{"id":1722,"date":"2026-11-07T18:54:54","date_gmt":"2026-11-07T10:54:54","guid":{"rendered":"https:\/\/people.utm.my\/arazin\/?page_id=1722"},"modified":"2026-11-07T18:54:55","modified_gmt":"2026-11-07T10:54:55","slug":"2d-formulation-and-discretization-explorer","status":"publish","type":"page","link":"https:\/\/people.utm.my\/arazin\/2d-formulation-and-discretization-explorer\/","title":{"rendered":"2D Formulation and Discretization Explorer"},"content":{"rendered":"
\n

<\/p>\n

\n

Interactive tool · From the PDE to the matrix system<\/p>\n

2D Formulation and Discretization Explorer<\/h1>\n
\nAbout this tool<\/summary>\n

Choose the problem, the integral form, how the geometry is represented and how the field is interpolated, and follow the derivation from the strong form to the matrix system, step by step, with the numbers of your model. The tool assembles the system but does not solve it: the point is to see how NURBS, finite elements and meshfree shape functions enter the same equations. For every choice, a detailed derivation with a worked example can be saved as a PDF or a LaTeX file, and a Python script rebuilds the same system, checks it against the tool and goes one step further: it solves the system and plots the field.<\/p>\n<\/details>\n<\/div>\n

<\/p>\n

\n

Alatan interaktif · Daripada PDE ke matrix system<\/p>\n

2D Formulation and Discretization Explorer<\/h1>\n
\nTentang alatan ini<\/summary>\n

Pilih masalah, integral form, cara geometri diwakili dan cara field diinterpolasi, dan ikuti derivation daripada strong form hingga matrix system, langkah demi langkah, dengan nombor model anda. Tool ini meng-assemble sistem tetapi tidak menyelesaikannya: tujuannya ialah melihat bagaimana NURBS, finite element dan shape functions meshfree masuk ke dalam persamaan yang sama. Bagi setiap pilihan, derivation terperinci dengan contoh berangka boleh disimpan sebagai PDF atau fail LaTeX, dan skrip Python membina semula sistem yang sama, menyemaknya dengan tool dan melangkah setapak lagi: menyelesaikan sistem itu dan memplot field.<\/p>\n<\/details>\n<\/div>\n

<\/p>\n

\n

Loading the interactive tool…<\/span>Memuatkan alatan interaktif…<\/span><\/p>\n<\/div>\n

\n<\/p>\n
\n
\nWhat this tool shows<\/summary>\n
\n

A numerical method for a PDE is built in four decisions, and textbooks usually fix three of them before the reader notices. This tool keeps all four open: the problem<\/b> (steady heat conduction or plane stress), the integral form<\/b> (the Galerkin weak form or the divergence theorem applied cell by cell), the geometry representation<\/b> (an exact NURBS patch, a Lagrange finite element mesh, or a meshfree map that interpolates the coordinates of the nodes by MLS or RBF) and the field interpolation<\/b> (NURBS, Lagrange or meshfree shape functions, built in the parameter space or in the physical space). For every combination the derivation on the right is rewritten from the strong form to the matrix system, with the numbers of the current model, and the drawings show what the method sees: the domain as the representation knows it, the integration cells and Gauss points, one cell’s contribution and the sparsity pattern of the matrix. The system is assembled but never solved: the point is to see how it comes about.<\/p>\n

One parameter square.<\/b> The idea that ties the representations together is that each one is a map from the same unit square (ξ, η) onto the domain. A NURBS patch is a rational map; a structured finite element mesh is a piecewise polynomial map, one element at a time; a meshfree map interpolates the coordinates of nodes placed on the geometry with MLS or RBF functions, of the lowest order that is worth it. The field can be built on that square (NURBS, Lagrange, or meshfree in ξ, η as in our N-RPIM and NDM) or directly in x, y (meshfree, EFG-type). This is why the pairs form a full table (three geometries; the meshfree field counted twice, built in ξ, η or in x, y), here with the names the pairings carry in the literature:<\/p>\n

\n\n\n\n\n\n\n\n
geometry \\ field<\/th>\nNURBS<\/th>\nLagrange FE<\/th>\nmeshfree in (ξ, η)<\/th>\nmeshfree in (x, y)<\/th>\n<\/tr>\n<\/thead>\n
NURBS patch<\/th>\nIGA
cells: CV-IGA<\/small><\/td>\n		exact-geometry FEM (GIFT-type)
cells: NDFEM<\/small><\/td>\n		N-RPIM (RPIM); MLS: no name
cells: NDM<\/small><\/td>\n		EFG on the exact geometry
cells: MLPG5-type<\/small><\/td>\n<\/tr>\n
FE mesh<\/th>\nB-spline field on a mesh
cells: no name<\/small><\/td>\n		isoparametric FEM
cells: CVFEM<\/small><\/td>\n		parameter-space meshfree
cells: NDM-type<\/small><\/td>\n		EFG with a background mesh
cells: MLPG5-type<\/small><\/td>\n<\/tr>\n
meshfree map<\/th>\nB-spline field on it
cells: no name<\/small><\/td>\n		Lagrange field on it
cells: CVFEM-type<\/small><\/td>\n		meshfree map and field
cells: NDM-type<\/small><\/td>\n		EFG
cells: MLPG5-type<\/small><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

First line: the Galerkin weak form. Second line (cells): the divergence theorem with cells around the nodes; with a regular grid of cells, not centred on the nodes, a Lagrange or spline field gives the subdomain method of weighted residuals instead.<\/p>\n

Every cell of the table can be built, because even the meshfree representation is a map from the parameter square: its nodes carry parameter coordinates, so a B-spline or a Lagrange field can live on it too. Some pairings are unusual (a B-spline field on a finite element map, a NURBS basis with cell balances) and are kept to make the point that the geometry and the field need not share a basis; step 4 says whether a pairing is isoparametric and what is lost when it is not.<\/p>\n

Names.<\/b> Several pairings have a name in the literature, and a few are easily confused. NEFEM (Sevilla, Fernández-Méndez and Huerta 2008) is not the NURBS × Lagrange pairing of this tool: NEFEM keeps the exact NURBS only on the boundary, uses standard elements inside and builds the field of a boundary element with polynomials in x, y. An exact NURBS map with a Lagrange field in the parameter space is closer to the Geometry-Independent Field approximaTion (GIFT) of Atroshchenko et al. (2018), where the geometry and the field are chosen independently. The cell balance with a Lagrange field and cells around the nodes is the control-volume finite element method (CVFEM, Baliga and Patankar 1980), on any geometry; with a NURBS field and cells midway between the Greville points it is the control volume isogeometric analysis (CV-IGA) of Kamber et al. (2022), who use Fup basis functions. MLS in x, y with one cell balance per node is of the MLPG5 type (meshless local Petrov–Galerkin with the Heaviside step function as test function, Atluri and Shen 2002). “Meshfree” refers to the approximation: every pairing here, EFG included, still integrates over cells.<\/p>\n

Galerkin or divergence.<\/b> The weak form multiplies the PDE by a weight function, integrates by parts and lets the natural boundary condition enter by itself; the resulting matrix is symmetric and the derivatives are shared between the weight functions and the unknown, so C0<\/sup> approximations suffice. The divergence form integrates the PDE over one cell per node and turns it into a flux balance (or, for plane stress, the equilibrium of the cell); no weight function, no symmetry, the full derivative of the unknown on the cell sides, and the essential condition imposed by a collocation row at the boundary nodes. With the cells built in the NURBS parameter space, every cell is a rectangle there and the normal on each side is simply (0, ±1) or (±1, 0): this is the NDM of Rasin et al. (2025).<\/p>\n

What the numbers mean.<\/b> The metrics line reports the number of unknowns, the size of the system actually shown (smaller after the direct partition, larger with Lagrange multipliers), the non-zero entries and the bandwidth. In step 3 the area of the representation is compared with the exact area: the NURBS map is exact, a Q4 mesh loses the segments between its chords, and a meshfree map approaches it with the order reported in step 3. In step 4 the value of a shape function at its own node tells whether the field interpolates (1 for Lagrange and RPIM; less for NURBS and MLS, whose coefficients are not nodal values). In step 5 the selected cell lists its Gauss points with the Jacobian determinant and the weight, and its local matrix with the rows and columns it is scattered into. In step 6 the essential condition is imposed by the direct partition, by a penalty term, by Nitsche’s method or by Lagrange multipliers, each with its effect on the size and the symmetry of the system.<\/p>\n

Model data.<\/b> Heat conduction: k = 1, prescribed temperature 1, inflow flux 1 on loaded edges, source 1 when switched on. Plane stress: E = 1, ν = 0.3, thickness 1, fixed edges, a normal pull of 1 on loaded edges, body force (0, −1) when switched on. The values only scale the right-hand side; the structure of the matrix does not depend on them.<\/p>\n

The detailed derivation.<\/b> The download section under the name of the method (open it with +) produces a longer document for the current choices, written for students who want every step: the energy balance or the equilibrium of a differential element, integration by parts (or the divergence theorem) line by line, the construction of the chosen shape functions and geometry map, the Jacobian and the Gauss rules, the essential-condition method, then a worked example of the selected cell or node, Gauss point by Gauss point with the numbers of the model, and a verification table (PASS when an identity holds, OPEN when it is not established). PDF<\/i> opens it as a print view (choose “Save as PDF”); LaTeX source<\/i> downloads a .tex file that compiles with pdflatex. The document is in English.<\/p>\n

The Python script.<\/b> Python script<\/i> downloads a .py file for the current choices that carries the derivation into code, one function per step of the document and with its equation numbers in the comments: the NURBS patch and its refinement, the geometry map, the shape functions, the assembly cell by cell (or row by row), the essential condition. It holds the code of the chosen pairing only, so a student reads one formulation, not a library. Run it with Python 3 and numpy: it first checks its own matrix against the numbers of the tool (size, non-zero entries, norm, right-hand side, the worked example; PASS or FAIL), then goes one step further than the tool, solves the system and plots the field with matplotlib. For the quarter annulus in heat conduction, with insulated straight edges, it also measures the error against the exact radial solution, so raising LEVEL<\/code> shows the convergence. The refinement level, the Gauss rule, the boundary conditions, the material and the loads can be changed at the top of the file.<\/p>\n

One set of symbols.<\/b> Every pairing is written with the same symbols, so that FEM, IGA, EFG, N-RPIM and NDM can be read side by side; the full list is step 0 of the derivation. In short: the notation of the MSNM textbooks (fraction derivatives, the operator vector {∂} and matrix [∂], the material matrix [E], matrices in [ ] and column vectors in { }); field shape functions NJ<\/sub> of every kind (the papers write ΦJ<\/sub> for RPIM and a modified N for MLS) and a geometry basis Ngeo<\/sup>k<\/sub>; unknowns {T̂}, {û} and prescribed values T̄, ū; global entries KIJ<\/sub> with row I and column J, local entries kij<\/sub>; Galerkin cells Ωe<\/sub> and divergence cells ΩI<\/sub>; one system [K]{T̂} = {R} for both forms (the NDM paper writes {F}), with {R} = {Q} + {B} and locally [k]{T̂e<\/sub>} = {q} + {b} = {r}; the NDM flux {H} = k[G]{∂T\/∂ξ  ∂T\/∂η}; [A] for the moment matrix of any meshfree support, so that [G] is kept for the NDM metric. Essential conditions are imposed by row–column condensation, [Kff<\/sub>]{T̂f<\/sub>} = {Rf<\/sub>} − [Kfc<\/sub>]{T̄c<\/sub>}, or by penalty, Nitsche’s method or Lagrange multipliers. Nodes and cells are numbered from 1.<\/p>\n

Integration cells of the divergence form.<\/b> One cell per unknown is all that the method needs; the cells need not be centred on the nodes. Around the nodes<\/i> builds the dual cells of the node layout (a vertex-centred finite volume: CVFEM with a Lagrange field, CV-IGA with a NURBS field, NDM with a meshfree field); regular grid<\/i> places an nξ<\/sub> × nη<\/sub> grid over the node indices and pairs cell (i, j) with node (i, j), as the NDM does: with evenly spaced nodes it is the regular grid of the parameter square, and with unequal knot spans it follows the node spacing, so that every node stays inside its own cell (otherwise the system can become singular); the size factor γ shrinks the cells (gaps) or enlarges them (overlap). Fields built in x, y keep their dual cells.<\/p>\n

Choices made in this tool.<\/b> Refinement level r gives 2r<\/sup> cells per knot span in each direction, and the nodes of the Lagrange, lattice and meshfree-map representations coincide, so the methods can be compared at equal unknowns. The meshfree map is built on the node coordinates in the index space of the node lattice, either by MLS (Gaussian weight, support 2.5 spacings, not interpolating) or by RBF (multiquadric with a polynomial, support 3 spacings, interpolating), with a linear or a quadratic basis: the optimum is the lowest order whose boundary error the next order does not at least halve, unless it is set by hand. Meshfree in the parameter space uses the index-space construction of our N-RPIM and NDM codes (MLS with a quadratic basis and Gaussian weight, or RPIM with multiquadrics and a linear polynomial; supports stop only at the C0<\/sup> knot lines of the unrefined patch, not at the knots inserted by refinement). Meshfree in the physical space uses MLS with a quadratic basis, a quartic spline weight and the visibility criterion, tested against the boundary of the geometry representation itself (sampled between the boundary nodes, so that the Gauss points next to a curved edge still see their nodes). The divergence cells are the rectangles between the mid-points of neighbouring node coordinates in the parameter space, or the dual cells of the node lattice in x, y. Dense storage is used because the systems are small. Not covered: solving, error estimates, time dependence, multi-patch geometry, Nitsche’s method.<\/p>\n<\/div>\n<\/details>\n

\nThings to try<\/summary>\n
\n
    \n
  1. Start with the quarter annulus, NURBS geometry and NURBS field (isogeometric analysis). Switch the geometry to a finite element mesh and watch step 3: the area drops below the exact value and the dashed exact outline no longer coincides with the boundary. Raise the refinement level and watch the ratio approach 1.<\/li>\n
  2. Keep the finite element mesh and choose a meshfree field in x, y: this is the element-free Galerkin method with a background mesh. Select a node near the hole of the plate; the support circle and the nodes inside it show which unknowns couple, and the bandwidth jumps.<\/li>\n
  3. Choose NURBS geometry, meshfree field in (ξ, η) and the divergence form: the NDM. Tap a node and read its cell in step 5: four sides, each with the known flux or an unknown one, and the balance as a row of [K]. Then switch the integration cells to the regular grid and move the size factor γ: the cells detach from the nodes, overlap or leave gaps, and the system stays square. Then tap a node on an essential edge: its row is a collocation condition.<\/li>\n
  4. With a Lagrange field, impose the essential condition directly in step 6 and note the smaller system; switch to Lagrange multipliers and watch the system grow, with a zero block in the sparsity pattern.<\/li>\n
  5. With a meshfree field (MLS or RPIM), where the direct partition is not available, compare the penalty method with Nitsche’s method in step 6: both keep the size and the symmetry, but Nitsche’s parameter βN<\/sub> is a few orders of magnitude smaller, because its extra boundary terms make the method consistent instead of relying on a large number.<\/li>\n
  6. Choose plane stress and the divergence form: each cell contributes two rows, the equilibrium of forces in x and y, and the matrix is built from 2 × 2 blocks.<\/li>\n
  7. Choose the meshfree map with MLS. On the square the optimum order is linear and the map is exact (the patch is affine); on the L-shaped domain it is quadratic and again exact (the patch is bilinear); on the quarter annulus the quadratic map is several times more accurate than the linear one. Switch to RBF: it passes through the nodes, and a quadratic polynomial gains it little, so the optimum often stays linear.<\/li>\n
  8. Choose a Lagrange Q9 field with 2 × 2 Gauss points and Lagrange multipliers, then open the detailed derivation: Section 9 reports a singular system and explains the hourglass mode of reduced integration. Raise the Gauss points to 3 and the check passes.<\/li>\n
  9. Set every edge to a zero natural condition. The derivation reports a singular matrix: [K]{1} = {0} for heat conduction, three rigid-body modes for plane stress. The tests of this tool check exactly these identities for every pairing.<\/li>\n<\/ol>\n<\/div>\n<\/details>\n
    \nSources<\/summary>\n
    \n