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From Physics to Surrogate Intelligence: A Unified Electro-Thermo-Optimization Framework for TSV Networks

Mohamed Gharib, Leonid Popryho, Inna Partin-Vaisband

Abstract

High-density through-substrate vias (TSVs) enable 2.5D/3D heterogeneous integration but introduce significant signal-integrity and thermal-reliability challenges due to electrical coupling, insertion loss, and self-heating. Conventional full-wave finite-element method (FEM) simulations provide high accuracy but become computationally prohibitive for large design-space exploration. This work presents a scalable electro-thermal modeling and optimization framework that combines physics-informed analytical modeling, graph neural network (GNN) surrogates, and full-wave sign-off validation. A multi-conductor analytical model computes broadband S-parameters and effective anisotropic thermal conductivities of TSV arrays, achieving $5\%-10\%$ relative Frobenius error (RFE) across array sizes up to $15x15$. A physics-informed GNN surrogate (TSV-PhGNN), trained on analytical data and fine-tuned with HFSS simulations, generalizes to larger arrays with RFE below $2\%$ and nearly constant variance. The surrogate is integrated into a multi-objective Pareto optimization framework targeting reflection coefficient, insertion loss, worst-case crosstalk (NEXT/FEXT), and effective thermal conductivity. Millions of TSV configurations can be explored within minutes, enabling exhaustive layout and geometric optimization that would be infeasible using FEM alone. Final designs are validated with Ansys HFSS and Mechanical, showing strong agreement. The proposed framework enables rapid electro-thermal co-design of TSV arrays while reducing per-design evaluation time by more than six orders of magnitude.

From Physics to Surrogate Intelligence: A Unified Electro-Thermo-Optimization Framework for TSV Networks

Abstract

High-density through-substrate vias (TSVs) enable 2.5D/3D heterogeneous integration but introduce significant signal-integrity and thermal-reliability challenges due to electrical coupling, insertion loss, and self-heating. Conventional full-wave finite-element method (FEM) simulations provide high accuracy but become computationally prohibitive for large design-space exploration. This work presents a scalable electro-thermal modeling and optimization framework that combines physics-informed analytical modeling, graph neural network (GNN) surrogates, and full-wave sign-off validation. A multi-conductor analytical model computes broadband S-parameters and effective anisotropic thermal conductivities of TSV arrays, achieving relative Frobenius error (RFE) across array sizes up to . A physics-informed GNN surrogate (TSV-PhGNN), trained on analytical data and fine-tuned with HFSS simulations, generalizes to larger arrays with RFE below and nearly constant variance. The surrogate is integrated into a multi-objective Pareto optimization framework targeting reflection coefficient, insertion loss, worst-case crosstalk (NEXT/FEXT), and effective thermal conductivity. Millions of TSV configurations can be explored within minutes, enabling exhaustive layout and geometric optimization that would be infeasible using FEM alone. Final designs are validated with Ansys HFSS and Mechanical, showing strong agreement. The proposed framework enables rapid electro-thermal co-design of TSV arrays while reducing per-design evaluation time by more than six orders of magnitude.

Paper Structure

This paper contains 28 sections, 26 equations, 13 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Electrothermal Analytical Model
  3. Electromagnetic Framework
  4. Electrothermal Framework Using Equivalent Thermal Conductivity
  5. Equivalent Thermal Conductivity Formulation
  6. Equivalent Volumetric Heat Capacity
  7. Array-Level Anisotropic Conductivities
  8. Heat Source Extraction from S-Parameters
  9. Steady-State Thermal Solution Using Ansys Mechanical
  10. ML-Assisted Optimization Framework
  11. Analytical Dataset Generation
  12. Graph Representation
  13. Node Representation
  14. Edge Representation
  15. Physics-Informed Graph Neural Network
  16. ...and 13 more sections

Figures (13)

  • Figure 1: Comparison of representative TSV modeling and optimization frameworks in terms of scalability, frequency range, and computational cost.
  • Figure 2: Analytical simulation flow, where a Python-based netlist generator creates an RLCG SPICE representation of the TSV array, which is simulated using PrimeSim HSPICE to extract wideband S-parameters.
  • Figure 3: Flowchart of the coupled electrothermal simulation framework. Electrical losses are extracted from S-parameters and mapped to thermal heat sources. The resulting temperature distribution is used to update temperature-dependent electrical parameters until convergence.
  • Figure 4: Extraction of Joule heating from S-parameters and mapping to volumetric heat sources.
  • Figure 5: Graph representation of a $3\times4$ TSV array. Red nodes represent Signal TSVs, and blue nodes represent Ground TSVs. Edges carry spatial distance features.
  • ...and 8 more figures