  We present a new Integrated Finite Element Neural Network framework (I-FENN),
with the objective to accelerate the numerical solution of nonlinear
computational mechanics problems. We leverage the swift predictive capability
of neural networks (NNs) and we embed them inside the finite element stiffness
function, to compute element-level state variables and their derivatives within
a nonlinear, iterative numerical solution. This process is conducted jointly
with conventional finite element methods that involve shape functions: the NN
receives input data that resembles the material point deformation and its
output is used to construct element-level field variables such as the element
Jacobian matrix and residual vector. Here we introduce I-FENN to the continuum
damage analysis of quasi-brittle materials, and we establish a new non-local
gradient-based damage framework which operates at the cost of a local damage
approach. First, we develop a physics informed neural network (PINN) to
resemble the non-local gradient model and then we train the neural network
offline. The network learns to predict the non-local equivalent strain at each
material point, as well as its derivative with respect to the local strain.
Then, the PINN is integrated in the element stiffness definition and conducts
the local to non-local strain transformation, whereas the two PINN outputs are
used to construct the element Jacobian matrix and residual vector. This process
is carried out within the nonlinear solver, until numerical convergence is
achieved. The resulting method bears the computational cost of the conventional
local damage approach, but ensures mesh-independent results and a diffused
non-local strain and damage profile. As a result, the proposed method tackles
the vital drawbacks of both the local and non-local gradient method,
respectively being the mesh-dependence and additional computational cost.
