# GPU Kernel Plan: Newton-Krylov-Anderson ## What We Learned from CPU Reference The complete solver pipeline (see `newton_krylov_anderson_cpu.jl`) has these operations: ### Outer Loop: Newton Iterations ```julia for iter in 1:max_newton_iter r = assemble_residual(u) # ← GPU KERNEL 1 du = gmres_solve(J, -r) # ← GMRES loop (see below) u_new = u + α * du # ← GPU vector op u = anderson_step(u_new, r) # ← CPU (small vectors) end ``` ### Middle Loop: GMRES Iterations ```julia for j in 1:max_gmres_iter w = J * v_j # ← GPU KERNEL 1 (matvec via FD) # Arnoldi orthogonalization # ← cuBLAS (dot products) # Givens rotations # ← CPU (small matrices) end ``` ### Inner Operation: Matrix-Free Matvec ```julia function apply_jacobian(v) r_pert = assemble_residual(u + ε*v) # ← GPU KERNEL 1 return (r_pert - r) / ε end ``` ## GPU Kernel Requirements ### **KERNEL 1: Element Residual Assembly** (The Workhorse) **What it does:** ```julia for elem in elements # ← parallelized over GPU threads for gp in gauss_points # 1. Compute Jacobian J = Σ dN_i ⊗ X_i # 2. Physical derivatives dN_dx = J^-1 · dN_dξ # 3. Strain ε = sym(Σ dN_i ⊗ u_i) # 4. Stress with plasticity (σ, state_new) = return_mapping(ε, state_old) # 5. Nodal forces f_i = dN_i · σ # 6. Scatter to global (atomics!) atomic_add!(r_global[dofs], f_i) end end ``` **Inputs:** (all on GPU) - `nodes::CuArray{Float64, 2}` - Shape (3, n_nodes) - `elements::CuArray{Int32, 2}` - Shape (4, n_elements) for Tet4 - `u::CuArray{Float64}` - Solution vector - `states::CuArray{PlasticState}` - Plastic states per GP **Outputs:** (all on GPU) - `r::CuArray{Float64}` - Residual vector - `states::CuArray{PlasticState}` - Updated states (in-place) **Key challenge:** Manual indexing inside kernel (no Tensors.jl) ### **KERNEL 2: Vector Operations** (Trivial) These are standard cuBLAS operations: - `y = α*x + β*y` (axpy) - `dot(x, y)` - `norm(x)` Already provided by CUDA.jl ## The GPU Implementation Strategy ### Phase 1: Single Kernel Test ```julia # Test just the residual assembly kernel r_cpu = assemble_residual_cpu(u) r_gpu = assemble_residual_gpu(u) # ← Implement this! @test r_cpu ≈ r_gpu ``` ### Phase 2: Matrix-Free Matvec Test ```julia # Test Jacobian-vector product Jv_cpu = apply_jacobian_cpu(u, v) Jv_gpu = apply_jacobian_gpu(u, v) # ← Just calls kernel twice @test Jv_cpu ≈ Jv_gpu ``` ### Phase 3: GMRES on GPU ```julia # Krylov.jl supports CuArrays! using Krylov, CUDA du_gpu = CuVector(zeros(n)) gmres!(du_gpu, J_gpu, -r_gpu) # ← Should work with CuArrays ``` ### Phase 4: Complete Newton-Krylov-Anderson on GPU ```julia u_gpu = CuVector(u) states_gpu = CuArray(states) for iter in 1:max_iter r_gpu = assemble_residual_gpu!(u_gpu, states_gpu) # ← GPU gmres!(du_gpu, J_gpu, -r_gpu) # ← GPU u_gpu .+= du_gpu # ← GPU u_cpu = Vector(u_gpu) # ← Copy to CPU u_cpu = anderson_step(u_cpu, Vector(r_gpu)) # ← CPU u_gpu = CuVector(u_cpu) # ← Copy to GPU end ``` **Note:** Anderson runs on CPU (small vectors, needs least-squares). This is fine! ## Key Implementation Details ### 1. Plastic State on GPU ```julia # CPU version (Tensors.jl) struct PlasticState ε_p::SymmetricTensor{2,3,Float64,6} α::Float64 end # GPU version (manual indexing) struct PlasticStateGPU ε_p::NTuple{6,Float64} # Voigt notation: (ε11, ε22, ε33, ε12, ε13, ε23) α::Float64 end ``` ### 2. Return Mapping on GPU Must be **completely manual** - no Tensors.jl: ```julia function return_mapping_gpu(ε_total, state_old, E, ν, σ_y) # All operations in Voigt notation # ε = (ε11, ε22, ε33, ε12, ε13, ε23) # Trial stress (manual Hooke's law) λ = E * ν / ((1 + ν) * (1 - 2ν)) μ = E / (2(1 + ν)) ε_e = ε_total .- state_old.ε_p tr_ε = ε_e[1] + ε_e[2] + ε_e[3] σ_trial = ( λ * tr_ε + 2μ * ε_e[1], # σ11 λ * tr_ε + 2μ * ε_e[2], # σ22 λ * tr_ε + 2μ * ε_e[3], # σ33 2μ * ε_e[4], # σ12 2μ * ε_e[5], # σ13 2μ * ε_e[6] # σ23 ) # Deviatoric stress (manual) p = (σ_trial[1] + σ_trial[2] + σ_trial[3]) / 3 s = (σ_trial[1] - p, σ_trial[2] - p, σ_trial[3] - p, σ_trial[4], σ_trial[5], σ_trial[6]) # von Mises stress σ_eq = sqrt(3/2 * (s[1]^2 + s[2]^2 + s[3]^2 + 2*(s[4]^2 + s[5]^2 + s[6]^2))) # Yield check f = σ_eq - σ_y if f <= 0 return (σ_trial, state_old, false) else # Radial return Δγ = f / (3μ) β = 1 - 2μ * Δγ / σ_eq σ_new = ( β * s[1] + p, β * s[2] + p, β * s[3] + p, β * s[4], β * s[5], β * s[6] ) # Update plastic strain n = s ./ σ_eq Δε_p = Δγ .* n ε_p_new = state_old.ε_p .+ Δε_p α_new = state_old.α + Δγ state_new = PlasticStateGPU(ε_p_new, α_new) return (σ_new, state_new, true) end end ``` ### 3. Element Kernel Structure ```julia function element_residual_kernel!(r_global, nodes, elements, u, states, E, ν, σ_y) elem_idx = (blockIdx().x - 1) * blockDim().x + threadIdx().x if elem_idx <= size(elements, 2) # Extract element nodes n1, n2, n3, n4 = elements[:, elem_idx] X1 = (nodes[1, n1], nodes[2, n1], nodes[3, n1]) X2 = (nodes[1, n2], nodes[2, n2], nodes[3, n2]) X3 = (nodes[1, n3], nodes[2, n3], nodes[3, n3]) X4 = (nodes[1, n4], nodes[2, n4], nodes[3, n4]) # Extract displacements u1 = (u[3*n1-2], u[3*n1-1], u[3*n1]) u2 = (u[3*n2-2], u[3*n2-1], u[3*n2]) u3 = (u[3*n3-2], u[3*n3-1], u[3*n3]) u4 = (u[3*n4-2], u[3*n4-1], u[3*n4]) # Shape derivatives (constant for Tet4) dN1 = (-1.0, -1.0, -1.0) dN2 = (1.0, 0.0, 0.0) dN3 = (0.0, 1.0, 0.0) dN4 = (0.0, 0.0, 1.0) # Jacobian (manual tensor product and sum) J11 = dN1[1]*X1[1] + dN2[1]*X2[1] + dN3[1]*X3[1] + dN4[1]*X4[1] J12 = dN1[1]*X1[2] + dN2[1]*X2[2] + dN3[1]*X3[2] + dN4[1]*X4[2] # ... (all 9 components) # Inverse Jacobian (manual 3x3 inverse) detJ = J11*(J22*J33 - J23*J32) - J12*(J21*J33 - J23*J31) + ... invJ11 = (J22*J33 - J23*J32) / detJ # ... (all 9 components) # Physical derivatives (manual matrix-vector multiply) dN1_dx = (invJ11*dN1[1] + invJ12*dN1[2] + invJ13*dN1[3], invJ21*dN1[1] + invJ22*dN1[2] + invJ23*dN1[3], invJ31*dN1[1] + invJ32*dN1[2] + invJ33*dN1[3]) # ... (for all 4 nodes) # Strain (manual tensor product, sum, symmetrize) gradu11 = dN1_dx[1]*u1[1] + dN2_dx[1]*u2[1] + dN3_dx[1]*u3[1] + dN4_dx[1]*u4[1] # ... (all 9 components) ε = (gradu11, gradu22, gradu33, (gradu12 + gradu21) / 2, (gradu13 + gradu31) / 2, (gradu23 + gradu32) / 2) # Plasticity state_old = states[elem_idx, 1] # 1 GP for Tet4 (σ, state_new, plastic) = return_mapping_gpu(ε, state_old, E, ν, σ_y) states[elem_idx, 1] = state_new # Forces (manual contraction) f1 = (dN1_dx[1]*σ[1] + dN1_dx[2]*σ[4] + dN1_dx[3]*σ[5], dN1_dx[1]*σ[4] + dN1_dx[2]*σ[2] + dN1_dx[3]*σ[6], dN1_dx[1]*σ[5] + dN1_dx[2]*σ[6] + dN1_dx[3]*σ[3]) # ... (for all 4 nodes) # Gauss weight * detJ wdetJ = (1.0/6.0) * detJ # Scatter to global (ATOMIC!) CUDA.@atomic r_global[3*n1-2] += f1[1] * wdetJ CUDA.@atomic r_global[3*n1-1] += f1[2] * wdetJ CUDA.@atomic r_global[3*n1] += f1[3] * wdetJ # ... (for all 4 nodes) end return nothing end ``` ## Next Steps 1. **Implement `element_residual_kernel!`** - The core GPU kernel (all manual indexing) 2. **Test kernel correctness** - Compare GPU vs CPU residual 3. **Wrap in matrix-free operator** - For GMRES 4. **Test GMRES with CuArrays** - Krylov.jl should handle it 5. **Complete Newton loop** - Anderson on CPU is fine The key insight: **Everything except Anderson can stay on GPU!**