1D Nodal Basis Function Implementation in MATLAB

A nodal basis function (hat function) has a value of 1 at its own node and drops to 0 at all other nodes. This property allows piecewise linear interpolation across finite element meshes.

1. MATLAB Script: Generate and Plot Global Hat Functions

This script sets up a 1D grid of node points, evaluates the basis functions, and plots the complete family of hat functions.

% Define node coordinates
nodes = [0.0, 0.25, 0.5, 0.75, 1.0];
numNodes = length(nodes);

% Fine grid for plotting smooth lines
xFine = linspace(nodes(1), nodes(end), 600);
figure('Color', 'w', 'Position', [100, 100, 750, 400]);
hold on;
colors = lines(numNodes);
% Compute and plot basis function for each node
for i = 1:numNodes
    basisValues = zeros(size(xFine));
    
    for k = 1:length(xFine)
        x = xFine(k);
        
        % Left slope of the hat
        if i > 1 && x >= nodes(i-1) && x <= nodes(i)
            basisValues(k) = (x - nodes(i-1)) / (nodes(i) - nodes(i-1));
        % Right slope of the hat
        elseif i < numNodes && x >= nodes(i) && x <= nodes(i+1)
            basisValues(k) = (nodes(i+1) - x) / (nodes(i+1) - nodes(i));
        end
    end
    
    plot(xFine, basisValues, 'LineWidth', 1.8, 'Color', colors(i,:), ...
        'DisplayName', sprintf('Node %d', i));
end
% Highlight nodes on plot
plot(nodes, ones(size(nodes)), 'ko', 'MarkerFaceColor', 'k', 'DisplayName', 'Peak (Value = 1)');
plot(nodes, zeros(size(nodes)), 'rs', 'MarkerFaceColor', 'r', 'DisplayName', 'Zeros (Value = 0)');
xlabel('Position X');
ylabel('Basis Value');
title('1D Piecewise Linear Nodal Basis Functions');
legend('Location', 'northeastoutside');
grid on;
ylim([-0.1, 1.1]);

2. MATLAB Function: Reusable Evaluator

Use this function to interpolate any nodal field values at arbitrary query points.

function uInterp = interpolate1D(nodes, nodalValues, xQuery)
% Interpolates field values using 1D nodal basis functions
% nodes: Vector of node positions [x1, x2, ..., xN]
% nodalValues: Field values at each node [u1, u2, ..., uN]
% xQuery: Points where you want the interpolated result

numNodes = length(nodes);
uInterp = zeros(size(xQuery));
for i = 1:numNodes
    phi_i = zeros(size(xQuery));
    for k = 1:length(xQuery)
        x = xQuery(k);
        if i > 1 && x >= nodes(i-1) && x <= nodes(i)
            phi_i(k) = (x - nodes(i-1)) / (nodes(i) - nodes(i-1));
        elseif i < numNodes && x >= nodes(i) && x <= nodes(i+1)
            phi_i(k) = (nodes(i+1) - x) / (nodes(i+1) - nodes(i));
        end
    end
    uInterp = uInterp + nodalValues(i) * phi_i;
end
end

3. Test Interpolation Example

% Node positions and recorded sensor values at each node
gridNodes = [0, 2, 4, 6, 8, 10];
sensorReadings = [10, 25, 40, 30, 15, 5];

% Query value at x = 3.5 (between node 2 and node 3)
xTest = 3.5;
estimatedValue = interpolate1D(gridNodes, sensorReadings, xTest);
fprintf('Interpolated value at x = %.1f is %.2f\n', xTest, estimatedValue);
% Output: Interpolated value at x = 3.5 is 36.25