variational_form / weak_form / functional TODO list

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Originally from the PR, this is a TODO list to track priorities and development progress. It will be regularly updated as new features are completed.

## To-do list:
(1: high priority, 5: low priority)
(✔️: written and working, ⁉️: some development, but untested or not working, ❌: not started)

### Function Spaces

| | H1 | Hcurl | Hdiv | DG |
| -------------- | ----- | ----- | ---- | ----- |
| Triangles | ⁉️ (4) | ⁉️ (4) | ❌(5) | ⁉️ (4) |
| Quadrilaterals | ✔️ | ✔️ | ❌(4) | ✔️ |
| Tetrahedra | ⁉️ (3) | ⁉️ (3) | ❌(5) | ⁉️ (3) |
| Hexahedra | ✔️ | ✔️ | ❌(4) | ✔️ |

### Domains of Integration (see picture below)

[](https://user-images.githubusercontent.com/61714427/114587521-9a8e8600-9c3a-11eb-8dfd-05f0e87ebca7.gif)

| | 1 | 2 | 3 |
| ---- | ------ | ------- | ---- |
| 1D | ❌(4-5) | n/a | n/a |
| 2D | ✔️ | ✔️ | n/a |
| 3D | ❌(4) | ✔️ | ✔️ |

- ❌ flux-type element calculations over all interfaces between neighboring elements (4-5)
- ❌ contact surfaces (4-5)

### Functional features

- ✔️ kernel template: residual evaluation w/ automatic differentiation
- ✔️ kernel template: action of gradient
- ✔️ kernel template: assemble gradient matrix
- ❌ support with scalar or tensor valued arguments (e.g. make time available to q-function) (3)
- ✔️ support QoIs for case where test space == `double`
- ✔️ kernel template: assemble gradient vector (QoI)
- ✔️ implement some basic material models for different physics
- ✔️ allow user-defined types to be passed in to certain integrals (e.g. material state for plasticity models)
- ⁉️ allow user-defined annotations on integrals (e.g. only provide temperature gradient for this element calculation) (4)
- ✔️ multiphysics support (i.e. support more than 1 trial space per weak form)

### Tensor / Dual Number Classes

- ✔️ base implementation of statically sized tensor class, and dual number/tensor classes
- ✔️ support arithmetic on tuples of tensors / dual numbers
- ✔️ support differentiation of functions with multiple inputs/outputs
- ✔️ automatically detect if an output does not depend on certain inputs, and optimize out derivative calculation
- ⁉️ automatically detect if an output is linear / nonlinear, and cache derivatives accordingly (4)
- ⁉️ automatically detect symmetries (isotropy, orthotropy, symmetry, antisymmetry, etc) in outputs and store accordingly (4-5)
- ✔️ support evaluation/differentiation of advanced tensor functions (`inv`, `det`, `polar_decomposition`, etc)

### Performance Testing

- ✔️ profile existing tensor / dual number classes
- ✔️ profile shape function value / gradient
- ⁉️ profile common kernels (diffusion, mass, etc) (3-4)
- ✔️ profile high-level residual evaluation
- ⁉️ run/profile each of the above on CUDA as well (1-2)
- ✔️ investigate performance benefits of alternative implementations

This issue is a broad TODO list covering function spaces, integration domains, functional features, tensor and dual-number classes, and performance testing; it names no source files, tests, or entry points. Choose a specific unchecked item only after locating its relevant implementation and tests, and define done as that item being implemented and verified.

Written by the indexing model from the issue text.