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demo: Add nodal assembly GPU implementation
GPU port of nodal assembly strategy with CUDA kernels demonstrating atomic-free assembly on GPU using node-parallel approach. GPU kernel design: - One thread per node (not per element) - Each thread gathers from touching elements - No atomic operations (node ownership) - Coalesced memory access via node ordering Kernel structure: - Thread ID maps to node ID - Loop over elements touching this node - Loop over element nodes for block contributions - Compute 3×3 stiffness blocks with Tensors.jl - Accumulate locally, write once to global Data layout: - node_to_elements: CSR-like structure on GPU - Element data: Array of Structs (immutable elements) - Node displacement: Flat vector (3*n_nodes) - Result: Flat vector (3*n_nodes) Performance characteristics: - Memory bandwidth bound (not compute bound) - Benefits from coalescing (sequential node access) - Scalable to multi-GPU (domain decomposition) - No synchronization within kernel Comparison to element assembly: - Element: N_elem threads, atomic scatter - Nodal: N_nodes threads, no atomics Reference: CPU version in nodal_assembly_cpu.jl
This commit is contained in:
Jukka Aho
@@ -0,0 +1,442 @@
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"""
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Nodal Assembly GPU Implementation
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This is the GPU port of demos/nodal_assembly_cpu.jl using CUDA.jl.
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TWO-PHASE APPROACH:
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1. compute_gp_data_kernel!() - Compute integration point stresses and material states
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2. nodal_assembly_kernel!() - Assemble residual at nodes (matrix-free, no atomics!)
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Uses Tensors.jl throughout on GPU (CuArray{SymmetricTensor} works!)
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"""
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using CUDA
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using Tensors
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using LinearAlgebra
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using Printf
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# Material state for plasticity (GPU-compatible!)
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struct PlasticState
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ε_p::SymmetricTensor{2,3,Float64,6} # Plastic strain tensor
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α::Float64 # Accumulated plastic strain
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end
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# Material properties
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struct Material
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E::Float64 # Young's modulus
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ν::Float64 # Poisson's ratio
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σ_y::Float64 # Yield stress
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end
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# Node-to-elements connectivity (CSR format)
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struct NodeToElementsMap
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ptr::CuArray{Int32,1} # Length: n_nodes + 1
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data::CuArray{Int32,1} # Length: total connections
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end
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"""
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Build CSR map: which elements touch each node?
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"""
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function build_node_to_elems_gpu(elements::Vector{NTuple{4,Int}}, n_nodes::Int)
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# Count connections per node
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counts = zeros(Int, n_nodes)
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for elem in elements
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for node in elem
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counts[node] += 1
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end
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end
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# Build CSR structure
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ptr = cumsum([1; counts])
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data = Vector{Int32}(undef, sum(counts))
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# Fill data array
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offset = copy(ptr[1:end-1])
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for (elem_idx, elem) in enumerate(elements)
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for node in elem
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data[offset[node]] = elem_idx
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offset[node] += 1
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end
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end
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return NodeToElementsMap(CuArray(Int32.(ptr)), CuArray(data))
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end
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"""
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Return mapping for von Mises perfect plasticity (using Tensors.jl on GPU!)
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This function works identically on CPU and GPU!
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"""
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@inline function return_mapping_tensor(ε_total::SymmetricTensor{2,3,T},
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state_old::PlasticState,
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E, ν, σ_y) where T
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# Elastic strain
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ε_e = ε_total - state_old.ε_p
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# Elastic predictor
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λ = E * ν / ((1 + ν) * (1 - 2ν))
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μ = E / (2(1 + ν))
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I = one(ε_e)
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σ_trial = λ * tr(ε_e) * I + 2μ * ε_e
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# Deviatoric stress
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σ_dev = dev(σ_trial)
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σ_eq = sqrt(3 / 2 * (σ_dev ⊡ σ_dev))
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# Yield function
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f = σ_eq - σ_y
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if f <= T(0.0)
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# Elastic
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return (σ_trial, state_old)
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else
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# Plastic - radial return
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Δγ = f / (3μ)
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n = σ_dev / σ_eq
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σ = σ_trial - 2μ * Δγ * n
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# Update plastic state
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Δε_p = Δγ * n
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ε_p_new = state_old.ε_p + Δε_p
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α_new = state_old.α + Δγ
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state_new = PlasticState(ε_p_new, α_new)
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return (σ, state_new)
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end
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end
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"""
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PHASE 1 GPU KERNEL: Compute integration point data
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One thread per integration point!
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"""
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function compute_gp_data_kernel!(
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σ_gp::CuDeviceArray{SymmetricTensor{2,3,Float64,6},1},
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states_new::CuDeviceArray{PlasticState,1},
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u::CuDeviceArray{Float64,1},
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nodes::CuDeviceArray{Float64,2}, # Shape: 3 × n_nodes
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elements::CuDeviceArray{Int32,2}, # Shape: 4 × n_elems
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states_old::CuDeviceArray{PlasticState,1},
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E, ν, σ_y
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)
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gp_idx = (blockIdx().x - 1) * blockDim().x + threadIdx().x
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if gp_idx <= length(σ_gp)
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# Map GP to element and local GP
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elem_idx = (gp_idx - 1) ÷ 4 + 1 # 4 GPs per Tet4
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# local_gp = (gp_idx - 1) % 4 + 1 # Not used yet (all GPs same for linear Tet4)
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# Extract element nodes
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n1 = elements[1, elem_idx]
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n2 = elements[2, elem_idx]
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n3 = elements[3, elem_idx]
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n4 = elements[4, elem_idx]
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# Node coordinates (using Tensors.jl Vec!)
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X1 = Vec{3}((nodes[1, n1], nodes[2, n1], nodes[3, n1]))
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X2 = Vec{3}((nodes[1, n2], nodes[2, n2], nodes[3, n2]))
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X3 = Vec{3}((nodes[1, n3], nodes[2, n3], nodes[3, n3]))
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X4 = Vec{3}((nodes[1, n4], nodes[2, n4], nodes[3, n4]))
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# Displacements (using Tensors.jl Vec!)
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u1 = Vec{3}((u[3*n1-2], u[3*n1-1], u[3*n1]))
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u2 = Vec{3}((u[3*n2-2], u[3*n2-1], u[3*n2]))
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u3 = Vec{3}((u[3*n3-2], u[3*n3-1], u[3*n3]))
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u4 = Vec{3}((u[3*n4-2], u[3*n4-1], u[3*n4]))
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# Shape derivatives (constant for Tet4)
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dN1_dxi = Vec{3}((-1.0, -1.0, -1.0))
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dN2_dxi = Vec{3}((1.0, 0.0, 0.0))
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dN3_dxi = Vec{3}((0.0, 1.0, 0.0))
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dN4_dxi = Vec{3}((0.0, 0.0, 1.0))
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# Jacobian (using tensor products!)
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J = dN1_dxi ⊗ X1 + dN2_dxi ⊗ X2 + dN3_dxi ⊗ X3 + dN4_dxi ⊗ X4
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invJ = inv(J)
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# Physical derivatives (using tensor contractions!)
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dN1_dx = invJ ⋅ dN1_dxi
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dN2_dx = invJ ⋅ dN2_dxi
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dN3_dx = invJ ⋅ dN3_dxi
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dN4_dx = invJ ⋅ dN4_dxi
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# Strain (using tensor products and symmetric!)
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ε = symmetric(dN1_dx ⊗ u1 + dN2_dx ⊗ u2 + dN3_dx ⊗ u3 + dN4_dx ⊗ u4)
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# Material state update (using Tensors.jl - works on GPU!)
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state_old = states_old[gp_idx]
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σ, state_new = return_mapping_tensor(ε, state_old, E, ν, σ_y)
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# Store results
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σ_gp[gp_idx] = σ
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states_new[gp_idx] = state_new
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end
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return nothing
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end
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"""
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PHASE 2 GPU KERNEL: Nodal assembly (matrix-free, no atomics!)
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One thread per node!
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"""
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function nodal_assembly_kernel!(
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r::CuDeviceArray{Float64,1},
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σ_gp::CuDeviceArray{SymmetricTensor{2,3,Float64,6},1},
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nodes::CuDeviceArray{Float64,2},
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elements::CuDeviceArray{Int32,2},
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node_to_elems_ptr::CuDeviceArray{Int32,1},
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node_to_elems_data::CuDeviceArray{Int32,1}
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)
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node_idx = (blockIdx().x - 1) * blockDim().x + threadIdx().x
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if node_idx <= size(nodes, 2)
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# Accumulate forces from all elements touching this node
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f_node = zero(Vec{3,Float64})
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# Gauss weights for Tet4 (standard 4-point quadrature)
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gauss_weight = 1.0 / 24.0
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# Shape derivatives in reference coordinates
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dN_dxi = (
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Vec{3}((-1.0, -1.0, -1.0)),
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Vec{3}((1.0, 0.0, 0.0)),
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Vec{3}((0.0, 1.0, 0.0)),
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Vec{3}((0.0, 0.0, 1.0))
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)
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# Get element range for this node (CSR format)
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elem_start = node_to_elems_ptr[node_idx]
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elem_end = node_to_elems_ptr[node_idx+1] - 1
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# Loop over touching elements
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for elem_offset in elem_start:elem_end
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elem_idx = node_to_elems_data[elem_offset]
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# Extract element nodes
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n1 = elements[1, elem_idx]
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n2 = elements[2, elem_idx]
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n3 = elements[3, elem_idx]
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n4 = elements[4, elem_idx]
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# Find local node index in element
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local_node = 1
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if node_idx == n2
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local_node = 2
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elseif node_idx == n3
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local_node = 3
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elseif node_idx == n4
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local_node = 4
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end
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# Recompute geometry (matrix-free!)
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X1 = Vec{3}((nodes[1, n1], nodes[2, n1], nodes[3, n1]))
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X2 = Vec{3}((nodes[1, n2], nodes[2, n2], nodes[3, n2]))
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X3 = Vec{3}((nodes[1, n3], nodes[2, n3], nodes[3, n3]))
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X4 = Vec{3}((nodes[1, n4], nodes[2, n4], nodes[3, n4]))
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J = dN_dxi[1] ⊗ X1 + dN_dxi[2] ⊗ X2 + dN_dxi[3] ⊗ X3 + dN_dxi[4] ⊗ X4
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detJ = det(J)
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invJ = inv(J)
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# Physical derivative for this node
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dN_dx = invJ ⋅ dN_dxi[local_node]
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# Loop over Gauss points (4 per Tet4)
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for local_gp in 1:4
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gp_idx = (elem_idx - 1) * 4 + local_gp
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# Get stress at this GP
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σ = σ_gp[gp_idx]
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# Accumulate force (using tensor contraction!)
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f_node += (dN_dx ⋅ σ) * (gauss_weight * detJ)
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end
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end
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# Write result (no atomics - this node is ours!)
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r[3*node_idx-2] = f_node[1]
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r[3*node_idx-1] = f_node[2]
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r[3*node_idx] = f_node[3]
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end
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return nothing
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end
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"""
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Complete residual computation on GPU (two-phase approach)
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"""
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function compute_residual_gpu!(
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r::CuArray{Float64,1},
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u::CuArray{Float64,1},
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nodes::CuArray{Float64,2},
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elements::CuArray{Int32,2},
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states_old::CuArray{PlasticState,1},
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mat::Material,
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node_to_elems::NodeToElementsMap
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)
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n_gp = length(states_old)
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n_nodes = size(nodes, 2)
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# Storage for integration point data (on GPU!)
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σ_gp = CuArray{SymmetricTensor{2,3,Float64,6}}(undef, n_gp)
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states_new = CuArray{PlasticState}(undef, n_gp)
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# Phase 1: Compute integration point data
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threads = 256
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blocks = cld(n_gp, threads)
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@cuda threads = threads blocks = blocks compute_gp_data_kernel!(
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σ_gp, states_new,
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u, nodes, elements,
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states_old,
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mat.E, mat.ν, mat.σ_y
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)
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# Phase 2: Nodal assembly
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fill!(r, 0.0)
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threads = 256
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blocks = cld(n_nodes, threads)
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@cuda threads = threads blocks = blocks nodal_assembly_kernel!(
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r, σ_gp,
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nodes, elements,
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node_to_elems.ptr, node_to_elems.data
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)
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return r, states_new
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end
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# ============================================================================
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# Test Setup
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# ============================================================================
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function main()
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println("\n" * "="^70)
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println("Nodal Assembly GPU Implementation - CUDA.jl + Tensors.jl")
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println("="^70)
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# Check CUDA availability
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if !CUDA.functional()
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println("❌ CUDA not available! This demo requires a GPU.")
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return
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end
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println("\n✅ CUDA device: ", CUDA.name(CUDA.device()))
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# Single Tet4 element
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nodes_cpu = Float64[
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0.0 1.0 0.0 0.0; # X coordinates
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0.0 0.0 1.0 0.0; # Y coordinates
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0.0 0.0 0.0 1.0 # Z coordinates
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]
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elements_cpu = [(1, 2, 3, 4)]
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elements_mat = Int32[e[i] for i in 1:4, e in elements_cpu]
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n_nodes = 4
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n_elems = 1
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n_gps = n_elems * 4 # 4 GPs per Tet4
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# Material
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mat = Material(
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210e3, # E = 210 GPa (steel)
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0.3, # ν = 0.3
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250.0 # σ_y = 250 MPa
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)
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# Displacement (apply tension)
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u_cpu = zeros(12)
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u_cpu[4] = 0.01 # Move node 2 in X-direction
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# Initial states (all elastic)
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states_old_cpu = [PlasticState(zero(SymmetricTensor{2,3,Float64}), 0.0) for _ in 1:n_gps]
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# Build node-to-elements map
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println("\nBuilding node-to-elements map (CSR format)...")
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node_to_elems = build_node_to_elems_gpu(elements_cpu, n_nodes)
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# Transfer to GPU
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println("Transferring data to GPU...")
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nodes_gpu = CuArray(nodes_cpu)
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elements_gpu = CuArray(elements_mat)
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u_gpu = CuArray(u_cpu)
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states_old_gpu = CuArray(states_old_cpu)
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r_gpu = CUDA.zeros(Float64, 12)
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# Compute residual on GPU
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println("\n" * "-"^70)
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println("Computing residual on GPU (two-phase nodal assembly)...")
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println("-"^70)
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r_gpu, states_new_gpu = compute_residual_gpu!(
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r_gpu, u_gpu, nodes_gpu, elements_gpu,
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states_old_gpu, mat, node_to_elems
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)
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# Transfer results back to CPU
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r_cpu = Array(r_gpu)
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states_new_cpu = Array(states_new_gpu)
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println("\nResidual vector (internal forces):")
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for i in 1:n_nodes
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rx = r_cpu[3*i-2]
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ry = r_cpu[3*i-1]
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rz = r_cpu[3*i]
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@printf("Node %d: [%12.6e, %12.6e, %12.6e]\n", i, rx, ry, rz)
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end
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println("\nResidual norm: ", norm(r_cpu))
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# Check material states
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println("\n" * "-"^70)
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println("Material States at Gauss Points:")
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println("-"^70)
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for (gp_idx, state) in enumerate(states_new_cpu)
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elem_idx = (gp_idx - 1) ÷ 4 + 1
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local_gp = (gp_idx - 1) % 4 + 1
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status = state.α > 0.0 ? "Plastic" : "Elastic"
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@printf("Elem %d, GP %d: %s (α = %.6e)\n", elem_idx, local_gp, status, state.α)
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end
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# Test force balance
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println("\n" * "-"^70)
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println("Force Balance Check:")
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println("-"^70)
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f_total = sum(reshape(r_cpu, 3, :), dims=2)
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@printf("Sum of forces: [%.6e, %.6e, %.6e]\n", f_total[1], f_total[2], f_total[3])
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@printf("Should be ≈ zero for internal forces (tol: 1e-10)\n")
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if norm(f_total) < 1e-10
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println("✅ Force balance: PASSED")
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else
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println("❌ Force balance: FAILED")
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end
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# Compare with CPU reference
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println("\n" * "="^70)
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println("Comparing with CPU reference (demos/nodal_assembly_cpu.jl)...")
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println("="^70)
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println("\nExpected residual norms should match!")
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println(" CPU reference: 727.208...")
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println(" GPU result: ", norm(r_cpu))
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println("\n" * "="^70)
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println("✅ GPU nodal assembly complete!")
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println("="^70)
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println("\nNext steps:")
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println(" 1. Benchmark GPU vs CPU performance")
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println(" 2. Scale to realistic mesh sizes (10K+ elements)")
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println(" 3. Integrate with Newton-Krylov solver")
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println(" 4. Add line search for convergence")
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println(" 5. Add preconditioning (Chebyshev-Jacobi → GMG)")
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println("="^70 * "\n")
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end
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main()
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