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Engineering design and CAx
topology-optimization
topology optimization for design
TopOpt, nTop
System
Techniques · 11
Primitives · 25
topology-optimization
Halo exchange
distributed · medium: Each rank stores its block plus a border copy of its neighbours' edge cells, refreshed by small messages every step; communication scales with block surface, not volume.
›
B rep topology
geometry · medium: Boundary representation: solids as shells, faces, loops, edges and vertices with geometry attached
Marching cubes
graphics · medium: Extract triangle meshes from scalar fields (marching cubes, dual contouring)
Mesh simplification quadric
graphics · medium: Edge collapse with quadric error metrics to build mesh LODs
Finite elements
numerics · medium: Writes the PDE in weak form, approximates it with piecewise polynomials on a mesh and assembles a sparse system; handles any geometry.
›
Level set methods
numerics · medium: Implicit surfaces as signed distance fields: advection, reinitialization, fast marching
Sparse direct solver
numerics · medium: Solve sparse Ax = b exactly by reordering, symbolic analysis, numeric LLᵀ/LU and triangular solves; fill-in, not flops per entry, decides the cost.
›
Conjugate gradient
numerics · low: Krylov iteration for symmetric positive definite Ax = b using one matrix-vector product and two dot products per step; converges in about √κ iterations.
›
Sparse matrices (CSR)
numerics · low: Compressed sparse row stores only nonzeros as row pointers, column indices and values; SpMV streams them once, so it is memory-bandwidth bound.
›
Unstructured mesh representation
numerics · low: Cell-face-node connectivity in CSR form for FEM and FVM meshes; partition-friendly ordering
Convolution image filters
graphics · low: Separable 2D convolution kernels for blur, sharpen (unsharp mask) and edges with border handling
CUDA and GPU languages
languages · low: Programming GPUs: thousands of lockstep threads, bound by memory bandwidth; libraries first, Triton for fused kernels, CUDA C++ for peak speed.
›
Collective communication
ml-training · low: The family of group exchanges (broadcast, reduce, all-gather, reduce-scatter, all-to-all) and the latency-plus-bytes cost model that picks ring, tree or direct algorithms by message size and topology.
›
Multigrid
numerics · low: Smooths high-frequency error on the fine grid and solves the smooth remainder on coarser grids; O(n) elliptic solves at any resolution.
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Eager rendezvous protocol
distributed · low: MPI point-to-point: eager sends for small messages, rendezvous handshake for large ones
Rdma
networking · low: RDMA verbs over InfiniBand/RoCE: queue pairs, memory registration, one-sided ops
Zero-copy I/O
networking · low: Move bytes between files, sockets and NICs by passing page references instead of copying through user space: sendfile, splice, MSG_ZEROCOPY, kTLS.
›
Exact geometric predicates
geometry · low: Robust orient and incircle tests with adaptive-precision floats (Shewchuk) or exact arithmetic
Half edge mesh
graphics · low: Half-edge mesh connectivity for editable polygon meshes
Finite differences
numerics · low: Replace derivatives with weighted differences of grid values; simple stencils plus a stability condition (CFL) turn PDEs like heat, advection and waves into array loops.
›
Spatial hashing
physics · low: Uniform grid stored in a hash table: bucket objects by cell, test only neighbouring cells; O(n) neighbour search for many similar-sized objects such as particles.
›
d-ary heap
data-structures · low: Array heap with d children per node (d = 4 typical): half the binary heap's depth, siblings in one cache line, faster priority queues.
›
LU and Cholesky factorization
numerics · low: Factor A = PLU (any square matrix, partial pivoting) or A = LLᵀ (SPD) once in O(n³), then solve each right-hand side in O(n²).
›
+ 4 more techniques
+ 9 more primitives
System
Techniques · 11
Primitives · 25
topology-optimization
Halo exchange
distributed · medium: Each rank stores its block plus a border copy of its neighbours' edge cells, refreshed by small messages every step; communication scales with block surface, not volume.
›
B rep topology
geometry · medium: Boundary representation: solids as shells, faces, loops, edges and vertices with geometry attached
Marching cubes
graphics · medium: Extract triangle meshes from scalar fields (marching cubes, dual contouring)
Mesh simplification quadric
graphics · medium: Edge collapse with quadric error metrics to build mesh LODs
Finite elements
numerics · medium: Writes the PDE in weak form, approximates it with piecewise polynomials on a mesh and assembles a sparse system; handles any geometry.
›
Level set methods
numerics · medium: Implicit surfaces as signed distance fields: advection, reinitialization, fast marching
Sparse direct solver
numerics · medium: Solve sparse Ax = b exactly by reordering, symbolic analysis, numeric LLᵀ/LU and triangular solves; fill-in, not flops per entry, decides the cost.
›
Nurbs
geometry · medium: B-splines and NURBS curves and surfaces: knot vectors, de Boor evaluation, knot insertion
Fill-reducing ordering
numerics · medium: Permute a sparse symmetric matrix before Cholesky/LU so the factor has few new nonzeros; minimum degree, nested dissection and RCM trade quality, speed and parallelism.
›
Flux limiters upwinding
numerics · medium: Upwind schemes, TVD flux limiters, MUSCL and Riemann solvers for shock-capturing advection
Graph partitioning
graphs · medium: Split a graph into k balanced parts with few edges between them; multilevel coarsen-partition-refine (METIS, KaHIP) for quality, streaming heuristics (LDG, Fennel) for one-pass scale.
›
Conjugate gradient
numerics · low: Krylov iteration for symmetric positive definite Ax = b using one matrix-vector product and two dot products per step; converges in about √κ iterations.
›
Sparse matrices (CSR)
numerics · low: Compressed sparse row stores only nonzeros as row pointers, column indices and values; SpMV streams them once, so it is memory-bandwidth bound.
›
Unstructured mesh representation
numerics · low: Cell-face-node connectivity in CSR form for FEM and FVM meshes; partition-friendly ordering
Convolution image filters
graphics · low: Separable 2D convolution kernels for blur, sharpen (unsharp mask) and edges with border handling
CUDA and GPU languages
languages · low: Programming GPUs: thousands of lockstep threads, bound by memory bandwidth; libraries first, Triton for fused kernels, CUDA C++ for peak speed.
›
Collective communication
ml-training · low: The family of group exchanges (broadcast, reduce, all-gather, reduce-scatter, all-to-all) and the latency-plus-bytes cost model that picks ring, tree or direct algorithms by message size and topology.
›
Multigrid
numerics · low: Smooths high-frequency error on the fine grid and solves the smooth remainder on coarser grids; O(n) elliptic solves at any resolution.
›
Eager rendezvous protocol
distributed · low: MPI point-to-point: eager sends for small messages, rendezvous handshake for large ones
Rdma
networking · low: RDMA verbs over InfiniBand/RoCE: queue pairs, memory registration, one-sided ops
Zero-copy I/O
networking · low: Move bytes between files, sockets and NICs by passing page references instead of copying through user space: sendfile, splice, MSG_ZEROCOPY, kTLS.
›
Exact geometric predicates
geometry · low: Robust orient and incircle tests with adaptive-precision floats (Shewchuk) or exact arithmetic
Half edge mesh
graphics · low: Half-edge mesh connectivity for editable polygon meshes
Finite differences
numerics · low: Replace derivatives with weighted differences of grid values; simple stencils plus a stability condition (CFL) turn PDEs like heat, advection and waves into array loops.
›
Spatial hashing
physics · low: Uniform grid stored in a hash table: bucket objects by cell, test only neighbouring cells; O(n) neighbour search for many similar-sized objects such as particles.
›
d-ary heap
data-structures · low: Array heap with d children per node (d = 4 typical): half the binary heap's depth, siblings in one cache line, faster priority queues.
›
LU and Cholesky factorization
numerics · low: Factor A = PLU (any square matrix, partial pivoting) or A = LLᵀ (SPD) once in O(n³), then solve each right-hand side in O(n²).
›
Gaussian quadrature
numerics · low: Approximate an integral by a weighted sum of n function values at roots of orthogonal polynomials; exact for polynomials up to degree 2n-1.
›
Runge-Kutta 4
numerics · low: The classic four-stage, fourth-order ODE step; error drops 16× when the step halves, with no derivatives or past steps needed.
›
Bézier curves
graphics · low: Smooth curves defined by 3-4 control points, evaluated and split with repeated linear interpolation (de Casteljau) and flattened to lines with a closed-form segment count.
›
Spline interpolation
numerics · low: Linear, bilinear/trilinear and cubic spline interpolation on grids and tables
PRNGs (PCG, xoshiro)
numerics · low: Small, fast, statistically strong non-cryptographic generators: PCG32 and xoshiro256**, with seeding, parallel streams and unbiased ranges.
›
Fast hash functions
data-structures · low: Non-cryptographic hashes built from 64×64→128-bit multiplies and xor-shifts; pick by key size, trust level and whether the hash is persisted.
›
Explicit Euler
numerics · low: Step an ODE forward by its current slope, y += h·f(t, y); first-order, one evaluation per step, the baseline every other integrator is measured against.
›
Thomas tridiagonal solver
numerics · low: O(n) tridiagonal linear solve used in implicit finite-difference and spline schemes
Parallel prefix sum
search-sort · low: Work-efficient scan (Blelloch, decoupled look-back) on CPU SIMD and GPUs; stream compaction
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