  The bilayers of Transition Edge Sensors (TESs) are often modified with
additional normal-metal features such as bars or dots. Previous device
measurements suggest that these features improve performance, reducing
electrical noise and altering response times. However, there is currently no
numerical model to predict and quantify these effects. Here we extend existing
techniques based on Usadel's equations to describe TESs with normal-metal
features. We show their influence on the principal TES characteristics, such as
the small-signal electrothermal parameters $\alpha$ and $\beta$ and the
superconducting transition temperature $T_{c}$. Additionally, we examine the
effects of an applied magnetic field on the device performance. Our model
predicts a decrease in $T_{c}$, $\alpha$ and $\beta$ as the number of lateral
metal structures is increased. We also obtain a relationship between the length
$L$ of a TES and its critical temperature, $T_{c} \propto L^{-0.7}$ for a
bilayer with normal-metal bars. We predict a periodic magnetic flux dependence
of $\alpha, \beta$ and $I_{c}$. Our results demonstrate good agreement with
published experimental data, which also show the reduction of $\alpha$, $\beta$
and $T_{c}$ with increasing number of bars. The observed Fraunhofer dependence
of critical current on magnetic flux is also anticipated by our model. The
success of this model in predicting the effects of additional structures
suggests that in the future numerical methods can be used to better inform the
design of TESs, prior to device processing.
