratio (t/L) is assumed, giving rise to a plane stress state (i.e. $\sigma_z$ = 0) and a uniform strain and stress distribution across the thickness of the element. The two-dimensional constitutive relationship can be expressed by the following equation in each integration point:

(1) $\begin{equation*}\{n_x, n_y, n_{xy}\} = [D_m] \{ \hat{\varepsilon}_x,\hat{\varepsilon}_y,\hat{\varepsilon}_{xy} \} \end{equation*}$

The stress and strain components shown in Equation (1) are schematically illustrated in Figure 3.

Figure 3: (a) Strains and (b) stresses at two-dimensional membrane element

To understand the differences between the NLFEA based on the cracked membrane model and the novel ML-NLFEA pipeline, Figure 4schematically illustrates the two workflows. The NLFEA itself (Figure 4 (a)) comprises several calculation steps, which are carried out in an iterative manner until the residual forces approach zero. The input parameters, which can later be optimised for an efficient structure, consist of variables describing the material properties (e.g. the Young’s Modulus $E$ , the Poisson ratio $\nu$ , the reinforcement ratio $\rho$ and the concrete class CC) as well as variables describing the geometrical properties of a given structure (e.g. thickness $t$ and length $L$ ) and its loading conditions (e.g. force $\mathbf{F}$ and supports). As outputs of the NLFEA, we receive displacements $\mathbf{u}$ , as well as generalised stress $\boldsymbol{\hat{\sigma}}$ and strain $\boldsymbol{\hat{\varepsilon}}$ distributions for any location in the analysed structure.

Figure 4: (a) Standard NLFEA framework (b) ML-NLFEA framework.

Figure 4 (b) highlights the benefits of using a surrogate model within the NLFEA framework: A previously not continuously differentiable pipeline is equipped with a material surrogate model, rendering it differentiable and thus optimisable with respect to the input parameters. The two major substitutions marked in turquoise transform the existing NLFEA into this differentiable ML-NLFEA framework. This implies that the surrogate model will substitute the relationship between generalised stresses ( $\boldsymbol{\hat{\sigma}}$ ) and generalised strains ( $\boldsymbol{\hat{\varepsilon}}$ ) and at the same time also needs to be able to capture the stiffness matrix $\mathbf{D}$ correctly, which is required to calculate the displacements $\mathbf{u}$ . Particularly this last part of predicting a correct stiffness matrix poses a challenge compared to the existing linear elastic framework already developed previously (Blogpost 2). These two major substitutions can also be formulated analytically:

(2) $\begin{align*} \boldsymbol{\hat{\sigma}} & = f_{NN}(\boldsymbol{\hat{\varepsilon}}) & \quad \text{instead of } \boldsymbol{\sigma_i} = \mathbf{D_P}\boldsymbol{\varepsilon_i} \text{ and } \mathbf{D_P} = f(\boldsymbol{\sigma_i},\boldsymbol{\varepsilon_i}) \text{ and } \boldsymbol{\hat{\sigma}} = \boldsymbol{\sigma_i} \cdot t \text{)}\\\mathbf{D} & = \frac{\partial \boldsymbol{\hat{\sigma}}}{\partial \boldsymbol{\hat{\varepsilon}}} = \frac{ f_{NN}(\boldsymbol{\hat{\varepsilon}})}{\partial \boldsymbol{\hat{\varepsilon}}} & \quad \text{instead of } \mathbf{D}=\mathbf{D_P}\cdot t \text{ and } \mathbf{D_P} = f(\boldsymbol{\sigma_i},\boldsymbol{\varepsilon_i}) \text{)}\end{align*}$

where $f_{NN}$ represents the differentiable surrogate material model and $\mathbf{D_P}$ represents the in-plane stiffness matrix.

Within this research, the surrogate material model used is a neural network (NN). Hence, data and a suitable NN architecture are required. While the last blogpost focused on the data acquisition, this one will primarily explain the utilised NN architecture and how it aids in creating a differentiable surrogate material model tailored to the specific problem statement outlined above. Nevertheless, data generation also played a crucial role in enabling good predictions of load-deformation curves of membranes loaded in plane within this ML-NLFEA hybrid framework.

Synthetic data (i.e. data collected from repeated simulations) was utilised for training the NN model. For this project of my research, the thickness was held constant at $t$ = 300 mm, the reinforcement ratio $\rho_x$ was set to 1% and the concrete material parameters corresponding to a concrete of strength C30/37 were also held constant. Therefore, the model contains four input parameters (the three generalised strains $\boldsymbol{\hat{\varepsilon}}$ and a varying reinforcement ratio $\rho_y$ ) and three output parameters (the three generalised stresses $\boldsymbol{\hat{\sigma}}$ ), as well as nine derivative values for each data point. The constant parameters pose a major limitation to the generalisability; however, they enable the development of a proof of concept for the ML-NLFEA framework.

The NN architecture was then constructed as visualised in Figure 5: Inputs and outputs are defined according to the needs of the surrogate material model. In between, a multi-layer perceptron (MLP) provides the learnable parameters (weights and biases) of the network, such that it reproduces the desired material behaviour. The loss function, which should be minimised during training, contains two separate terms:

$\mathcal{L}_1$ , the standard loss for the generalised stresses: This loss determines the difference between the true generalised stresses from the dataset and the prediction of the NN for the generalised stresses.
$\mathcal{L}_2$ , an additional loss (“Sobolev loss”) for the stiffnesses: Using automatic differentiation (AD), a derivative of the outputs ( $\boldsymbol{\hat{\sigma}}$ ) with respect to the inputs ( $\boldsymbol{\hat{\varepsilon}}$ ) can be calculated. Again, these values are compared to the ones available in the dataset to ensure a correct prediction of the derivative by the learned function $f_{NN}$ .

Figure 5: NN architecture for physics-based surrogate material model

After setting up the NN architecture, the hyperparameters $W$ (width) and $d$ (depth)of the MLP, as well as the Sobolev weights ( $w_1,w_2$ in Figure 5)and additional parameters such as the type of activation function, the learning rate, etc., are tuned in an iterative process to find the best possible training of the surrogate material model. To check the impact of the additional Sobolev loss $\mathcal{L}_2$ on the prediction accuracy, we took a closer look at its influence when training on different amounts of data. The prediction accuracy was hereby measured as a root mean squared error (RMSE), where lower values correspond to a higher accuracy. Confirming the observations from literature, for small amounts of data ( $N$ < 1,000), the additional loss $\mathcal{L}_2$ leads to an improved prediction of the generalised stresses $\boldsymbol{\hat{\sigma}}$ . However, after a turning point at $N\approx$ 3,000, the additional Sobolev loss $\mathcal{L}_2$ does not aid the prediction of $\boldsymbol{\hat{\sigma}}$ . This observation will need to be explored in future research in more detail. The trained model “NN v480” (including the Sobolev loss and trained on $N$ = 3’000’000 data points) was selected to be used in the ML-NLFEA framework due to its accurate predictions of the generalised stresses and because it still retains its differentiability based on the Sobolev loss.

Figure 6: Comparison of predictions of generalised stresses when including or excluding the Sobolev loss in training.

Let us now get back to the case study from the beginning of the blogpost: The membrane in pure shear. For every finite element in the 100 elements shown in Figure 2, the NN now makes predictions of the generalised stresses and the stiffness matrix. These values are then utilised in each iteration of the NLFEA framework (Figure 4). These iterations are carried out for multiple prescribed load levels, in this case $n_{xy}$ , and result in the load-displacement curves shown in Figure 7 (a). The results of the ML-NLFEA framework (marked with “NN” in Figure 7) coincide very well with the results of the NFLEA framework, as indicated by the small deviations of the calculated values from the diagonal in Figure 7 (b), proving the feasibility of the ML-NLFEA framework.

Figure 7: Load-deformation curves for pure shear. e represents the root mean squared error with respect to the generalised strains.

A paper was submitted to the Journal “Computer Methods in Applied Mechanics and Engineering” [1]. In it, twelve additional case studies were analysed for different load situations to demonstrate the generalisability of the method. This paper will be made available within this blog post as soon as an official version is published – until then, feel free to reach out to me, I’d be happy to explain more. Stay tuned for the next findings in machine learning for finite element analyses of reinforced concrete structures!

Vera Balmer

Referenzen

V. Balmer, K. Thoma, A. Näsbom, M.A. Kraus, R. Bischof, W. Kaufmann, S. Coros: “A Differentiable Material Surrogate Model for Nonlinear Reinforced Concrete Membrane Elements in Plane Stress State”, submitted to Computer Methods in Applied Mechanics and Engineering, submitted May 2026 (Under Review).