OPTIMUM-DERAM: Adaptive Mesh & Distributed Memory

• The paper demonstrates a dual-method system where dynamic adaptive mesh refinement yields near-optimal topology designs with up to 50% computational cost reduction.
• It employs RLNC with consistent hashing and a blockchain oracle to ensure atomic, secure, and scalable shared memory operations in distributed systems.
• Combined, these methodologies offer strong theoretical guarantees and practical resilience under adversarial conditions and high-load scenarios.

OPTIMUM-DERAM denotes two distinct and highly specialized methodologies in academic research: (1) a topology optimization framework based on dynamic adaptive mesh refinement and derefinement in computational design (Wang et al., 2010), and (2) a distributed atomic memory protocol utilizing Random Linear Network Codes (RLNC), consistent hashing, and a blockchain oracle for scalable, secure, and consistent multi-object memory (Nicolaou et al., 19 Jan 2026). Both systems demonstrate rigorous algorithmic integration, strong theoretical guarantees, and significant empirical performance advances in their respective domains.

1. Dynamic Adaptive Mesh Refinement for Topology Optimization

OPTIMUM-DERAM (“OPTimization Using dyNamic DERefinements And Mesh‐refinements”) addresses the mesh resolution challenge in topology optimization, wherein a modest volume fraction of material renders uniform fine-mesh discretization computationally inefficient. The framework assigns a piecewise-constant density field $\rho(x) \in [\rho_0, 1]$ over $\Omega \subset \mathbb{R}^d$, targeting compliance minimization subject to a prescribed maximum solid volume $V^*$:

$\min_{\rho} c(\rho) = f^T u$

subject to

$$K(\rho) u = f, \quad \sum_e \rho_e V_e \leq V^*, \quad \rho_0 \leq \rho_e \leq 1 \ \forall e$$

with $K(\rho)$ assembled by a SIMP penalty law ($p \approx 3$).

The mesh adaptivity mechanism eschews global a posteriori error estimators, relying instead on a density-proximity indicator: for each element $e$,

$$\eta_e := \max \left\{ \rho_e,\, \max_{d:\|\mathbf{x}_d - \mathbf{x}_e\| \leq r} \rho_d \right\}$$

Elements are refined if $\eta_e > \rho_s$ (typically $\rho_s=0.5$), or derefined if $\eta_e < \rho_s$ and all neighbors within radius $r$ have $\rho < \rho_s$. AMR compatibility is strictly enforced: only level-one hanging node incompatibilities are permitted, with level-$\geq$2 triggering additional local mesh refinement.

2. Distributed, Atomic, and Secure Shared Memory using RLNC

OPTIMUM-DERAM in distributed systems is a decentralized, reconfigurable, atomic read/write shared memory protocol. Its key components are:

• Flexnodes: A dynamic set $I$ of servers each storing coded fragments.
• Clients: Issue read$(v)$ or write$(v)$ requests to any flexnode.
• Blockchain oracle (SMR): Maintains total order of node membership changes, supporting dynamic joins and departures.

Each object $o \in O$ is represented as a length-$k$ vector over a large finite field $\mathbb{F}_q$ and stored in encoded form on a cluster $C \subset I$. RLNC is central:

• Encoding: The value $v$ is split, and a random $(n \times k)$ matrix $A$ is used to generate fragments $c = A \cdot v$. Each flexnode stores $(\vec{\alpha}_i, c_i)$, with $\vec{\alpha}_i$ the $i$th row of $A$.
• Recoding: In-network mixing of coded pairs, generating new fragments as random linear combinations.
• Decoding: Any $k$ linearly independent fragments allow recovery of $v$ via matrix inversion.

Object placement and lookup leverage consistent hashing: both nodes and objects hash into a $2^{256}$-bit identifier space, enabling clusters $D_{\text{suc}}(o,S)$ for efficient, consistent, and scalable sharding.

3. Mesh and Membership Adaptivity Protocols

In computational optimization, the OPTIMUM-DERAM loop dynamically alternates between optimization and mesh updates:

Initialize mesh ℳ₀; set densities; repeat: 1. Assemble K(ρ), solve for u 2. Compute sensitivities, optionally filter 3. Update densities by OC/MMA 4. If convergence and mesh-update thresholds met, mark for (de)refinement using ηₑ, enforce mesh hierarchy project densities and displacement onto new mesh

Dynamic membership in distributed OPTIMUM-DERAM is coordinated via a blockchain oracle encoding join/leave events in a SMR. Joins and departures trigger data fetch, recoding, push, and consistency updates over quorums, leveraging the globally replicated and totally ordered SMR for consistency.

4. Consistency, Atomicity, and Byzantine Fault Tolerance

Atomic read/write operations in the distributed OPTIMUM-DERAM protocol utilize RLNC fragments and digital signatures. Protocols are built on quorums:

• For a cluster size $n$, code dimension $k$, and up to $b$ Byzantine nodes, quorums of size $q = \lceil (2n + k)/3 \rceil$ are constructed, ensuring any two intersect in $\geq b + k$ correct nodes.
• Read and write operations have two phases, guaranteeing that reads see the highest committed tag, and writes propagate with unique tags.

Byzantine resilience is achieved under the constraint $b < (n - k)/3$. All fragments and meta-information are signed; Byzantine nodes may only replay or withhold data, but cannot forge or inject new valid fragments. The protocol is shown to ensure atomicity, liveness, and consistency under asynchrony and adversarial network conditions.

5. Numerical and Empirical Performance Evaluation

In computational design, OPTIMUM-DERAM demonstrates strong empirical efficiency:

Study	Uniform Elements	Final Elements (AMR)	Rel. L¹ Diff.	CPU Time Ratio
2D Cantilever	65,536	~25,229	0.17%	≈0.33×
3D Cantilever	131,072	~57,173	0.091%	≈0.33×
3D Cross	40,960	19,736	2.58%	≈0.30×

Final designs match uniform-mesh optima within $1$–$3$ ‰ in compliance and density norms, requiring only $30$–$50\%$ of the computational cost.

For the distributed protocol, extensive geo-distributed testbed evaluations reveal:

• For $8$MB objects, read/write latency is ≈$25\%$ lower than MWABD ($n=5$); at $16$MB, up to $50\%$ lower.
• Latency remains flat as the number of supported objects scales to $1000$.
• Memory usage is nearly halved compared to full replication.
• Performance is invariant up to $52$ nodes, with stable behavior under heavy read/write concurrency.

6. Implementation and Best Practices

Computational topology optimization uses hierarchical meshes permitting only one level of hanging-node incompatibility, requiring solution projection across mesh refinements. Filtering must remain consistent across AMR steps; stencils are updated per mesh change. Stiffness matrices $K(\rho)$ are preconditioned via diagonal scaling and solved with iterative Krylov methods (MINRES/RMINRES).

Distributed OPTIMUM-DERAM mandates:

• Each fragment is signed and stored with accompanying tag data.
• Membership change protocols are atomic and strictly ordered.
• Proactive data transfer and recoding on node join/departure ensure state continuity.
• DHT routing tables and consistent hashing maintain object discovery and placement across network changes.
• Quorum construction and signature verification are enforced for all protocol steps.

7. Theoretical Guarantees and Practical Significance

OPTIMUM-DERAM in topology optimization guarantees convergence to uniform-mesh optima under proper mesh coverage, continuous adaptation, and filtering constraints. In distributed systems, proof sketches establish atomicity, liveness, and robustness against Byzantine faults, given quorum and code dimension requirements.

In both domains, OPTIMUM-DERAM exemplifies the integration of algorithmic rigor, scalability, and efficiency—demonstrating that properly designed adaptivity and coding/sharding mechanisms can deliver significant improvements over traditional static or replication-based models, with strong guarantees on optimality, consistency, and adversarial resilience (Wang et al., 2010, Nicolaou et al., 19 Jan 2026).