Chong Peng

A boundary-condition closure is presented for renormalised, least-squares corrected meshless Laplacian operators in strong-form collocation on point clouds.Unlike the dominant particle-based treatments, the closure does not pad the domain with auxiliary particles, represent the boundary with a level set, or enforce the conditions weakly.A single code path folds Neumann, Dirichlet, and Robin conditions directly into the discrete collocation row, organised by condition type, so that a prescribed flux enters the right-hand side while a value-dependent condition contributes to the diagonal.The Neumann flux is imposed through a tangential projection in physical space before renormalisation, and a general Gram projector treats non-orthogonal corners exactly.A complementary algebraic-ghost closure completes the truncated boundary stencil by reflecting the local support across the nearby boundary planes, requiring no extra unknowns.The free surface is detected from the support-deficiency geometry of the cloud, and its enforcement strength reduces exactly to the total kernel weight, so no penalty coefficients or case-dependent thresholds are introduced.The resulting method, named FIRM (Flux-Integrated Renormalised Meshless), is verified in two dimensions against analytical solutions and irregular-domain Poisson benchmarks, on regular and highly irregular point clouds, alongside a second-order generalised finite difference baseline.The value closure exhibits supraconvergence, the algebraic-ghost closure reduces the Neumann-limited error by about an order of magnitude, and with the sum normalisation the method approaches second-order convergence at straight boundaries.The renormalised operators are also consistent for a single Helmholtz--Hodge decomposition, whose residual divergence equals the composition defect of the operators and vanishes under refinement.

A defining property of Lagrangian and semi-Lagrangian methods is that their stability is not governed by the Courant--Friedrichs--Lewy (CFL) condition. The computational stencil is carried with the flow, and uniform translation cannot corrupt the discrete operators. Numerous contemporary studies of these methods prescribe time-step limits based on the classical CFL number, i.e. on the absolute velocity and discretisation size. This paper first clarifies the physical meaning of the time-step criteria relevant to incompressible Lagrangian advection, where the constraining mechanisms are the strain of flow and relative motion of neighbouring computational points. It proposes a new geometric criterion, which bounds the exact finite-step change of every neighbour distance. It is the finite-step counterpart of Monaghan's collision constraint, enhanced for transverse relative motion and integrator-matched higher-order forms (up to an exact streamline form) that remove the spurious limits of straight-line displacements on rotational motion. Numerical experiments on analytic flows show stability at equivalent classical CFL numbers between 6 and 38, while strain-based Lagrangian measures remain of order unity. Driving the adaptive time step of an incompressible Lagrangian flow on violent free-surface dam break validates that the geometric criterion matches the incompressibility control of the strain-based bound while taking 3.3 times larger steps.

This research addresses and resolves current challenges in meshless Lagrangian methods for simulating viscoelastic materials. A split-step scheme, or pressure Poisson reformulation of the Navier–Stokes equations, is introduced for incompressible viscoelastic flows in a Lagrangian context. The Lagrangian differencing dynamics (LDD) method, which is a thoroughly validated Lagrangian method for Newtonian and non-Newtonian incompressible flows, is extended to solve the introduced split-step scheme to simulate viscoelastic flows based on the Oldroyd-B constitutive model. To validate and evaluate the new method’s capabilities, the following benchmarks were used: lid-driven cavity flow, droplet impact response, 4:1 planar sudden contraction, and die swelling. These findings highlight the LDD method’s effectiveness in accurately simulating viscoelastic flows and capturing large deformations and memory effects. Even though the extra stress was directly modeled without any regularization approach, the method produced stable simulations for high Weissenberg numbers. The stability and performance of the the Lagrangian numerics for complex temporal evolution of material properties and stress responses encourage its use for industrial problems dealing with polymers.

This study investigates the impact of fluid loads on the elastic deformation and dynamic response of linear structures. A weakly coupled modal solver is presented, which involves the solution of a dynamic equation of motion with external loads. The mode superposition method is used to find the dynamic response, utilizing predetermined mode shapes and natural frequencies associated with the structure. These essential parameters are pre-calculated and provided as input for the simulation. Integration of the weakly coupled modal solver is accomplished with the Lagrangian Differencing Dynamics (LDD) method. This method can directly use surface mesh as boundary conditions, so it is much more convenient than other meshless CFD methods. It employs Lagrangian finite differences, utilizing a strong formulation of the Navier–Stokes equations to model an incompressible free-surface flow. The elastic deformation of the structure, induced by fluid forces obtained from the flow solver, is computed within the modal coupling algorithm through direct numerical integration. Subsequently, this deformation is introduced into the flow solver to account for changes in geometry, resulting in updated flow pressure and velocity fields. The flow particles and vertices of the structure are advected in Lagrangian coordinates, resulting in Lagrangian–Lagrangian coupling in spaces with weak or explicit coupling in time. The two-way coupling between fluid and structure is successfully validated through various FSI benchmark cases. The efficiency of the LDD method is highlighted as it operates directly on surface meshes, streamlining the simulation setup. Direct coupling of structural deformation eliminates the conventional step of mapping fluid results onto the structural mesh and vice versa.

This paper proposes a regularized smoothed particle hydrodynamics (SPH) scheme to solve the problems of tensile instability and stress noise in large‐deformation soil–structure interaction. Particle shifting and stress regularization are combined to maintain uniform stresses and particle distributions. A generalized frictional boundary condition is presented to model soil–structure interfaces with abitrary friction coefficients. A modified particle shifting technique is developed to regularize particles near the frictional boundaries. Furthermore, GPU acceleration algorithm is adopted to improve the efficiency. The proposed scheme is compared with the conventional SPH methods using numerical examples, including the collapse of three‐dimensional granular column, the collapse of the cohesive soil, and pipe penetrates into soft clay. The comparison indicates that the proposed regularized SPH model can give highly accurate results for large‐deformation soil–structure interaction problems with arbitrary friction coefficient.