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Charles Masamed Marcus (born October 8, 1962) is an American physicist and professor. Currently a professor at the University of Washington and the Niels Bohr Institute, he previously worked at both Stanford and Harvard universities. He was elected to the National Academy of Sciences in 2018 for his contributions to condensed matter and mesoscopic physics. He has also been recognized with the H. C. Ørsted Gold Medal for his contributions to quantum computing, spin qubits, and superconducting qubits.

Charles_M._Marcus

https://en.wikipedia.org/wiki/Charles_M._Marcus

Marcus was born on October 8, 1962, in Pittsburgh, Pennsylvania, and grew up in Sonoma, California. He was the valedictorian of Sonoma Valley High School 's class of 1980, and attended Stanford University, graduating with a Bachelor of Science degree in physics. He later received Master of Arts and Doctor of Philosophy degrees in physics from Harvard University. His doctoral thesis, published in 1990, is entitled Dynamics of Analog Neural Networks.

Charles_M._Marcus

https://en.wikipedia.org/wiki/Charles_M._Marcus

In 1992, Marcus began working as an assistant professor at Stanford University. He was promoted to associate professor in 1999, but left the next year for a professor position at Harvard University. He worked there until 2012, when he moved to the Niels Bohr Institute at the University of Copenhagen to serve as Villum Kahn Rasmussen Professor. He continues to hold his professorship in Copenhagen, but since 2023 has served as professor and Boeing Johnson Endowed Chair at the University of Washington. Marcus stated that he was "excited to shepherd exchange between the UW and University of Copenhagen" as he continues to hold appointments at both institutions.

Charles_M._Marcus

https://en.wikipedia.org/wiki/Charles_M._Marcus

Honors won by Marcus include fellowship in the American Physical Society and the American Association for the Advancement of Science, as well as election to the National Academy of Sciences and the Royal Danish Academy of Sciences and Letters.

Charles_M._Marcus

https://en.wikipedia.org/wiki/Charles_M._Marcus

This article about an American physicist is a stub. You can help Wikipedia by expanding it.

Charles_M._Marcus

https://en.wikipedia.org/wiki/Charles_M._Marcus

Landau–Peierls instability refers to the phenomenon in which the mean square displacements due to thermal fluctuatuions diverge in the thermodynamic limit and is named after Lev Landau (1937) and Rudolf Peierls (1934). This instability prevails in one-dimensional ordering of atoms/molecules in 3D space such as 1D crystals and smectics and also in two-dimensional ordering in 2D space such as a monomolecular adsorbed filsms at the interface between two isotrophic phases. The divergence is logarthmic, which is rather slow and therefore it is possible to realize substances (such as the smectics) in practice that are subject to Landau–Peierls instability.

Landau–Peierls_instability

https://en.wikipedia.org/wiki/Landau–Peierls_instability

Consider a one-dimensionally ordered crystal in 3D space. The density function is then given by ρ ρ = ρ ρ ( z ) . Since this is a 1D system, only the displacement u along the z -direction due to thermal fluctuations can smooth out the density function; displacements in other two directions are irrelevant. The net change in the free energy due to the fluctuations is given by

Landau–Peierls_instability

https://en.wikipedia.org/wiki/Landau–Peierls_instability

where F 0 is the free energy without flcutuations. Note that F cannot depend on u or be a linear function of ∇ ∇ u because the first case corresponds to a simple uniform translation and the second case is unstable. Thus, F must be quadratic in the derivatives of u . These are given by

Landau–Peierls_instability

https://en.wikipedia.org/wiki/Landau–Peierls_instability

where C , λ λ 1 and λ λ 2 are material constants; in smectics, where the symmetry z ↦ ↦ − − z must be obeyed, the second term has to be set zero, i.e., λ λ 1 = 0 . In the Fourier space (in a unit volume), the free energy is just

Landau–Peierls_instability

https://en.wikipedia.org/wiki/Landau–Peierls_instability

From the equipartition theorem (each Fourier mode, on average, is allotted an energy equal to k B T / 2 ), we can deduce that

Landau–Peierls_instability

https://en.wikipedia.org/wiki/Landau–Peierls_instability

The mean square displacement is then given by

Landau–Peierls_instability

https://en.wikipedia.org/wiki/Landau–Peierls_instability

where the integral is cut off at a large wavenumber that is comparable to the linear dimension of the element undergoing deformation. In the thermodynamic limit, L → → ∞ ∞ , the integral diverges logarthmically. This means that an element at a particular point is displaced through very large distances and therefore smoothes out the function ρ ρ ( z ) , leaving ρ ρ = constant as the only solution and destroying the 1D ordering.

Landau–Peierls_instability

https://en.wikipedia.org/wiki/Landau–Peierls_instability

Curved structures are constructions generated by one or more generatrices (which can be either curves or surfaces) through geometrical operations. They traditionally differentiate from the other most diffused construction technology, namely the post and lintel, which results from the addition of regular and linear architectural elements.

Curved_structures

https://en.wikipedia.org/wiki/Curved_structures

They have been exploited for their advantageous characteristics since the first civilisations and for different purposes. The materials, the shapes and the assemblage techniques followed the technological and cultural evolution of the societies over time. Curved structures have been preferred to cover large spaces of public buildings. In spite of their sensitivity to earthquakes, they work well from the structural static point of view.

Curved_structures

https://en.wikipedia.org/wiki/Curved_structures

From the geometrical point of view, curved structures are three-dimensional solids. They are generated starting from genetratrices which undergo the geometrical operations of extrusion or revolution. The three classes of structures stated previously can be explained as follows:

Curved_structures

https://en.wikipedia.org/wiki/Curved_structures

More complex shapes can be generated by boolean operations on a set of interacting volumes. The simplest examples, resulting from the intersection of two or more vaults and the successive subtraction of the excess volumes, are:

Curved_structures

https://en.wikipedia.org/wiki/Curved_structures

The actions performed to make these solids are the same needed to generate them in a CAD or – to some extent – in a FEM software to analyse them.

Curved_structures

https://en.wikipedia.org/wiki/Curved_structures

Differently from the post and lintel construction, which capacity depends on the resistance of the single members, curved structures can rely on their shape too. However, single curvature structures (that is, simple vaults) show less capacity than double curvature ones (e.g., domes, domical and cloister and saddle). This is because a simple vault – from a geometric point of view – corresponds to a developable surface, which has null Gaussian curvature, therefore it can be flattened to a planar surface with no distortion. Dome-like and saddle structures have respectively a positive and a negative Gaussian curvature, being shape-resistant structures par excellence.

Curved_structures

https://en.wikipedia.org/wiki/Curved_structures

All the typologies of arches, vaults and domes come from the operations stated in the previous section. They are comprehensively collected and explained in each correspondant Wikipedia article. Curved shapes were used in the past for covering large rooms in buildings, as happened for example in the Domus Aurea of Emperor Nero, the Basilica of Maxentius, the Pantheon, Rome, or Hagia Sophia. However, they could be used for infrastructures too. For instance, the Ancient Roman civilization exploited curved structures for bridges, aqueducts, sewage ducts, and arch-dam. The main materials of such constructions were Masonry and Roman concrete.

Curved_structures

https://en.wikipedia.org/wiki/Curved_structures

With the Industrial Revolution, the material chosen were more likely wrought, cast iron or, later, reinforced concrete. In this way, also the shape of the infrastructures started to change. Some example of curved structures were the Palm House, Kew Gardens by Richard Turner and Decimus Burton and the The Crystal Palace by Joseph Paxton, or on the infrastructures side, the Garabit viaduct by Gustave Eiffel. Later in 20 century, Pier Luigi Nervi started studying the possibilities of reinforced concrete, building his famous ribbed hangars.

Curved_structures

https://en.wikipedia.org/wiki/Curved_structures

Many other structures have been built by exploiting curved surface. For instance, the Philips Pavilion in Brussels by Le Corbusier and L'Oceanogràfic in Valencia by Félix Candela and Alberto Domingo are two examples of exploitation of the hyperbolic paraboloid shapes.

Curved_structures

https://en.wikipedia.org/wiki/Curved_structures

Because of their nature, curved structures cannot stand alone until their completion, especially vaults and arches. Therefore, the construction of a supporting structure (referred to as centring) is almost always necessary. These are temporary falsework which stay in place until the keystone has been set down and the arch is stabilised.

Curved_structures

https://en.wikipedia.org/wiki/Curved_structures

However, there are a few cases in which, by some expedient and careful design of the construction process, some structures have been erected without any centring. A widely known example is the domical vault of the Florence Cathedral, built by Filippo Brunelleschi in the 15 century. He achieved such a challenge by building a massive structure, mechanically behaving like a spherical dome, but with large ribs and exploiting the masonry herringbone bond to lean and fix every new layer on the previous one. Each layer of the structure seems to be composed by many small arches. The vault is also double-skin, with an intermediate hollow space hosting the staircases, through which air can flow to avoid humidity concentration. To resist parallel tensile stresses which may separate the fuses of the vault, Brunelleschi arranged sandstone chain along some parallel plane.

Curved_structures

https://en.wikipedia.org/wiki/Curved_structures

Another example of structure built with no formwork is the Global Vipassana Pagoda, located in the North of Mumbai, between the Gorai Creek and the Arabian Sea. It is a meditation hall covered by the largest masonry dome in the world, with an inner diameter at ground level of about 85m. The absence of centring was possible thanks to the double curvature of the dome and the special shape given to the carved sandstone blocks constituting the skin.

Curved_structures

https://en.wikipedia.org/wiki/Curved_structures

The boundary conditions that would cause bending and shear stress in a post and lintel structure, in a curved structure cause just axial stress in the unit elements. Indeed, according to Professor de:Jacques Heyman, in the case of masonry curved structures (he referred especially to Gothic architecture), the assumptions of unlimited compressive resistance, null tensile and shear resistance and under the hypothesis of small displacements, it can be assumed that a structure is safe and stable until the funicolar polygon stays within the middle third of the cross section. This method has been widely used in the past because of its simplicity and effectiveness. However it is still studied by some scholars, and adapted to the three-dimensional case for double curvature.

Curved_structures

https://en.wikipedia.org/wiki/Curved_structures

Traditional masonry curved structures are often the result of the assemblage of many units, the Voussoirs. The resistance of an arch then, neglecting the possibility of a material failure, depends on the equilibrium of the voussoirs. Given the shape of vaults and domes instead, the double curvature plays a positive role in terms of stability as well as the arrangement of the single units (interlocking).

Curved_structures

https://en.wikipedia.org/wiki/Curved_structures

Studying the problem of a hemispherical membrane in a gravitational field, it can be demonstrated that the membrane undergoes compressive stress in its upper part, while it is subjected to hoop tensile stresses in the lower part (under 52° from the vertical axis of symmetry). This leads to the formation of meridian craks, which tend to divide the dome in slices.

Curved_structures

https://en.wikipedia.org/wiki/Curved_structures

Daylighting is usually guaranteed by openings at the end of vaulted bays, as happens in Gloucester Cathedral, Chartres Cathedral, or Sainte-Chapelle (Paris), and specifically in the lunettes (where the vaults end against a wall) like in the Church of Santa Maria del Suffragio in L'Aquila (Italy) and in the Church of San Paolo in Albano Laziale (Italy).

Curved_structures

https://en.wikipedia.org/wiki/Curved_structures

Another structurally relevant place for an opening is the top of the domes, where in many cases an oculus can be found. Sometimes it is bare, as in the Roman Pantheon, while often is covered by another architectural element referred to as Lantern, as happens – for instance – in the Florence Cathedral.

Curved_structures

https://en.wikipedia.org/wiki/Curved_structures

Some double curvature structures are known for the echo or the reverberation phenomena they create. These are due to the size of the spaces and the materials exploited for the structure or the finishing (usually hard and with small pores). The shape does a lot in preventing or enhancing the effect. Cross or cloister vaults do not generate an echo. Pointed domes easily create reverberation more than echo. At the same time, spherical surfaces are highly reflective due to their concavity. Indeed, hemispheres, paraboloid, or similar surfaces are effective at reflecting and redirecting sound, sometimes constituting a whispering gallery. Examples of whispering galleries can be found in well-known architectures like St Paul's Cathedral in London, where the phenomenon has been studied by Lord Rauyleigh or the Archbasilica of Saint John Lateran in Rome, but also in caves like the Ear of Dionysius in Syracuse, Sicily, which has been treated by Wallace Clement Sabine.

Curved_structures

https://en.wikipedia.org/wiki/Curved_structures

The existing variety of domes is due to the assignation of symbolic meanings related to history and cultures, ranging from funerary to palatine and religious architecture, but also the response to practical problems. Indeed, a recent study addressed how in the Baroque staircase of the Royal Palace of Caserta (Italy), designed by Luigi Vanvitelli, the double dome could make the listeners feel as if they were enveloped by the music. Thus enhancing the marvel typically researched by baroque architects.

Curved_structures

https://en.wikipedia.org/wiki/Curved_structures

A modern example of architecture thought to respond and participate to sound was the Philips Pavilion designed by Le Corbusier and Iannis Xenakis for the Expo 58 in Brussels.

Curved_structures

https://en.wikipedia.org/wiki/Curved_structures

In mathematics, a normalized solution to an ordinary or partial differential equation is a solution with prescribed norm, that is, a solution which satisfies a condition like ∫ ∫ R N | u ( x ) | 2 d x = 1. In this article, the normalized solution is introduced by using the nonlinear Schrödinger equation. The nonlinear Schrödinger equation (NLSE) is a fundamental equation in quantum mechanics and other various fields of physics, describing the evolution of complex wave functions. In Quantum Physics, normalization means that the total probability of finding a quantum particle anywhere in the universe is unity.

Normalized_solutions_(nonlinear_Schrödinger_equation)

https://en.wikipedia.org/wiki/Normalized_solutions_(nonlinear_Schrödinger_equation)

In order to illustrate this concept, consider the following nonlinear Schrödinger equation with prescribed norm:

Normalized_solutions_(nonlinear_Schrödinger_equation)

https://en.wikipedia.org/wiki/Normalized_solutions_(nonlinear_Schrödinger_equation)

where Δ Δ is a Laplacian operator, N ≥ ≥ 1 , λ λ ∈ ∈ R is a Lagrange multiplier and f is a nonlinearity. If we want to find a normalized solution to the equation, we need to consider the following functional : Let I : H 0 1 ( R N ) → → R be defined by

Normalized_solutions_(nonlinear_Schrödinger_equation)

https://en.wikipedia.org/wiki/Normalized_solutions_(nonlinear_Schrödinger_equation)

with the constraint

Normalized_solutions_(nonlinear_Schrödinger_equation)

https://en.wikipedia.org/wiki/Normalized_solutions_(nonlinear_Schrödinger_equation)

where H 0 1 ( R N ) is the Hilbert space and F ( s ) is the primitive of f ( s ) .

Normalized_solutions_(nonlinear_Schrödinger_equation)

https://en.wikipedia.org/wiki/Normalized_solutions_(nonlinear_Schrödinger_equation)

A common method of finding normalized solutions is through variational methods, i.e., finding the maxima and minima of the corresponding functional with the prescribed norm. Thus, we can find the weak solution of the equation. Moreover, if it satisfies the constraint, it's a normalized solution.

Normalized_solutions_(nonlinear_Schrödinger_equation)

https://en.wikipedia.org/wiki/Normalized_solutions_(nonlinear_Schrödinger_equation)

On a Euclidean space R 3 , we define a function f : R 2 → → R :

Normalized_solutions_(nonlinear_Schrödinger_equation)

https://en.wikipedia.org/wiki/Normalized_solutions_(nonlinear_Schrödinger_equation)

f ( x , y ) = ( x + y ) 2 with the constraint x 2 + y 2 = 1 .

Normalized_solutions_(nonlinear_Schrödinger_equation)

https://en.wikipedia.org/wiki/Normalized_solutions_(nonlinear_Schrödinger_equation)

By direct calculation, it is not difficult to conclude that the constrained maximum is f = 2 , with solutions ( x , y ) = ( 2 2 , 2 2 ) and ( x , y ) = ( − − 2 2 , − − 2 2 ) , while the constrained minimum is f = 0 , with solutions ( x , y ) = ( − − 2 2 , 2 2 ) and ( x , y ) = ( 2 2 , − − 2 2 ) .

Normalized_solutions_(nonlinear_Schrödinger_equation)

https://en.wikipedia.org/wiki/Normalized_solutions_(nonlinear_Schrödinger_equation)

The exploration of normalized solutions for the nonlinear Schrödinger equation can be traced back to the study of standing wave solutions with prescribed L 2 -norm. Jürgen Moser firstly introduced the concept of normalized solutions in the study of regularity properties of solutions to elliptic partial differential equations (elliptic PDEs). Specifically, he used normalized sequences of functions to prove regularity results for solutions of elliptic equations, which was a significant contribution to the field. Inequalities developed by Emilio Gagliardo and Louis Nirenberg played a crucial role in the study of PDE solutions in L p spaces. These inequalities provided important tools and background for defining and understanding normalized solutions.

Normalized_solutions_(nonlinear_Schrödinger_equation)

https://en.wikipedia.org/wiki/Normalized_solutions_(nonlinear_Schrödinger_equation)

For the variational problem, early foundational work in this area includes the concentration-compactness principle introduced by Pierre-Louis Lions in 1984, which provided essential techniques for solving these problems.

Normalized_solutions_(nonlinear_Schrödinger_equation)

https://en.wikipedia.org/wiki/Normalized_solutions_(nonlinear_Schrödinger_equation)

For variational problems with prescribed mass, several methods commonly used to deal with unconstrained variational problems are no longer available. At the same time, a new critical exponent appeared, the L 2 -critical exponent. From the Gagliardo-Nirenberg inequality, we can find that the nonlinearity satisfying L 2 -subcritical or critical or supercritical leads to a different geometry for functional. In the case the functional is bounded below, i.e., L 2 subcritical case, the earliest result on this problem was obtained by Charles-Alexander Stuart using bifurcation methods to demonstrate the existence of solutions. Later, Thierry Cazenave and Pierre-Louis Lions obtained existence results using minimization methods. Then, Masataka Shibata considered Schrödinger equations with a general nonlinear term.

Normalized_solutions_(nonlinear_Schrödinger_equation)

https://en.wikipedia.org/wiki/Normalized_solutions_(nonlinear_Schrödinger_equation)

In the case the functional is not bounded below, i.e., L 2 supcritical case, some new difficulties arise. Firstly, since λ λ is unknown, it is impossible to construct the corresponding Nehari manifold. Secondly, it is not easy to obtain the boundedness of the Palais-Smale sequence. Furthermore, verifying the compactness of the Palais-Smale sequence is challenging because the embedding H 1 ( R N ) ↪ ↪ L 2 ( R N ) is not compact. In 1997, Louis Jeanjean using the following transform:

Normalized_solutions_(nonlinear_Schrödinger_equation)

https://en.wikipedia.org/wiki/Normalized_solutions_(nonlinear_Schrödinger_equation)

Thus, one has the following functional:

Normalized_solutions_(nonlinear_Schrödinger_equation)

https://en.wikipedia.org/wiki/Normalized_solutions_(nonlinear_Schrödinger_equation)

Then,

Normalized_solutions_(nonlinear_Schrödinger_equation)

https://en.wikipedia.org/wiki/Normalized_solutions_(nonlinear_Schrödinger_equation)

which corresponds exactly to the Pokhozhaev's identity of equation. Jeanjean used this additional condition to ensure the boundedness of the Palais-Smale sequence, thereby overcoming the difficulties mentioned earlier. As the first method to address the issue of normalized solutions in unbounded functional, Jeanjean's approach has become a common method for handling such problems and has been imitated and developed by subsequent researchers.

Normalized_solutions_(nonlinear_Schrödinger_equation)

https://en.wikipedia.org/wiki/Normalized_solutions_(nonlinear_Schrödinger_equation)

In the following decades, researchers expanded on these foundational results. Thomas Bartsch and Sébastien de Valeriola investigate the existence of multiple normalized solutions to nonlinear Schrödinger equations. The authors focus on finding solutions that satisfy a prescribed L 2 norm constraint. Recent advancements include the study of normalized ground states for NLS equations with combined nonlinearities by Nicola Soave in 2020, who examined both subcritical and critical cases. This research highlighted the intricate balance between different types of nonlinearities and their impact on the existence and multiplicity of solutions.

Normalized_solutions_(nonlinear_Schrödinger_equation)

https://en.wikipedia.org/wiki/Normalized_solutions_(nonlinear_Schrödinger_equation)

In bounded domain, the situation is very different. Let's define f ( s ) = | s | p − − 2 s where p ∈ ∈ ( 2 , 2 ∗ ∗ ) . Refer to Pokhozhaev's identity,

Normalized_solutions_(nonlinear_Schrödinger_equation)

https://en.wikipedia.org/wiki/Normalized_solutions_(nonlinear_Schrödinger_equation)

The boundary term will make it impossible to apply Jeanjean's method. This has led many scholars to explore the problem of normalized solutions on bounded domains in recent years. In addition, there have been a number of interesting results in recent years about normalized solutions in Schrödinger system, Choquard equation, or Dirac equation.

Normalized_solutions_(nonlinear_Schrödinger_equation)

https://en.wikipedia.org/wiki/Normalized_solutions_(nonlinear_Schrödinger_equation)

Let's consider the nonlinear term to be homogeneous, that is, let's define f ( s ) = | s | p − − 2 s where p ∈ ∈ ( 2 , 2 ∗ ∗ ) . Refer to Gagliardo-Nirenberg inequality: define

Normalized_solutions_(nonlinear_Schrödinger_equation)

https://en.wikipedia.org/wiki/Normalized_solutions_(nonlinear_Schrödinger_equation)

then there exists a constant C N , p such that for any u ∈ ∈ H 1 ( R N ) , the following inequality holds:

Normalized_solutions_(nonlinear_Schrödinger_equation)

https://en.wikipedia.org/wiki/Normalized_solutions_(nonlinear_Schrödinger_equation)

Thus, there's a concept of mass critical exponent,

Normalized_solutions_(nonlinear_Schrödinger_equation)

https://en.wikipedia.org/wiki/Normalized_solutions_(nonlinear_Schrödinger_equation)

From this, we can get different concepts about mass subcritical as well as mass supercritical. It is also useful to get whether the functional is bounded below or not.

Normalized_solutions_(nonlinear_Schrödinger_equation)

https://en.wikipedia.org/wiki/Normalized_solutions_(nonlinear_Schrödinger_equation)

Let X be a Banach space and I : X → → R be a functional. A sequence ( u n ) n ⊂ ⊂ X is called a Palais-Smale sequence for I at the level c ∈ ∈ R if it satisfies the following conditions:

Normalized_solutions_(nonlinear_Schrödinger_equation)

https://en.wikipedia.org/wiki/Normalized_solutions_(nonlinear_Schrödinger_equation)

1. Energy Bound: sup n I ( u n ) < ∞ ∞ .

Normalized_solutions_(nonlinear_Schrödinger_equation)

https://en.wikipedia.org/wiki/Normalized_solutions_(nonlinear_Schrödinger_equation)

2. Gradient Condition: ⟨ ⟨ I ′ ( u n ) , u n − − u ⟩ ⟩ → → 0 as n → → ∞ ∞ for some u ∈ ∈ X .

Normalized_solutions_(nonlinear_Schrödinger_equation)

https://en.wikipedia.org/wiki/Normalized_solutions_(nonlinear_Schrödinger_equation)

Here, I ′ denotes the Fréchet derivative of I , and ⟨ ⟨ ⋅ ⋅ , ⋅ ⋅ ⟩ ⟩ denotes the inner product in X . Palais-Smale sequence named after Richard Palais and Stephen Smale.

Normalized_solutions_(nonlinear_Schrödinger_equation)

https://en.wikipedia.org/wiki/Normalized_solutions_(nonlinear_Schrödinger_equation)

Momentum mapping format is a key technique in the Material Point Method (MPM) for transferring physical quantities such as momentum, mass, and stress between a material point and a background grid.

Momentum_mapping_format

https://en.wikipedia.org/wiki/Momentum_mapping_format

The Material Point Method (MPM) is a numerical technique using a mixed Eulerian-Lagrangian description. It discretises the computational domain with material points and employs a background grid to solve the momentum equations. Proposed by Sulsky et al. in 1994.

Momentum_mapping_format

https://en.wikipedia.org/wiki/Momentum_mapping_format

MPM has since been expanded to various fields such as computational solid dynamics. Currently, MPM features several momentum mapping schemes, with the four main ones being PIC (Particle-in-cell), FLIP (Fluid-Implicit Particle), hybrid format, and APIC (Affine Particle-in-Cell). Understanding these schemes in-depth is crucial for the further development of MPM.

Momentum_mapping_format

https://en.wikipedia.org/wiki/Momentum_mapping_format

MPM represents materials as collections of material points (or particles). Unlike other particle methods such as SPH (Smoothed-particle hydrodynamics) and DEM (Discrete element method), MPM also uses a background grid to solve the momentum equations arising from particle interactions. MPM can be categorized as a mixed particle/ grid method or a mixed Lagrangian-Eulerian method. By combining the strengths of both frameworks, MPM aims to be the most effective numerical solver for large deformation problems. It has been further developed and applied to various challenging problems such as high-speed impact (Huang et al., 2011), landslides (Fern et al., 2019), saturated porous media (He et al., 2024), and fluid-structure interaction (Li et al., 2022).

Momentum_mapping_format

https://en.wikipedia.org/wiki/Momentum_mapping_format

The Material Point Method (MPM) community has developed several momentum mapping schemes, among which PIC, FLIP, the hybrid scheme, and APIC are the most common. The FLIP scheme is widely used for dynamic problems due to its energy conservation properties, although it can introduce numerical noise and instability (Bardenhagen, 2002), potentially leading to computational failure. Conversely, the PIC scheme is known for numerical stability and is advantageous for static problems, but it suffers from significant numerical dissipation (Brackbill et al., 1988), which is unacceptable for strongly dynamic responses. Nairn et al. combined FLIP and PIC linearly (Nairn, 2015) to create a hybrid scheme, adjusting the proportion of each component based on empirical rather than theoretical analysis. Hammerquist and Nairn (2017) introduced an improved scheme called XPIC-m (for eXtended Particle-In-Cell of order m), which addresses the excessive filtering and numerical diffusion of PIC while suppressing the noise caused by the nonlinear space in FLIP used in MPM. XPIC-1 (eXtended Particle-In-Cell of order 1) is equivalent to the standard PIC method. Jiang et al. (2017, 2015) introduced the Affine Particle In Cell (APIC) method, where particle velocities are represented locally affine, preserving linear and angular momentum during the transfer process. This significantly reduces numerical dissipation and avoids the velocity noise and instability seen in FLIP. Fu et al. (2017) introduced generalized local functions into the APIC method, proposing the Polynomial Particle In Cell (PolyPIC) method. PolyPIC views G2P (Grid-to-Particle) transfer as a projection of the particle's local grid velocity, preserving linear and angular momentum, thereby improving energy and vorticity retention compared to the original APIC. Additionally, PolyPIC retains the filtering properties of APIC and PIC, providing robustness against noise.

Momentum_mapping_format

https://en.wikipedia.org/wiki/Momentum_mapping_format

In the PIC scheme, particle velocities during the Grid-to-Particle (G2P) substep are directly overwritten by extrapolating the nodal velocities to the particles themselves:

Momentum_mapping_format

https://en.wikipedia.org/wiki/Momentum_mapping_format

V p n + 1 = ∑ ∑ I S I p n V I n + 1

Momentum_mapping_format

https://en.wikipedia.org/wiki/Momentum_mapping_format

In the FLIP scheme, the material point velocities are updated by interpolating the velocity increments of the grid nodes over the current time step:

Momentum_mapping_format

https://en.wikipedia.org/wiki/Momentum_mapping_format

a I n = f I i n t , n + f I e x t , n m I n

Momentum_mapping_format

https://en.wikipedia.org/wiki/Momentum_mapping_format

V p n + 1 = V p n + ∑ ∑ I S I p n a I n Δ Δ t

Momentum_mapping_format

https://en.wikipedia.org/wiki/Momentum_mapping_format

The hybrid scheme's momentum mapping can be mathematically represented as:

Momentum_mapping_format

https://en.wikipedia.org/wiki/Momentum_mapping_format

V p n + 1 = ( 1 − − α α F L I P ) V p P I C , n + 1 + α α F L I P ) V p F L I P , n + 1

Momentum_mapping_format

https://en.wikipedia.org/wiki/Momentum_mapping_format

where the parameters are defined as shown here below

Momentum_mapping_format

https://en.wikipedia.org/wiki/Momentum_mapping_format

Based on the idea of "providing the local velocity field around the material point to the background grid by transferring the material point's velocity gradient," Jiang et al. (2015) proposed the APIC method. In this method, the particle velocity is locally affine, mathematically expressed as:

Momentum_mapping_format

https://en.wikipedia.org/wiki/Momentum_mapping_format

V p a f f i n e = V p + C p ( x − − x p )

Momentum_mapping_format

https://en.wikipedia.org/wiki/Momentum_mapping_format

where the parameters are defined as shown here below:

Momentum_mapping_format

https://en.wikipedia.org/wiki/Momentum_mapping_format

PIC (Particle-In-Cell), FLIP (Fluid-Implicit Particle), hybrid (hybrid solution) and APIC (Affine) The different numerical methods used in Particle-In-Cell fluid simulation greatly show how they map momentum and time integrals between material points and grids, and how they differ from each other. The typical time integration schemes for PIC, FLIP, hybrid, and APIC schemes have their own unique characteristics. The evolution of momentum on the grid under each scheme is identical. Despite the differences among these four-momentum mapping formats, their common points are still dominant. In the P2G process, the momentum mapping in PIC, FLIP, and hybrid schemes is the same. The material point positions are updated in the same manner across all four schemes. During the G2P stage, PIC transfers the updated momentum on grid nodes directly back to the material points, FLIP uses incremental mapping, and the hybrid scheme linearly combines FLIP and PIC using a coefficient. APIC mapping maintains an additional affine matrix on top of the PIC mapping.

Momentum_mapping_format

https://en.wikipedia.org/wiki/Momentum_mapping_format

Numerical tests on ring collision highlight the performance of different momentum mapping schemes in dynamic problems. The mean stress distribution and total energy evolution curve at typical time are the key contents of researchers' attention. Due to the PIC mapping scheme canceling out velocities in opposite directions, significant energy loss occurs, preventing effective conversion of kinetic energy into strain energy. GIMP_FLIP (Generalized Interpolation Material Point - Fluid Implicit Particle) shows notable numerical noise and instability, with severe oscillations in mean stress, leading to numerical fracture. GIMP_FLPI0.99 exhibits improved stability but still carries the risk of numerical fracture. Tests indicate that increasing the PIC component enhances numerical stability, with stress distribution becoming more uniform and regular, and the probability of numerical fracture decreasing. However, energy loss also becomes more pronounced. GIMP_APIC (Generalized Interpolation Material Point - Affine Particle-In-Cell) demonstrates the best performance, providing a stable and smooth stress distribution while maintaining excellent energy conservation characteristics.

Momentum_mapping_format

https://en.wikipedia.org/wiki/Momentum_mapping_format

Recently, Qu et al. proposed PowerPIC (Qu et al., 2022), a more stable and accurate mapping scheme based on optimization, which also maintains volume and uniform particle distribution characteristics.

Momentum_mapping_format

https://en.wikipedia.org/wiki/Momentum_mapping_format

Anchor losses are a type of damping commonly highlighted in micro-resonators. They refer to the phenomenon where energy is dissipated as mechanical waves from the resonator attenuate into the substrate.

Anchor_losses

https://en.wikipedia.org/wiki/Anchor_losses

In physical systems, damping is the loss of energy of an oscillating system by dissipation. In the field of micro-electro-mechanicals, the damping is usually measured by a dimensionless parameter Q factor (Quality factor). A higher Q factor indicates lower damping and reduced energy dissipation, which is desirable for micro-resonators as it leads to lower energy consumption, better accuracy and efficiency, and reduced noise.

Anchor_losses

https://en.wikipedia.org/wiki/Anchor_losses

Several factors contribute to the damping of micro-electro-mechanical resonators, including fluid damping and solid damping. Anchor losses are a type of solid damping observed in resonators operating in various environments. When a resonator is fixed to a substrate, either directly or via other structures such as tethers, mechanical waves propagate into the substrate through these connections. The wave traveling through a perfectly elastic solid would have a constant energy and an isolated perfectly elastic solid once set into vibration would continue to vibrate indefinitely. Actual materials do not show such behavior and dissipation will happen due to some imperfection of elasticity within the body. In typical micro-resonators, the substrate dimensions are significantly larger than those of the resonator itself. Consequently, it can be approximated that all waves entering the substrate will attenuate without reflecting back to the resonator. In other words, the energy carried by the waves will dissipate, leading to damping. This phenomenon is referred to as anchor losses.

Anchor_losses

https://en.wikipedia.org/wiki/Anchor_losses

Standard theories of structural mechanics permit the expression of concentrated forces and couples exerted by the structure on the support.These generally include a constant component (due, for instance, to pre-stresses or initial deformation) and a sinusoidal varying contribution. Some researchers have investigated some simple geometries following this idea, and one example is the anchor losses of a cantilaver beam connected to a 3-D semi-infinite region:

Anchor_losses

https://en.wikipedia.org/wiki/Anchor_losses

Q a n c h o r ≈ ≈ C ( L W ) ( L H ) 4

Anchor_losses

https://en.wikipedia.org/wiki/Anchor_losses

where L is the length of the beam,H is the in-plane (curvature plane) thickness, W is the out-of-plane thickness, C is a constant depending on the Poisson's coefficient, with C = 3.45 for ν = 0.25, C = 3.23 for ν = 0.3, C = 3.175 for ν = 0.33.

Anchor_losses

https://en.wikipedia.org/wiki/Anchor_losses

Due to the complexity of geometries and the anisotropy or inhomogeneities of materials, usually it is difficult to use analytical method to estimate the anchor losses of some devices. Numerical methods are more widely applied for this issue. An artifical boundary or an artifical absorbing layer is applied to the numerical model to prevent the wave reflection. One such method is the perfectly matched layer, initially developed for electromagnetic wave transmission and later adapted for solid mechanics. Perfectly matched layers act as special elements where wave attenuation occurs through a complex coordinate transformation, ensuring all waves entering the layer are absorbed, thus simulating anchor losses.

Anchor_losses

https://en.wikipedia.org/wiki/Anchor_losses

To determine the Q factor from a Finite Element Method model with perfectly matched layers, two common approaches are used:

Anchor_losses

https://en.wikipedia.org/wiki/Anchor_losses

Q = R e ( ω ω ) 2 I m ( ω ω )

Anchor_losses

https://en.wikipedia.org/wiki/Anchor_losses

where R e ( ω ω ) and I m ( ω ω ) is the real and imaginary part of the complex eigenfrequency.

Anchor_losses

https://en.wikipedia.org/wiki/Anchor_losses

Anchor losses are highly dependent on the geometry of the resonator. How to anchor the resonator or the size of the tether has a strong effect on the anchor losses. Some common methods to eliminate anchor losses are summarized as followings.

Anchor_losses

https://en.wikipedia.org/wiki/Anchor_losses

A common method is to fix the resonator at the nodal points, where the motion amplitude is minimum. From the definitions of anchor losses, now the wave magnitude into the substrate will be minimized and less energy will dissipate. However, this method may not apply to certain resonators, in which the nodal points are not around the resonator edges, causing difficulty in tether designs.

Anchor_losses

https://en.wikipedia.org/wiki/Anchor_losses

Quarter wavelength tether is an effective approach to minimize the energy loss through these tethers. Similar to the theory used for transmission lines, quarter wavelength tether is assumed as the best acoustic isolation, since the complete in phase reflection occurs as the tether length equals to a quarter acoustic wavelength, or λ/4. Therefore, there is hardly any energy dissipation to the substrate through the tethers. However, quarter-wavelength design results in extremely long tether structures, usually in tens to hundreds of micrometers, which is counter to minimization and leads to a decrease in the mechanical stability of the devices.

Anchor_losses

https://en.wikipedia.org/wiki/Anchor_losses

The resonator structure and anchoring stem are made with different materials. The acoustic impedance mismatch between these two suppresses the energy from the resonator to the stem, thus reducing anchor losses and allowing high Q factor.

Anchor_losses

https://en.wikipedia.org/wiki/Anchor_losses

The basic mechanism is to reflect back a portion of the elastic waves at the anchor boundary due to the discontinuity in the acoustic impedance caused by the acoustic cavity (the etching trenches).

Anchor_losses

https://en.wikipedia.org/wiki/Anchor_losses

Phonon crystal tether is a promising way to restrain the acoustic wave propagation in the supporting tethers, since they can arouse complete band gaps in which the transmissions of the elastic waves are prohibited. Thus, the vibration energy is retained in the resonator body, reducing the anchor losses into the substrate. Besides the phonon crystal tether, some other kinds of metamaterial could be applied to the anchor and surrounding regions to prohibit the wave transmission. A key drawback of this method is the challenge to the fabrication process.

Anchor_losses

https://en.wikipedia.org/wiki/Anchor_losses

Anchor losses are highly sensitive to the geometry of the anchors. Features such as fillets, curvature, sidewall inclination, and other detailed geometric aspects can affect anchor losses. By carefully optimizing these geometric configurations, anchor losses can be significantly reduced.

Anchor_losses

https://en.wikipedia.org/wiki/Anchor_losses

SAGE (Soviet–American Gallium Experiment, or sometimes Russian-American Gallium Experiment) is a collaborative experiment devised by several prominent physicists to measure the solar neutrino flux.

Gallium_anomaly

https://en.wikipedia.org/wiki/Gallium_anomaly

SAGE was devised to measure the radio-chemical solar neutrino flux based on the inverse beta decay reaction, Ga + ν ν e → → e − − + Ge. The target for the reaction was 50-57 tonnes of liquid gallium metal stored deep (2100 meters) underground at the Baksan Neutrino Observatory in the Caucasus mountains in Russia. The laboratory containing the SAGE-experiment is called gallium-germanium neutrino telescope (GGNT) laboratory, GGNT being the name of the SAGE experimental apparatus. About once a month, the neutrino induced Ge is extracted from the Ga. Ge is unstable with respect to electron capture ( t 1 / 2 = 11.43 days) and, therefore, the amount of extracted germanium can be determined from its activity as measured in small proportional counters.

Gallium_anomaly

https://en.wikipedia.org/wiki/Gallium_anomaly

The experiment had begun to measure the solar neutrino capture rate with a target of gallium metal in December 1989 and continued to run in August 2011 with only a few brief interruptions in the timespan. As of 2013 is the experiment was described as "being continued" with the latest published data from August 2011. As of 2014 it was stated that the SAGE experiment continues the once-a-month extractions. The SAGE experiment continued in 2016. As of 2017, the SAGE-experiment continues.

Gallium_anomaly

https://en.wikipedia.org/wiki/Gallium_anomaly

The experiment has measured the solar neutrino flux in 168 extractions between January 1990 and December 2007. The result of the experiment based on the whole 1990-2007 set of data is 65.4 +3.1 −3.0 (stat.) −2.8 (syst.) SNU. This represents only 56%-60% of the capture rate predicted by different Standard Solar Models, which predict 138 SNU. The difference is in agreement with neutrino oscillations.

Gallium_anomaly

https://en.wikipedia.org/wiki/Gallium_anomaly

The collaboration has used a 518 k Ci Cr neutrino source to test the experimental operation. The energy of these neutrinos is similar to the solar Be neutrinos and thus makes an ideal check on the experimental procedure. The extractions for the Cr experiment took place between January and May 1995 and the counting of the samples lasted until fall. The result, expressed in terms of a ratio of the measured production rate to the expected production rate, is 1.0 ± 0.15. This indicates that the discrepancy between the solar model predictions and the SAGE flux measurement cannot be an experimental artifact. Also calibrations with a Ar neutrino source had been performed.

Gallium_anomaly

https://en.wikipedia.org/wiki/Gallium_anomaly