physicsbase docs

# solve a single-bar problem
curl -X POST https://api.physicsbase.dev/solve \
  -H "content-type: application/json" \
  -d '{
    "nodes": [[0,0],[2,0]],
    "materials": {"steel": {"E": 200e9, "nu": 0.3}},
    "elements": [{"type":"truss2d","nodes":[0,1],
                  "material":"steel","section":{"A":0.01}}],
    "supports": [{"node":0,"dofs":["ux","uy"]},
                 {"node":1,"dofs":["uy"]}],
    "loads": [{"node":1,"fx":50000}]
  }'

# 1. discover — learn the schema before building anything
→ fem_capabilities()
← { "element_types": { "beam2d": "…section {A, I}", … }, "analyses": [ … ] }

# 2+3. translate the words into a problem and solve
→ fem_solve({
    "nodes": [[0,0],[1,0],[2,0],[3,0]],
    "materials": { "steel": { "E": 210e9, "nu": 0.3 } },
    "elements": [
      {"type":"beam2d","nodes":[0,1],"material":"steel","section":{"A":0.01,"I":8e-6}},
      {"type":"beam2d","nodes":[1,2],"material":"steel","section":{"A":0.01,"I":8e-6}},
      {"type":"beam2d","nodes":[2,3],"material":"steel","section":{"A":0.01,"I":8e-6}}
    ],
    "supports": [ {"node":0,"dofs":["ux","uy","rz"]} ],
    "loads": [ {"node":3,"fy":-5000} ]
  })
← { "displacements": [ …, {"uy": -0.0268, "rz": -0.0134} ],
    "reactions": [ {"fy": 5000, "mz": 15000}, … ],
    "elements": [ {"moment_i": 15000, …}, … ] }

# 4. reason on the returned numbers (no physics guessed by the agent)
tip_drop = 26.8 mm            # from displacements[-1].uy
M_wall   = 15 kN·m            # from reactions[0].mz / elements[0].moment_i
sigma    = M_wall * c / I = 94 MPa
verdict  = 94 MPa < 250 MPa yield  ->  safe, factor of safety 2.7

{
  "displacements": [ { "ux": 0.0, "uy": 0.0 }, … ],   // per node, by DOF
  "reactions":     [ { "fx": -50000.0, "fy": 0.0 }, … ], // per node
  "elements": [
    { "id": "span_1", "type": "beam2d",
      "moment_i": 10000.0, "shear_i": 5000.0, … }  // depends on type
  ],
  "summary": { "n_dof": 9, "max_abs_displacement": 0.0079,
                "solve_ms": 1.2 }
}

const res = await fetch("https://api.physicsbase.dev/solve", {
  method: "POST",
  headers: { "content-type": "application/json" },
  body: JSON.stringify(problem)
});
const out = await res.json();
console.log(out.displacements.at(-1).uy); // tip deflection

// MCP client config — remote endpoint
{
  "mcpServers": {
    "physicsbase": {
      "url": "https://<your-app>.up.railway.app/mcp"
    }
  }
}

{
  "mcpServers": {
    "physicsbase": {
      "command": "python",
      "args": ["-m", "femengine.mcp.server"]
    }
  }
}

from femengine import Model, solve
from femengine.materials import Material
from femengine.client import FEMClient

# fluent model builder
m = Model()
m.add_nodes([[0,0], [2,0]])
m.add_material("steel", Material(E=200e9, nu=0.3))
m.add_element("truss2d", [0,1], "steel", {"A": 0.01})
m.add_support(0, ["ux","uy"]); m.add_support(1, ["uy"])
m.add_load(1, fx=50_000)
res = solve(m)
print(res.displacements[1]["ux"])   # 5e-05

# …or solve a JSON spec via the client
out = FEMClient().solve(problem_dict)          # in-process
out = FEMClient("https://api.physicsbase.dev").solve(problem_dict)

{
  "frequencies_hz": [ 4.79, 30.0, 84.1, … ],
  "angular_frequencies": [ 30.1, 188.5, 528.3, … ],
  "mode_shapes": [ [ {"ux":0.0,"uy":0.0,"rz":0.0}, … ], … ],
  "summary": { "num_modes": 6, "analysis": "modal" }
}

{
  "nodes": [[0],[0.25],[0.5],[0.75],[1]],
  "materials": { "steel": { "k": 45 } },   // W/m·K
  "elements": [ /* field_line2 chain */ ],
  "element_sources": [ /* {element:i, Q:1e5} heat gen per element */ ],
  "values": [ {"node":0,"value":20}, {"node":4,"value":20} ]
}
// -> { "values": [20, …, T_mid, …, 20], "fluxes": [ … ] }

{
  "nodes": [ /* 1D chain 0 … 1 */ ],
  "materials": { "m": { "k": 1.0 } },   // diffusivity D
  "elements": [ /* field_line2 */ ],
  "velocity": [1.0],                     // advection speed
  "values": [ {"node":0,"value":0}, {"node":-1,"value":1} ]
}

from femengine import thermo_mechanical

res = thermo_mechanical(field_model, struct_model, T_ref=20.0)
res.structural["elements"][0]["axial_stress"]   # thermal stress
res.element_dT                                  # ΔT applied per element

{
  "analysis": "damage", "k0": 1e-4, "kf": 5e-4, "increments": 30,
  "nodes": [[0,0],[1,0]],
  "materials": { "concrete": { "E": 30e9, "nu": 0.2 } },
  "elements": [ {"type":"truss2d","nodes":[0,1],"material":"concrete","section":{"A":0.01}} ],
  "supports": [ {"node":0,"dofs":["ux","uy"]},
                 {"node":1,"dofs":["ux","uy"],"values":[3e-4,0]} ]
}
// -> { "history":[{load_factor,max_damage,…}], "elements":[{damage,…}] }

cd fem-engine
pip install -e ".[api,mcp,dev]"      # core is just numpy + scipy + pydantic

from femengine import Model, solve
from femengine.materials import Material

m = Model()
m.add_nodes([[0,0], [3,0]])
m.add_material("steel", Material(E=210e9, nu=0.3))
m.add_element("beam2d", [0,1], "steel", {"A":0.01, "I":8e-6})
m.add_support(0, ["ux","uy","rz"])
m.add_load(1, fy=-5000)
print(solve(m).displacements[-1])      # tip deflection

femengine-api --port 8000                 # or: uvicorn femengine.api.server:app

curl -s localhost:8000/capabilities        # discover elements & analyses
curl -s -X POST localhost:8000/solve \
     -H "content-type: application/json" \
     -d @examples/cantilever_beam.json         # solve a saved problem

python -m femengine.mcp.server               # tools: fem_solve, fem_modal,
                                             # fem_buckling, fem_transient, …

cd fem-engine
pytest -q                     # all suites
pytest tests/test_validation.py -q     # core elements
pytest tests/test_advanced.py -q       # 3D frames, modal, buckling
pytest tests/test_textbook.py -q       # solids, thermal, axisym, transient

def test_buckling_pinned():
    m, p = _column("pinned")              # pin-ended steel column
    res = buckling(m, num_modes=1)
    expected = math.pi**2 * p["E"] * p["I"] / p["L"]**2   # Euler's Pcr
    assert res.critical_loads[0] == pytest.approx(expected, rel=1e-2)

{
  "title": "Warren truss, 3 x 60 kN",
  "nodes": [[0,0],[4,0],[8,0],[12,0],[16,0],
            [2,3],[6,3],[10,3],[14,3]],
  "materials": { "steel": { "E": 210e9, "nu": 0.3 } },
  "elements": [
    { "type":"truss2d", "nodes":[0,1], "material":"steel", "section":{"A":2e-3} },
    // …bottom & top chords, diagonals (15 members total)
  ],
  "supports": [ {"node":0,"dofs":["ux","uy"]}, {"node":4,"dofs":["uy"]} ],
  "loads": [ {"node":1,"fy":-60000}, {"node":2,"fy":-60000},
             {"node":3,"fy":-60000} ]
}

{
  "title": "Portal frame, 40 kN lateral",
  "nodes": [[0,0],[0,4],[6,4],[6,0]],
  "materials": { "steel": { "E": 210e9, "nu": 0.3 } },
  "elements": [
    { "type":"beam2d", "nodes":[0,1], "material":"steel", "section":{"A":0.012,"I":3e-4} },
    { "type":"beam2d", "nodes":[1,2], "material":"steel", "section":{"A":0.012,"I":3e-4} },
    { "type":"beam2d", "nodes":[2,3], "material":"steel", "section":{"A":0.012,"I":3e-4} }
  ],
  "supports": [ {"node":0,"dofs":["ux","uy","rz"]}, {"node":3,"dofs":["ux","uy","rz"]} ],
  "loads": [ {"node":1,"fx":40000} ]
}

nx, ny, Lx, Ly = 24, 8, 6.0, 2.0
nid = lambda ix, iy: iy*(nx+1) + ix
problem = {
  "materials": {"steel": {"E": 200e9, "nu": 0.3}},
  "nodes": [[x, y] for y in linspace(0,Ly,ny+1)
                    for x in linspace(0,Lx,nx+1)],
  "elements": [
    {"type":"quad4",
     "nodes":[nid(ix,iy),nid(ix+1,iy),nid(ix+1,iy+1),nid(ix,iy+1)],
     "material":"steel", "section":{"t":0.02,"kind":"plane_stress"}}
    for iy in range(ny) for ix in range(nx)
  ],
  "supports": [{"node":nid(0,iy),"dofs":["ux","uy"]} for iy in range(ny+1)],
  "loads": [{"node":nid(nx,iy),"fy":-200000/(ny+1)} for iy in range(ny+1)]
}

import httpx
YIELD, TARGET = 250e6, 2.0          # Pa, desired factor of safety
I = 4e-6
for _ in range(12):
    problem = build_cantilever(I=I)         # agent's own translation step
    out = httpx.post(API, json=problem).json()
    M = max(abs(e["moment_i"]) for e in out["elements"])
    sigma = M * c / I
    fos = YIELD / sigma
    if fos >= TARGET:
        break
    I *= fos / TARGET * 1.1                 # grow the section, try again
print(f"sized: I={I:.2e} m^4, FoS={fos:.2f}")

{
  "title": "3D frame, 30 kN corner wind load",
  "nodes": [[0,0,0],[6,0,0],[6,0,5],[0,0,5],
            [0,4,0],[6,4,0],[6,4,5],[0,4,5]],
  "materials": { "steel": { "E": 210e9, "nu": 0.3 } },
  "elements": [
    // 4 columns (0-4,1-5,2-6,3-7) + 4 roof beams (4-5,5-6,6-7,7-4)
    { "type":"frame3d", "nodes":[0,4], "material":"steel",
      "section":{"A":8e-3,"Iy":4e-5,"Iz":4e-5,"J":6e-5} }
    // …7 more members, same section
  ],
  "supports": [
    {"node":0,"dofs":["ux","uy","uz","rx","ry","rz"]},
    {"node":1,"dofs":["ux","uy","uz","rx","ry","rz"]},
    {"node":2,"dofs":["ux","uy","uz","rx","ry","rz"]},
    {"node":3,"dofs":["ux","uy","uz","rx","ry","rz"]}
  ],
  "loads": [ {"node":7,"fx":30000} ]
}

{
  "title": "Cantilever, 4 kN/m UDL + self-weight",
  "nodes": [[0,0],[1,0],[2,0],[3,0]],
  "materials": { "steel": { "E": 210e9, "nu": 0.3, "rho": 7850 } },
  "elements": [
    { "type":"beam2d", "nodes":[0,1], "material":"steel", "section":{"A":0.01,"I":8e-6} },
    { "type":"beam2d", "nodes":[1,2], "material":"steel", "section":{"A":0.01,"I":8e-6} },
    { "type":"beam2d", "nodes":[2,3], "material":"steel", "section":{"A":0.01,"I":8e-6} }
  ],
  "element_loads": [
    {"element":0,"w":[0,-4000]}, {"element":1,"w":[0,-4000]},
    {"element":2,"w":[0,-4000]}
  ],
  "gravity": [0, -9.81],
  "supports": [ {"node":0,"dofs":["ux","uy","rz"]} ]
}

{
  "title": "Pin-ended column buckling",
  "analysis": "buckling", "num_modes": 3,
  "nodes": [[0,0],[0.375,0],[0.75,0], /* …down to */ [3,0]],
  "materials": { "steel": { "E": 210e9, "nu": 0.3 } },
  "elements": [ /* chain of beam2d, section {A:0.02, I:8e-6} */ ],
  "supports": [ {"node":0,"dofs":["ux","uy"]},
                 {"node":8,"dofs":["uy"]} ],
  "loads": [ {"node":8,"fx":-1000} ]   // reference compression
}
// -> { "load_factors": [1842.x, …], "critical_loads": [1.842e6, …] }

curl -X POST https://api.physicsbase.dev/modal \
  -d '{ "analysis":"modal", "num_modes":3,
        "nodes": [[0,0], … ,[3,0]],
        "materials": {"steel":{"E":210e9,"nu":0.3,"rho":7850}},
        "elements": [ /* beam2d chain, {A:0.01,I:8e-6} */ ],
        "supports": [{"node":0,"dofs":["ux","uy","rz"]}] }'
# -> { "frequencies_hz": [9.1, 57.0, 159.6], … }

{
  "title": "Two-span continuous beam, 5 kN/m",
  "nodes": [[0,0],[1,0],[2,0],[3,0],[4,0],
            [5,0],[6,0],[7,0],[8,0]],
  "materials": { "steel": { "E": 210e9, "nu": 0.3 } },
  "elements": [ /* 8 beam2d spans, section {A:0.01, I:2e-5} */ ],
  "element_loads": [ /* {element:i, w:[0,-5000]} for i in 0..7 */ ],
  "supports": [ {"node":0,"dofs":["ux","uy"]},
                 {"node":4,"dofs":["uy"]},
                 {"node":8,"dofs":["uy"]} ]
}
// reactions[4].fy is the large interior reaction; the span-0 element
// moment_j over the middle support is negative (hogging).

{
  "title": "2-storey moment frame, wind",
  // columns at x=0 and x=6; floors at y=0,3,6
  "nodes": [[0,0],[0,3],[0,6],[6,0],[6,3],[6,6]],
  "materials": { "steel": { "E": 210e9, "nu": 0.3 } },
  "elements": [
    {"type":"beam2d","nodes":[0,1],"material":"steel","section":{"A":0.012,"I":3e-4}},
    {"type":"beam2d","nodes":[1,2],"material":"steel","section":{"A":0.012,"I":3e-4}},
    {"type":"beam2d","nodes":[3,4],"material":"steel","section":{"A":0.012,"I":3e-4}},
    {"type":"beam2d","nodes":[4,5],"material":"steel","section":{"A":0.012,"I":3e-4}},
    {"type":"beam2d","nodes":[1,4],"material":"steel","section":{"A":0.012,"I":3e-4}},
    {"type":"beam2d","nodes":[2,5],"material":"steel","section":{"A":0.012,"I":3e-4}}
  ],
  "supports": [ {"node":0,"dofs":["ux","uy","rz"]},
                 {"node":3,"dofs":["ux","uy","rz"]} ],
  "loads": [ {"node":1,"fx":20000}, {"node":2,"fx":20000} ]
}

{
  "title": "Lattice tower bay",
  // base square z=0 (nodes 0-3), top square z=4 (nodes 4-7)
  "nodes": [[0,0,0],[2,0,0],[2,2,0],[0,2,0],
            [0,0,4],[2,0,4],[2,2,4],[0,2,4]],
  "materials": { "steel": { "E": 210e9, "nu": 0.3 } },
  "elements": [ /* 4 legs (0-4,1-5,2-6,3-7), top ring, base ring, diagonals;
                    every one: type truss3d, section {A:1e-3} */ ],
  "supports": [ /* pin all four base nodes: dofs ["ux","uy","uz"] */ ],
  "loads": [ {"node":4,"fz":-50000,"fx":15000} ]
}

k_soil = 2.0e6            # N/m per node (subgrade modulus × tributary length)
problem = {
  "nodes": [[x, 0] for x in range(11)],        # 10 m beam, 1 m spacing
  "materials": {"c": {"E": 30e9, "nu": 0.2}},
  "elements": [ {"type":"beam2d","nodes":[i,i+1],"material":"c",
                 "section":{"A":0.09,"I":6.75e-4}} for i in range(10) ],
  "springs": [ {"node":i,"dof":"uy","k":k_soil} for i in range(11) ],
  # one horizontal restraint stops rigid sliding
  "supports": [ {"node":0,"dofs":["ux"]} ],
  "loads": [ {"node":5,"fy":-120000} ]   # central column load
}

"supports": [
  { "node": 0, "dofs": ["ux","uy"] },
  { "node": 4, "dofs": ["uy"], "values": [-0.010] },  // 10 mm settlement
  { "node": 8, "dofs": ["uy"] }
]

{
  "title": "Restrained bar, +50°C",
  "nodes": [[0,0],[1,0]],
  "materials": { "steel": { "E": 200e9, "nu": 0.3, "alpha": 12e-6 } },
  "elements": [ {"type":"truss2d","nodes":[0,1],"material":"steel","section":{"A":0.01}} ],
  "supports": [ {"node":0,"dofs":["ux","uy"]}, {"node":1,"dofs":["ux","uy"]} ],
  "thermal": [ {"element":0,"dT":50} ]
}
// elements[0].axial_stress = -E*alpha*dT = -120 MPa (compression)

{
  "nodes": [[0,0,0],[1,0,0],[1,1,0],[0,1,0],
            [0,0,1],[1,0,1],[1,1,1],[0,1,1]],
  "materials": { "steel": { "E": 200e9, "nu": 0.3 } },
  "elements": [ {"type":"hex8","nodes":[0,1,2,3,4,5,6,7],"material":"steel"} ],
  "supports": [ /* clamp the x=0 face: nodes 0,3,4,7 in ux,uy,uz */ ],
  "loads": [ /* tension on the x=1 face nodes 1,2,5,6 */ ]
}
// elements[0].stress = [sxx, syy, szz, sxy, syz, szx],  elements[0].von_mises

{
  // nodes are [r, z]; keep r > 0 (off the axis)
  "nodes": [[1.0,0],[1.2,0],[1.2,1],[1.0,1]],
  "materials": { "steel": { "E": 200e9, "nu": 0.3 } },
  "elements": [ {"type":"axiquad4","nodes":[0,1,2,3],"material":"steel"} ],
  "supports": [ /* restrain uz on z=0 edge; radial free */ ],
  "loads": [ /* internal pressure as radial ring loads on inner nodes */ ]
}
// stress = [srr, szz, s_hoop, srz] per element

curl -X POST https://api.physicsbase.dev/transient \
  -d '{ "analysis":"transient", "dt":2e-4, "n_steps":150,
        "damping":[0.5, 1e-5], "monitor":[1,"uy"],
        "nodes": [[0,0], … ], "materials": {"s":{"E":210e9,"nu":0.3,"rho":7850}},
        "elements": [ /* beam2d chain */ ],
        "supports": [{"node":0,"dofs":["ux","uy","rz"]}],
        "loads": [{"node":1,"fy":-5000}] }'
# -> { "peak_displacement": …, "peak_time": …,
#      "monitor": { "history": [ … ] }, "frames": [ … ] }

{
  "title": "Fixed-fixed beam: load + heating",
  "nodes": [[0,0],[1,0],[2,0],[3,0],[4,0]],
  "materials": { "steel": { "E": 210e9, "nu": 0.3, "alpha": 12e-6 } },
  "elements": [ /* 4 beam2d spans, section {A:0.01, I:8e-6} */ ],
  "supports": [ {"node":0,"dofs":["ux","uy","rz"]},
                 {"node":4,"dofs":["ux","uy","rz"]} ],
  "loads": [ {"node":2,"fy":-8000} ],
  "thermal": [ /* {element:i, dT:40} for each span -> axial thrust */ ]
}
// reactions carry both the bending from the load and the axial
// thrust from restrained thermal expansion, in one solve.

  "elements": [
    {"type":"beam2d","nodes":[0,1],"material":"s","section":{"A":0.012,"I":3e-4}},
    {"type":"beam2d","nodes":[1,2],"material":"s","section":{"A":0.012,"I":3e-4}},
    {"type":"beam2d","nodes":[3,2],"material":"s","section":{"A":0.012,"I":3e-4}},
    {"type":"truss2d","nodes":[0,2],"material":"s","section":{"A":6e-4}},  // diagonal brace
    {"type":"truss2d","nodes":[3,1],"material":"s","section":{"A":6e-4}}   // diagonal brace
  ]