Search Results (70)

Multibody dynamics (MBD) simulations involving hundreds to thousands of bodies give rise to large-scale, sparse, and structurally indefinite linear systems. Traditional direct solvers incur prohibitive memory and computational costs, while iterative methods suffer from slow convergence due to severe ill-conditioning. This paper proposes HPI-MBD, a hybrid preconditioned iterative framework. It combines an algebraic multigrid (AMG) for global error smoothing with a block Jacobi preconditioner tailored to the kinematic constraint graph. The framework exploits graph topology to construct a block-diagonal Schur complement approximation, incorporates Tikhonov regularisation for redundant constraints, and maintains $\mathcal{O}\left(n\right)$ work per iteration, where n is the number of degrees of freedom. A rigorous spectral analysis supports the problem-size independent convergence of the Minimal Residual (MINRES) solver. Evaluated on five benchmark systems with ${10}^4$ to ${10}^6$ degrees of freedom, the HPI-MBD achieves speedups up to $12.7\times$ and memory reductions up to 68% against MA57, with comparable gains against PARDISO. All solutions maintain relative residuals below ${10}^{-6}$. Comparisons against ILU(0)-preconditioned Generalised Minimal Residual (GMRES), Finite Element Tearing and Interconnecting method (FETI-1), and a block-Jacobi-only variant confirm the essential role of AMG. The framework exhibits near-linear scalability and strong parallel efficiency on up to 32 processors, along with robust performance under redundant constraints and varying time step sizes. These results position HPI-MBD as a scalable, memory-efficient alternative for real-time simulation in virtual prototyping, robotics, and biomechanics. Full article

This article reviews the linear solvers available in OpenFOAM and assesses their impact on the convergence behaviour of the SIMPLE algorithm. The discretisation of transport equations in CFD results in large and sparse linear systems, for which the choice of linear solver strongly influences the computational time. Although the solver does not change the final discrete solution, the difference in speed and robustness between the solvers can be more than one order of magnitude. A brief overview is given concerning how the velocity and pressure fields are decoupled in OpenFOAM, followed by a detailed review of the main linear solver families, including direct methods, basic iterative methods, multigrid methods and Krylov subspace methods, with attention to their practical strengths and weaknesses. The performance of the most advanced solvers is evaluated on a full-scale non-reacting kiln case consisting of 2.3 million cells. The pressure-corrector equation is identified as the main bottleneck in the SIMPLE algorithm. The conjugate gradient (CG) solver with a multigrid (MG) preconditioner is found to be the fastest and most stable method, achieving speed-ups of up to a factor of 7 compared to the slower advanced methods. Using MG as a preconditioner also improves the robustness of the Bi-CGStab method. Full article

This paper is concerned with the numerical solution of the variable-order time fractional advection–diffusion–reaction equation (VO-TFADRE) in two space dimensions. We first propose a Crank–Nicolson (C-N) discretization scheme based on central difference operators and L1 formula for space and time variables, respectively. Then, we apply the C-N scheme to construct a new algorithm, namely the explicit group (EG) method, for the model problem under consideration. The EG method utilizes the idea of small fixed-size groups of mesh points and comes with computational merits as compared with the C-N scheme. Stability and convergence analyses are given in this work. The resulting discretization leads to large sparse linear systems, which are solved using the Bi-CGSTAB iterative method. Numerical experiments demonstrate that both the C–N and EG schemes achieve accurate approximations, while the EG method significantly reduces computational time. To economize further on the computational cost, we propose a parallelized version of the EG method for solving the VO-TFADRE. Carried out numerical simulations reveal that the parallel algorithm is more efficient than the serial algorithm for solving the problem under consideration. Full article

In order to overcome the computational challenges associated with block preconditioners for Krylov subspace methods, particularly those arising from Schur complement systems, this paper proposes an improved new block (INB) preconditioner for solving 3 × 3 block saddle point problems. A detailed semi-convergence analysis of the iterative scheme induced by the INB preconditioner is provided. Moreover, the spectral properties of the preconditioned matrix are analyzed, revealing strong eigenvalue clustering around one. Efficient formulas for selecting quasi-optimal parameters are derived based on Frobenius-norm minimization. Extensive numerical experiments demonstrate that the proposed INB preconditioner significantly reduces iteration counts and CPU time compared with several existing block preconditioners. Full article

The increasing complexity of modern Very Large-Scale Integration (VLSI) circuits, combined with unavoidable variations in physical and manufacturing parameters, poses significant challenges for accurate and efficient circuit simulation. Parametric model order reduction (PMOR) provides a viable solution by enabling the construction of compact reduced-order models that remain valid across a prescribed parameter space. However, the computational cost of generating such models can become prohibitive for large-scale circuits, particularly when high-fidelity projection subspaces are required. In this work, we present an efficient PMOR framework based on the Asymmetric Extended Krylov Subspace (AEKS). The proposed approach exploits structural sparsity imbalances between system matrices to guide the subspace expansion toward computationally favorable directions, thereby significantly reducing the cost of repeated linear system solves. By integrating AEKS within a concatenation-of-basis PMOR strategy, this method enables the rapid construction of accurate parametric reduced-order models for large-scale circuit systems. The proposed AEKS-PMOR framework is evaluated on industrial power distribution network benchmarks, where it demonstrates substantial reductions in model construction time compared to conventional EKS-based PMOR, while maintaining high approximation accuracy over the entire parameter space. Full article

In this work, we proposed a dynamic inverse solution with spatio-temporal constraints of the nonlinear heat diffusion problem in 1D and 2D based on a regularized Gauss–Newton and Krylov subspace with a preconditioner. The preconditioner is computed by approximating the Jacobian of the nonlinear system at each Gauss–Newton iteration. The proposed approach is used for estimation of the initial value from measurements of the last value by considering spatial and spatio-temporal constraints. The system is compared to a dynamic Tikhonov inverse solution and generalized minimal residual method (GMRES) with and without a preconditioner. The system is evaluated under noise conditions in order to verify the robustness of the proposed approach. It can be seen that the proposed spatio-temporal regularized Gauss–Newton method with GMRES and a preconditioner shows better estimation results than the other methods for both spatial and spatio-temporal constraints. Full article

In this work, we propose a nonlinear dynamic inverse solution to the diffusion problem based on Krylov Subspace Methods with spatiotemporal constraints. The proposed approach is applied by considering, as a forward problem, a 1D diffusion problem with a nonlinear diffusion model. The dynamic inverse problem solution is obtained by considering a cost function with spatiotemporal constraints, where the Krylov subspace method named the Generalized Minimal Residual method is applied by considering a linearized diffusion model and spatiotemporal constraints. In addition, a Jacobian-based preconditioner is used to improve the convergence of the inverse solution. The proposed approach is evaluated under noise conditions by considering the reconstruction error and the relative residual error. It can be seen that the performance of the proposed approach is better when used with the preconditioner for the nonlinear diffusion model under noise conditions in comparison with the system without the preconditioner. Full article

We present an advanced restrictively preconditioned biconjugate gradient-stabilized (RPBiCGSTAB) algorithm specifically designed to improve the convergence speed of Krylov subspace methods for nonlinear systems characterized by a structured 5-by-5 block configuration. This configuration frequently arises from cell-centered finite difference discretizations employed in solving image deblurring problems governed by mean curvature dynamics. The RPBiCGSTAB method is crafted to exploit this block structure, thereby optimizing both computational efficiency and convergence behavior in complex image processing tasks. Analyzing the spectral characteristics of preconditioned matrices often reveals a beneficial distribution of eigenvalues, which plays a critical role in accelerating the convergence of the RPBiCGSTAB algorithm. Furthermore, our numerical experiments validate the computational efficiency and practical applicability of the method in addressing nonlinear systems commonly encountered in image deblurring. Our analysis also extends to the spectral properties of the preconditioned matrices, noting a pronounced clustering of eigenvalues around 1, which contributes to enhanced stability and convergence performance.Through numerical simulations that focus on mean curvature-driven image deblurring, we highlight the superior performance of the RPBiCGSTAB method in comparison to other techniques in this specialized field. Full article

Due to the coupling of DC and AC components, the ion flow field of HVDC and HVAC transmission lines in the same corridor or even the same tower is complex and time-dependent. In order to effectively analyze the ground-level electric field of hybrid transmission lines, the Krylov subspace methods with pre-conditioning treatment are used to solve the discretization equations. By optimizing the coefficient matrix, the calculation efficiency of the iterative process of the electric field in the time domain is greatly increased. Based on the limit of electric field, radio interference and audible noise applied in China, the main factor influencing the design of hybrid transmission lines is determined in terms of electromagnetic environment. After the ground-level electric field of transmission lines with different configurations is analyzed, the minimum height and corridor width of double-circuit 500 kV HVAC lines and one-circuit ±800 kV HVDC lines in the same corridor are obtained. The research provides valuable practical recommendations for optimal tower configurations, minimum heights, and corridor widths under various electromagnetic constraints. Full article

As semiconductor technologies continue to scale aggressively, electromigration (EM) has become critical in modern VLSI design. Since traditional EM assessment methods fail to accurately capture the complex behavior of multi-segment interconnects, recent physics-based models have been developed to provide a more accurate representation of EM-induced stress evolution. However, numerical methods for these models result in large-scale systems, which are computationally expensive and impractical for complex interconnect structures. Model order reduction (MOR) has emerged as a key enabler for scalable EM analysis, with moment-matching (MM) techniques offering a favorable balance between efficiency and accuracy. However, conventional Krylov-based approaches often suffer from limited frequency resolution or high computational cost. Although the extended Krylov subspace (EKS) improves frequency coverage, its symmetric structure introduces significant overhead in large-scale scenarios. This work introduces a novel MOR technique based on the asymmetric extended Krylov subspace (AEKS), which improves upon the conventional EKS by incorporating a sparsity-aware and computationally efficient projection strategy. The proposed AEKS-based moment-matching framework dynamically adapts the Krylov subspace construction according to matrix sparsity, significantly reducing runtime without sacrificing accuracy. Experimental evaluation on IBM power grid benchmarks demonstrates the high accuracy of our method in both frequency-domain and transient EM simulations. The proposed approach delivers substantial runtime improvements of up to 15× over full-order simulations and 100× over COMSOL, while maintaining relative errors below 0.5%, even under time-varying current inputs. Full article

Model order reduction (MOR) is crucial for efficiently simulating large-scale RLCk models extracted from modern integrated circuits. Among MOR methods, balanced truncation offers strong theoretical error bounds but is computationally intensive and does not preserve passivity. In contrast, moment matching (MM) techniques are widely adopted in industrial tools due to their computational efficiency and ability to preserve passivity in RLCk models. Typically, MM approaches based on the rational Krylov subspace (RKS) are employed to produce reduced-order models (ROMs). However, the quality of the reduction is influenced by the selection of the number of moments and expansion points, which can be challenging to determine. This underlines the need for advanced strategies and reliable convergence criteria to adaptively control the reduction process and ensure accurate ROMs. This article introduces a frequency-aware multi-point MM (MPMM) method that adaptively constructs an RKS by closely monitoring the ROM transfer function. The proposed approach features automatic expansion point selection, local and global convergence criteria, and efficient implementation techniques. Compared to an established MM technique, MPMM achieves up to 16.3× smaller ROMs for the same accuracy, over 99.18% reduction in large-scale benchmarks, and up to 4× faster runtime. These advantages establish MPMM as a strong candidate for integration into industrial parasitic extraction tools. Full article

The most popular iterative methods for solving nonsymmetric linear systems are Krylov methods. Recently, an optimal Quasi-ORthogonal (Q-OR) method was introduced, which yields the same residual norms as the Generalized Minimum Residual (GMRES) method, provided GMRES is not stagnating. In this paper, we study how to introduce matrix sketching in this algorithm. It allows us to reduce the dimension of the problem in one of the main steps of the algorithm. Full article

Tempered fractional diffusion equations constitute a critical class of partial differential equations with broad applications across multiple physical domains. In this paper, the Crank–Nicolson method and the tempered weighted and shifted Grünwald formula are used to discretize the tempered fractional diffusion equations. The discretized system has the structure of the sum of the identity matrix and a diagonal matrix multiplied by a symmetric positive definite (SPD) Toeplitz matrix. For the discretized system, we propose a structure approximation-based preconditioning method. The structure approximation lies in two aspects: the inverse approximation based on the row-by-row strategy and the SPD Toeplitz approximation by the $\tau$ matrix. The proposed preconditioning method can be efficiently implemented using the discrete sine transform (DST). In spectral analysis, it is found that the eigenvalues of the preconditioned coefficient matrix are clustered around 1, ensuring fast convergence of Krylov subspace methods with the new preconditioner. Numerical experiments demonstrate the effectiveness of the proposed preconditioner. Full article

For topology optimization problems under harmonic excitation in a frequency band, a large number of displacement and adjoint displacement vectors for different frequencies need to be computed. This leads to an unbearable computational cost, especially for large-scale problems. An effective approach, the Second-Order Arnoldi (SOAR) method, effectively solves the response and adjoint equations by projecting the original model to a reduced order model. The SOAR method generalizes the well-known Krylov subspace in a specified frequency point and can give accurate solutions for the frequencies near the specified point by using only a few basis vectors. However, for a wide frequency band, more expansion points are needed to obtain the required accuracy. This brings up the question of how many points are needed for an arbitrary frequency band. The traditional reduced order method improves the accuracy by uniformly increasing the expansion points. However, this leads to the redundancy of expansion points, as some frequency bands require more expansion points while others only need a few. In this paper, a bisection-based adaptive SOAR method (ASOAR), in which the points are added adaptively based on a local error estimation function, is developed to solve this problem. In this way, the optimal number and position of expansion points are adaptively determined, which avoids the insufficient efficiency or accuracy caused by too many or too few points in the traditional strategy where the expansion points are uniformly distributed. Compared to the SOAR, the ASOAR can deal with wide low/mid-frequency bands both for response and adjoint equations with high precision and efficiency. Numerical examples show the validation and effectiveness of the proposed method. Full article

After nearly two decades of development, transient electromagnetic (TEM) 3D forward modeling technology has significantly improved both numerical precision and computational efficiency, primarily through advancements in mesh generation and the optimization of linear equation solvers. However, the dominant approach still relies on direct solvers, which require substantial memory and complicate the modeling of electromagnetic responses in large-scale models. This paper proposes a new method for solving large-scale TEM responses, building on previous studies. The TEM response is expressed as a matrix exponential function with an analytic initial field for a step-off source, which can be efficiently solved using the Shift-and-Invert Krylov (SAI-Krylov) subspace method. The Arnoldi algorithm is used to construct the orthogonal basis for the Krylov subspace, and the preconditioned conjugate gradient (PCG) method is applied to solve large-scale linear equations. The paper further explores how dividing the off-time and optimizing parameters for each time interval can enhance computational efficiency. The numerical results show that this parameter optimization strategy reduces the iteration count of the PCG method, improving efficiency by a factor of 5 compared to conventional iterative methods. Additionally, the proposed method outperforms direct solvers for large-scale model calculations. Conventional approaches require numerous matrix factorizations and thousands of back-substitutions, whereas the proposed method only solves about 300 linear equations. The accuracy of the approach is validated using 1D and 3D models, and the propagation characteristics of the TEM field are studied in large-scale models. Full article