Parallel FEM Solver for 2D Beam/Frame Structures

This HPSC course project implements, verifies, profiles, and parallelises a finite element solver for static and free-vibration analysis of 2D beam/frame structures using OpenMP.

The main report PDF is:

report/white_paper.pdf

GitHub repository:

https://github.com/Praveenjhas/hpsc_project

Project Highlights

• 2D Euler-Bernoulli frame element with 3 DOF per node: u, v, and theta
• Static structural solve using K u = F
• Free-vibration solve using K phi = omega^2 M phi
• First natural frequency computed by inverse iteration
• Sparse global stiffness matrix stored in CSR format
• Lumped diagonal mass matrix
• Jacobi-preconditioned Conjugate Gradient solver
• OpenMP parallelisation for element stiffness, assembly paths, mass assembly, sparse matrix-vector multiplication, vector updates, and dot-product reductions
• Cantilever validation against analytical beam formulas
• Portal-frame structural demonstration
• Stage-wise profiling, strong scaling, and weak scaling
• Publication-style white paper with figures, result tables, limitations, and reproducibility details

Key Results

The generated run reported:

Metric	Result
200-element cantilever tip-deflection error	1.41913e-05%
200-element cantilever first-frequency error	0.00733476%
Dominant profiled stage	vibration_inverse_iteration
Dominant-stage runtime share	81.5273%
Best OpenMP strong-scaling speedup	5.7283x on 8 threads
Best strong-scaling efficiency	0.7160
Weak-scaling efficiency at 8 threads	0.5650

Demonstration Cases

Case	Inputs	Purpose
Cantilever beam	L = 10 m, E = 200 GPa, A = 0.003 m^2, I = 8.0e-6 m^4, rho = 7850 kg/m^3, end load -1000 N	Verify static deflection and first natural frequency against beam theory
Portal frame	Width 8 m, height 5 m, E = 200 GPa, A = 0.004 m^2, I = 1.2e-5 m^4, rho = 7850 kg/m^3, top load (Fx, Fy) = (3000, -12000) N	Demonstrate frame behaviour for vertical columns and a horizontal beam

Solver Settings

Solver	Limit	Stopping criterion
Static CG	5000 iterations	Relative residual <= 1e-10
Inner CG inside inverse iteration	5000 iterations	Relative residual <= 1e-10
Cantilever inverse iteration	100 iterations	Relative eigenvalue change < 1e-9
Portal-frame inverse iteration	100 iterations	Relative eigenvalue change < 1e-8

Build

Using Make:

make

On Windows, the direct compiler command is:

g++ -O3 -std=c++17 -Wall -Wextra -Iinclude -fopenmp src/main.cpp src/fem.cpp src/sparse_matrix.cpp src/solver.cpp src/benchmark.cpp -o fem_solver.exe

Using CMake:

cmake -S . -B build
cmake --build build --config Release

Run

Cantilever demo:

.\fem_solver.exe demo 100 4

Validation sweep:

.\fem_solver.exe validation

Portal-frame demo:

.\fem_solver.exe portal 4

Stage-wise profiling:

.\fem_solver.exe profiling

Strong-scaling benchmark:

.\fem_solver.exe benchmark

Weak-scaling benchmark:

.\fem_solver.exe weak-scaling

Larger benchmark for stronger machines:

.\fem_solver.exe benchmark-large

Generate figures:

python scripts\plot_results.py

One-command Windows Workflow

For a clean full rerun:

powershell -ExecutionPolicy Bypass -File .\run_all.ps1 -Clean

This builds the C++ solver, runs validation and demos, profiles the solver, runs strong/weak scaling, generates plots, and builds report outputs.

Report and PDF Workflow

Main report files:

report/white_paper.tex
report/white_paper.pdf
report/PROJECT_REPORT.md
report/PROJECT_REPORT.pdf
report/PRESENTATION_OUTLINE.md

The final white paper PDF can also be regenerated through the browser-print workflow used on this machine:

powershell -ExecutionPolicy Bypass -File .\scripts\fix_runtime_share_figure.ps1
powershell -ExecutionPolicy Bypass -File .\scripts\whitepaper_to_html.ps1
& 'C:\Program Files (x86)\Microsoft\Edge\Application\msedge.exe' --headless --disable-gpu --no-pdf-header-footer --print-to-pdf='C:\Users\HI\OneDrive\Desktop\assignment\hpsc_project\report\white_paper.pdf' 'file:///C:/Users/HI/OneDrive/Desktop/assignment/hpsc_project/report/white_paper.html'

Outputs

Generated data:

results/validation.csv
results/benchmark.csv
results/profiling.csv
results/weak_scaling.csv
results/displacement.csv
results/mode_shape.csv
results/portal_displacement.csv
results/portal_mode_shape.csv

Generated figures:

report/figures/cantilever_validation_errors.png
report/figures/cantilever_tip_displacement_convergence.png
report/figures/cantilever_frequency_convergence.png
report/figures/cantilever_deflection.png
report/figures/cantilever_first_mode_shape.png
report/figures/portal_frame_deflection.png
report/figures/portal_frame_first_mode_shape.png
report/figures/stage_runtime_profile.png
report/figures/stage_runtime_share.png
report/figures/openmp_speedup.png
report/figures/openmp_efficiency.png
report/figures/element_kernel_runtime.png
report/figures/weak_scaling_runtime.png
report/figures/weak_scaling_efficiency.png

Repository Layout

include/                 C++ headers
src/                     FEM solver, sparse matrix, solvers, and benchmarks
scripts/                 Plotting and report-generation helpers
results/                 Generated CSV outputs
report/                  White paper, final PDF, report figures, and outline
run_all.ps1              One-command Windows workflow
Makefile                 Make build entry point
CMakeLists.txt           CMake build entry point

Notes

The numerical model is a 2D linear Euler-Bernoulli beam/frame formulation. It uses a lumped diagonal mass matrix and computes the first natural frequency rather than a full modal spectrum. These assumptions are documented in the final white paper under the limitations and future-work section.