A high-order multi-scale method and its error analysis for nonlocal-in-time fractionally damped wave problems in composite materials

• DNS is computationally demanding because fine spatial meshes must resolve repeated material interfaces while the simulation accounts for temporal history.The demand arises from combining spatial heterogeneity with nonlocal-in-time damping.Source: §1 · p. 2
• Existing multiscale and homogenization approaches address composite materials, but HOMS research on composite wave problems with classical inertia and Caputo fractional damping remains limited.Prior work covers heterogeneous waves, temporal-memory models, and fractional time discretizations, but the paper identifies this combination as underexplored.Source: §1 · p. 3
• The central challenge is reconstructing microscopic wave fields and gradients from a homogenized solution while accounting for spatial heterogeneity, temporal memory, boundary layers, and cell-discretization errors.The fully discrete scheme additionally requires stability and convergence analysis.Source: §1 · p. 3
• Two- and three-dimensional examples demonstrate accurate reconstruction at reduced computational cost, while a large composite-plate example demonstrates feasibility when DNS exceeds available resources.The fully discrete method also has stated displacement and velocity error estimates under the paper's assumptions.Source: §1 · p. 4

2.1 Statement of the multi-scale problem

• The model combines classical inertia, Caputo fractional damping, and stiffness in a scalar displacement problem with periodic coefficients that may jump across material interfaces.The fractional order lies between 1 and 2, and the prescribed source represents distributed force density.Source: §2.1 · p. 5
• Under the stated coefficient, boundary, initial-data, and source assumptions, the governing problem has a unique weak solution.The proof uses a homogeneous-boundary lifting and an energy estimate.Source: §2.1 · pp. 5–7

2.2 High-order multi-scale asymptotic analysis

• The asymptotic expansion separates microscopic and macroscopic variables and represents the displacement through terms up to second order in ε.The leading-order field is independent of the microscopic variable.Source: §2.2 · p. 7
• Averaging the leading-order equation over the periodic cell gives the macroscopic homogenized wave problem with effective material parameters.The homogenized problem retains the prescribed boundary and initial data.Source: §2.2 · p. 8
• Cell functions restore stiffness-driven gradient and flux fluctuations, while second-order terms correct local inertia, fractional damping, and curvature effects.The HOMS solution adds these terms to the first-order LOMS approximation, canceling its leading residual and yielding a formal O(ε) local strong residual under stated regularity conditions.Source: §2.2 · p. 9

3.1 Error analysis of HOMS solutions in the pointwise sense

• The LOMS residual retains a formal O(1) strong source as ε tends to zero, so its local physical balance defect does not vanish.The analysis identifies this leading residual as the defect addressed by the second-order HOMS corrections.Source: §3.1 · p. 10
• O(ε) pointwise error is achieved by second-order HOMS corrections, which cancel the leading local balance defect remaining in LOMS.The pointwise analysis assumes sufficient homogenized-solution regularity, including the stated endpoint conditions.Source: §3.1 · pp. 9–10
• For α>1, the t^-α endpoint factor is not integrable near zero, while the remaining t^(1-α) term is in L2 only for α<3/2.When α<3/2, the condition u0,ttt(0)=0 may be relaxed.Source: §3.1 · p. 11

3.2 Error analysis of HOMS solution in the integral sense

• O(ε) global energy-norm error follows under complete periodic-cell, coefficient-symmetry, and homogeneous Dirichlet cell-function assumptions.The estimate uses the stated regularity assumptions and constants independent of ε.Source: §3.1 · p. 9, §3.2 · pp. 11, 13

4.1 Computation for microscopic cell functions and homogenized material parameters

• First- and second-order cell functions are computed on interface-fitted meshes with homogeneous Dirichlet conditions and used to evaluate homogenized material parameters.The cell mesh is triangular in two dimensions or tetrahedral in three dimensions.Source: §4.1 · pp. 13–14

4.3 Computation and reconstruction for the HOMS solution

• The HOMS reconstruction uses the computed macroscopic displacement and shifted cell fields to form the corrected solution at each time step.The procedure solves for the macroscopic displacement, recovers velocity and acceleration, and evaluates the shifted fields and Q term.Source: §4.3 · p. 16

5.1 Shifted-energy stability for the fully discrete scheme at macro-scale

• The fully discrete macroscopic scheme has unconditional shifted-energy stability; coercivity gives a unique solution at each time step.The shifted-energy argument includes a nonnegative energy term for zero initial data.Source: §5 · p. 16, §5.1 · p. 17

5.2 Error analysis for microscopic cell functions and macroscopic homogenized material parameters

• The macroscopic homogenized material parameter is estimated from the cell-function discretization and computed using finite element approximations of first-order auxiliary cell functions.The parameter estimate follows from the cell-function analysis.Source: §5.2 · p. 20, §5.3 · p. 20

5.3 Error analysis for fully discrete method at the macro-scale

• A total macroscopic displacement-and-velocity error bound combines coefficient-perturbation and fully discrete errors, with C_T independent of hY, h, and τ under uniform bounds.The analysis combines the coefficient-perturbation estimate with the fully discrete error estimate.Source: §5.3 · p. 24

6 Numerical examples and results

• HOMS achieved single-run speedups over DNS of approximately 9.51, 5.32, 2.53, and 4.94 in Examples 1–4, respectively.Across those examples, HOMS also had lower relative errors over most displayed time intervals or after the initial peak.Source: §6.1.1 · p. 26, §6.1.2 · p. 28, §6.1.3 · p. 31

6.1.1 Example 1: The nonlocal-in-time fractional wave problem in 2D heterogeneous structures with circular inclusions

• HOMS maintained the lowest relative errors over most of the displayed time interval and showed no numerical blow-up through t = 2.0 s, supporting numerical robustness over that simulated interval.The HOMS computation achieved a single-run speedup of approximately 9.51 over DNS.Source: §6.1.1 · p. 26

6.1.2 Example 2: The nonlocal-in-time fractional wave problem in 2D heterogeneous structures with diamond-shaped inclusions

• In the diamond-inclusion composite, HOMS had the lowest displacement and gradient errors over most of the displayed interval, while DNS used approximately 56.98 times as many elements as the homogenized mesh.The HOMS computation achieved a single-run speedup of approximately 5.32 over DNS; absolute errors concentrated near diamond-shaped interfaces and decreased with expansion order.Source: Figure 3 · p. 27, §6.1.2 · p. 28, Figure 4 · p. 28

6.1.3 Example 3: The nonlocal-in-time fractional wave problem in 2D heterogeneous structures with layered inclusions

• For layered inclusions, the second-order correction reduced displacement errors, and HOMS better reproduced layer-dependent DNS variations than lower-order approximations.DNS used 256 times as many elements as the homogenized mesh, and HOMS achieved a single-run speedup of approximately 2.53.Source: Figure 7 · p. 29, §6.1.3 · pp. 30–31, Figure 8 · p. 30

6.1.4 Example 4: The nonlocal-in-time fractional wave problem in 3D heterogeneous structures with cubic inclusions

• After the initial peak, HOMS had the smallest relative L2-errors and relative H1-seminorm errors; the DNS mesh contained approximately 110.59 times as many elements as the homogenized mesh.The HOMS computation achieved a single-run speedup of approximately 4.94 over DNS.Source: §6.1.4 · p. 31

6 Numerical examples and results

• Across Examples 1–4, homogenization captured the smooth macroscopic response, LOMS recovered some microscale variation, and HOMS more closely reproduced DNS displacement fields.Example 4 used mixed boundary conditions, while the stated error estimates concern Dirichlet conditions.Source: §6 · p. 24, §6.1.4 · p. 31

6.2.1 Example 5: Simulation of a nonlocal-in-time fractional wave problem in a 3D heterogeneous composite plate with a large number of cubic inclusions

• For the large 3D composite plate, DNS mesh requirements exceeded available resources, so no converged reference solution was obtained.The example therefore reports estimated DNS mesh sizes alongside cell and homogenized mesh sizes.Source: Table 2 · p. 25, §6.2.1 · p. 33

7 Conclusions

• Example 4 displays displacement fields, absolute displacement errors, relative-error evolution, and displacement fields on vertical sections, including DNS.Source: Figure 15 · p. 34, Figure 14 · p. 34, Figure 16 · p. 35
• The HOMS solution combines the homogenized equation with first- and second-order auxiliary cell problems, cancels the LOMS leading residual, preserves local physical balance, and converges at O(ε) in the energy norm.Source: §7 · pp. 33, 35, 36
• Reported DNS speedups range from 2.53 to 9.51, while HOMS relative errors are smaller than homogenized and LOMS solutions over most of the displayed interval.The examples also demonstrate computational feasibility and robustness for composite structures with massive unit cells.Source: §7 · p. 36
• The error estimates rely on high temporal regularity; weaker-regularity estimates are identified as future work.Source: §7 · p. 36

Declarations

• The listed authors contributed to conceptualization, methodology, formal analysis, writing, and other roles, with responsibilities varying by author.Source: Declarations · p. 36
• The research lists support from national, Shaanxi, Xidian University, and central-university funding programs.Source: Declarations · pp. 36–37
• Study data are available from the corresponding author on reasonable request.Source: Declarations · p. 37
• The authors declare no conflicts of interest.Source: Declarations · p. 38