+
+This class is used to specify materials interpolated from a discrete Cartesian grid. A class object is passed as the
+`material` argument of a [`GeometricObject`](#GeometricObject) or the `default_material` argument of the `Simulation`
+constructor (similar to a material function).
+
+
+
+
+
+
+
+```python
+def __init__(self,
+ grid_size,
+ medium1,
+ medium2,
+ design_parameters=None,
+ grid_type='U_DEFAULT',
+ do_averaging=False,
+ beta=0,
+ eta=0.5):
+```
+
+
+
+Creates a `MaterialGrid` object.
+
+The input are two materials `medium1` and `medium2` which are linearly interpolated at each grid point using
+a NumPy array `design_parameters` of size `grid_size` (a 3-tuple or `Vector3` of integers) with floating-point values in
+the range [0,1] used to define a linear weight (i.e., 0 is `medium1` and 1 is `medium2`). Currently, only two material
+types are supported: (1) frequency-independent isotropic $\varepsilon$ or $\mu$ and (2) `LorentzianSusceptibility`.
+`medium1` and `medium2` must both be the same type.
+
+[Subpixel smoothing](Subpixel_Smoothing.md) can be enabled by specifying `do_averaging=True`. If you want to use a
+material grid to define a (nearly) discontinuous, piecewise-constant material that is either `medium1` or `medium2`
+almost everywhere, you can optionally enable a (smoothed) *projection* feature by setting the parameter `beta` to a
+positive value. When the projection feature is enabled, the design parameters $u(x)$ can be thought of as a [level-set
+function](https://en.wikipedia.org/wiki/Level-set_method) defining an interface at $u(x)=\eta$ with a smoothing factor
+$\beta$ ($\beta=\infty$ gives an unsmoothed, discontinuous interface). The projection operator is $\(tanh(\beta*\eta)
++ \tanh(\beta*(u-\eta))) / (\tanh(\beta*\eta) + \tanh(\beta*(1-\eta)))$ involving the parameters `beta`
+($\beta$: "smoothness" of the turn on) and `eta` ($\eta$: erosion/dilation). The level set provides a general approach for
+defining a *discontinuous* function of the otherwise continuously varying (via the bilinear interpolation) grid values.
+The subpixel smoothing is based on an adaptive quadrature scheme with properties `subpixel_maxeval` and `subpixel_tol` which
+can be specified using the [`Simulation`](#Simulation) constructor.
+
+Grids which are symmetric (e.g., mirror, rotation) must be explicitly defined. One way to implement this is by overlapping
+a given `MaterialGrid` object with a symmetrized copy of itself. In the case of spatially overlapping `MaterialGrid`
+objects (with no intervening objects), any overlapping points are combined using the method `grid_type` which is one of
+`"U_MIN"` (minimum of the overlapping grid values), `"U_PROD"` (product), `"U_SUM"` (mean), `"U_DEFAULT"`
+(topmost material at grid point).
+
+
+
+
+
+```python
+def update_parameters(self, x):
+```
+
+
+
+Reset the design grid `design_parameters` to `x`.
+
+
+
+