LPS-H: A 99-Line Homogenization Code for Lattice-skin Plate Structures

Abstract

Recent years have seen growing application potential for Lattice-skin Plate Structures in advanced manufacturing fields such as aerospace and automotive engineering. For multiscale performance evaluation of such structures, conventional homogenization methods for lattice-filled volume structures are often used for equivalent analysis. However, in finite-thickness Lattice-skin Plate Structures, periodic boundary conditions imposed along the three orthogonal directions of the representative cell cannot adequately capture the boundary effect of the free surfaces in the thickness direction, which introduces bias into the prediction of effective properties.

To reduce this bias, this study develops and open-sources a homogenization method for Lattice-skin Plate Structures, forming an open-source computational framework for this class of structures. Representative numerical examples show that the framework can stably extract effective plate/shell stiffness matrices and can be extended to predict multiphase material properties and analyze steady-state heat conduction. The tool provides an open and reusable analysis foundation for the high-fidelity design of multifunctional lightweight structures.

Framework

The companion page follows the manuscript and uses the original figures from the paper. The central idea is to replace direct three-dimensional simulation of a finite-thickness lattice-skin structure with an equivalent plate or shell model while preserving the physically meaningful membrane, bending, and coupling responses captured by the ABD stiffness matrix.

Equivalent plate shell characterization and ABD matrix — Figure 2. Equivalent plate/shell mechanical characterization of Lattice-skin Plate Structures and schematic illustration of the ABD stiffness matrix. (a) Dimensional reduction from a Lattice-skin Plate Structure to an equivalent homogeneous plate/shell. (b) Composition and definition of the ABD stiffness matrix. (c) Six basic loading forms used to extract the effective stiffness components. (d-f) Physical meanings of the stiffness components in the A, B, and D matrices.

Plate and shell reduction

The method targets finite-thickness lattice-skin structures whose mechanical behavior is more faithfully described by equivalent plate and shell kinematics than by volume-only periodic homogenization.

Loop-free tensor mapping

The implementation relies on multidimensional tensor slicing to construct local-to-global mappings efficiently and keep the core homogenization routine compact.

Reusable computation pipeline

The same framework supports stiffness extraction, multiphase material settings, and steady-state heat conduction analysis in a consistent open-source workflow.

Loop-free assembly logic for edofMat — Figure 3. Schematic of the loop-free assembly logic for the local-to-global DOF mapping tensor `edofMat` based on multidimensional tensor slicing.

Code

The manuscript provides a compact 99-line implementation. The following excerpt highlights the stress recovery and ABD extraction routine, where np.einsum is used to avoid explicit loops over tensor contractions.

lps_homogenization.py View Full Source

# Core computation: microscopic stress recovery and effective ABD extraction
ABD = np.zeros((6, 6))

# Accumulate over the eight hexahedral Gauss points
for i_gp in range(8):
    # Physical stress recovery: Sigma = C * (E_macro - B * U)
    Sigma = np.einsum('ij,kjl->kil', C, E_macro -
            np.einsum('ij,kjl->kil', Bs[i_gp], U[edofMat, :]))

    # Zeroth-order in-plane integration for the A and B matrices
    ABD[0:3, :] += np.sum(Sigma[:, [0, 1, 5], :], axis=0) * detJ / plate_area

    # First-order integration through the thickness for the D matrix
    ABD[3:6, :] += np.sum(
        Sigma[:, [0, 1, 5], :] * z_active[:, None, None],
        axis=0
    ) * detJ / plate_area

# Symmetrization to satisfy Maxwell-Betti reciprocity
ABD = (ABD + ABD.T) / 2.0
return ABD

Results

The paper validates the framework with representative thin-walled metamaterial cases, including TPMS and BCC sandwich structures, thickness size effects, multimaterial settings, and static-response comparisons against full-scale finite element models.

Four analysis cases — Figure 4. Schematic illustration of the four analysis cases. (a) TPMS sandwich structure without skins. (b) TPMS sandwich structure with skins. (c) BCC sandwich structure without skins. (d) BCC sandwich structure with skins.

Figure 5. Mesh convergence analysis of the sheet-type Primitive structure. (a) Evolution of the in-plane membrane stiffness A11 with respect to the resolution N. (b) Evolution of the bending stiffness D11 with respect to the resolution N.

Comparison between LVS-H and LPS-H on sheet-type TPMS structures — Figure 6. Heat-map comparison of effective ABD stiffness predictions from LVS-H and LPS-H for sheet-type TPMS structures, together with the corresponding relative-error distributions.

Thickness size-effect analysis of the Gyroid structure — Figure 7. Thickness size-effect analysis of the sheet-type Gyroid structure. (a) Schematic of the geometric evolution as the number of layers through the thickness, Nz, increases. (b) Convergence curves of the normalized in-plane membrane stiffness Aij versus Nz. (c) Convergence curves of the normalized bending stiffness Dij versus Nz.

Bimaterial laminated metamaterial plate shell — Figure 8. Schematic of a bimaterial laminated metamaterial plate/shell.

Static response comparison of full-scale model, LPS-H, and LVS-H — Figure 9. Comparison of the static responses predicted by the full-scale finite element model, LPS-H, and LVS-H. All three models use the same boundary conditions, with the upper boundary fixed and a 1 mm displacement load applied to the lower boundary, and the support reaction is used to characterize the overall stiffness.

Citation

If this code or the homogenization framework is useful in your research, please consider citing the paper.

@article{ji202699line, title={A 99-Line Homogenization Code for Lattice-skin Plate Structures}, author={Ji, Zhongkai and Li, Dawei and Zhao, Yong and Liao, Wenhe}, journal={arXiv preprint arXiv:2604.23181}, year={2026}, url={https://arxiv.org/abs/2604.23181} }