The project attempts to advance the solution procedures for strong-form
meshless methods. This kind of meshless methods is easy to formulate,
since it tackles the governing equations in strong-form. They can easily
work with complex geometries and are easy to implement in an dimension
agnostic manner.

The strong form meshless methods have been in recent years successfully
applied to several problems involving solidification. Among these are
complex multi-physics models of direct-chill casting of aluminium
which considered a comprehensive model composed of coupled solvers for
fluid flow, solid mechanics, grain growth and dendritic growth. This
model has been joint by a multiphysics model of continuous casting of
steel which considers a 3D model of turbulent fluid flow, specieces and
enthalpy. Further, these type of methods has been applied to several
models of dendritic solidification
On top of this, the methods have been also applied to a range of
multiphase flows from Rayleigh-Taylor instability to
flow-focusing nozzles. All the developments
above have been facilitated by relative simplicity of implementation of
meshless models in comparison to conventional methods such as finite
volume method or finite element method and the ease with which they can
describe the geometry of the computational domain. Additionally, they
are capable of obtaining more accurate results than conventional methods
for problems where mesh induced anisotropy plays a significant role in
the evolution of the system under observation, for example for dendritic
growth.

Modeling the solidification process requires the coupling of multiple
physical phenomena with strong gradients in material properties coming
from the multiphase nature of the process which is at present mitigated
by simplifying the geometry, smoothing the sharp gradients in material
properties and similar heuristic procedures.

The developed methods have found their use in the industry but in the
meantime the industry requirements changed from mainly using offline
models with runtimes measured in days with the aim to design the casting
device and establish the casting parameters envelope to requiring models
that are fast enough to be used as an live online model that helps set
the parameters of the physical casting device in the real time.

The co-financer of this project is Danieli Automation, which provides
automation equipment and engineering for the complete spectrum of
Danieli products with sales amounting to 4 billion euro in year 2023. In
the area of steelmaking this includes various kinds of casting equipment
and furnaces, rolling mills and quality control devices. Our
collaboration with Danieli Automation has been on going for over decade
and has in the past resulted in models for vertical semi-continuous
casting of steel for the first such device in the world and in heat
transfer models for the process of continuous casting of steel that have
been based on strong-form meshless methods developed in the applicant’s
group. Danieli Automation is primarily interested in fast models for
metal processing that can be used either in real time to control process
parameters or to train simpler, but even faster models such as reduced
order models or models based on machine learning techniques.

These needs motivate our search for new solution procedures that would
help us to reduce the computational times of the models and improve the
model accuracy. The models we are in particular interested are related
to the continuous casting of steel. They are a traveling slice model of
thermomechanics and a model of fluid flow and transport phenomena in the
mold. There are two main approaches that we want to explore:

1.  Oversampled methods which solve the least-squares formulation of the
    equations with the goal of getting the results that are provably
    robust and less sensitive to the discretization near the boundaries.

2.  Development of multilevel solvers which would allow us to solve
    continuum mechanics faster and with greater accuracy.

Both approaches above are applicable to fluid flow and solid mechanics.
A solid mechanics model formulated using these approaches would have
several advantages. Firstly, improved robustness would allow calculation
of the mechanical state earlier in the solidification process when the
volume of mechanically coherent material is still low. Secondly, it
would accelerate the solution procedure, since multilevel solvers are
known to be among the fastest solution procedures for this class of
problems.

When these approaches are applied to a fluid flow model, we expect to
see similar improvements. The solver would be more robust and the
handling of complex geometries would become easier. On top of that, the
multilevel solution procedure would make it possible to solve the
Navier-Stokes system with a higher accuracy. More accurate calculation
of the velocity fields would improve the accuracy of all involved
transport equations, especially the solute transport equation and thus
provide a better picture of solidification conditions.

All proposed approaches advance the state of the art in the field of
meshless models of continuum mechanics. The most interesting one among
them is the combination of the oversampling with the multilevel solution
procedures. Such a solution procedure would help significantly reduce
the computational time of oversampled methods, while retaining all the
robustness improvements.

Problem identification

There are several challenges this project attempts to address:

1.  Classical solution procedures for solid mechanics are often slow and
    not useful for real-time calculations.

2.  Even though multilevel solution procedures are common in classical
    numerical methods they are not used in connection with strong-form
    meshless methods.

3.  Strong form meshless methods can be sensitive to node positioning,
    especially near the boundaries.

4.  Operator splitting based methods for fluid flow offer reduced
    computational accuracy because of the splitting error.

Objectives

The main objectives of the project are the following:

1.  To develop solution procedures with reduced computational time while
    simultaneously retaining model complexity and accuracy.

2.  To develop solution procedures based on oversampling to improve
    robustness of continuum mechanics solivers based on strong form
    meshless methods.

3.  To develop a multilevel solver specialized for strong form meshless
    methods.

4.  To demonstrate the feasibility of such solvers by providing two
    solver prototypes: a model of solid mechanics during continuous
    casting of steel and a model of fluid flow in mold area during
    continuous casting of steel.

WP 1: Development of oversampled RBF-FD methods 

WP 1.1 : Reworking the existing solver framework to allow for
oversampled methods
Status: Completed.

WP 1.2 : Preconditioners for oversampled RBF-FD methods
Status: In Progress.

WP 1.3 : Oversampled RBF-FD methods for moving boundaries
Status: In progress.

WP 1.4 : Implementation of efficient solution procedures for HPC
hardware
Status: In progress.

WP 2: Development of meshless multilevel solvers 

WP 2.1 : Development of multilevel framework
Status: In progress.

WP 2.2 : Smoothing procedures for solid mechanics solvers
Status: In progress.

WP 2.3 : Smoothing procedures for fluid flow solvers
Status: In progress.

WP 2.4 : Structured refinement procedures
Status: In progress.

WP 2.5 : Implementation of efficient solution procedures for HPC
hardware
Status: In progress.

WP 3: Solver prototypes

WP 3.1 : Prototype solver for traveling slice model of thermomechanics
during CC
Status: The developed multilevel algorithm is being tested on
realistic solvers.

WP 3.2 : Prototype of fluid flow solver in the mold for CC
Status: In progress.

WP 4: Dissemination

WP 4.1 : Preparation of journal publications
Status: The paper will be prepared in H1 of 2026.

WP 4.2 : Organization of a workshop related to the project
Status: In progress.

WP 4.3 : Establishing open repositories to share the code
Status: In progress.

MAVRIČ, Boštjan, DOBRAVEC, Tadej, ŠARLER, Božidar. General travelling slice model for thermal simulations of the continuous casting process. Ljubljana: Faculty of Mechanical Engineering, 2026. [18] str., ilustr. [COBISS.SI-ID 272420611]